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Kernels for Nonparametric Curve Estimation
Abstract
The choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated. Explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error, IMSE (the present proof of the optimality of the latter kernels is restricted up to order k = 5). These kernels are also of interest for the nonparametric estimation of probability densities and spectral densities. A finite sample study indicates that higher order kernels – asymptotically improving the rate of convergence – may become attractive for realistic finite sample size. Suitably modified kernels are considered for estimating at the extremities of the data, in a way which allows to retain the order of the bias found for interior points.
... Several methods have been proposed for solving the density estimation problem within a bounded interval. These methods include reflection [3], pseudo-data generation [4], boundary kernel [5,6], and data transformation [7,8]. Among these, only the reflection method provides an exact kernel function when applied to a one-sided bounded domain problem. ...
... The solution consists of two steps. In the first step, we utilize the heat Equation (3) and boundary condition (5) to derive the heat kernel function. In the above example, the heat kernel is Gaussian, ...
... These two methods are the closest to the approach used in the present study. Additional methods include the boundary kernel [5,6] and transformation methods [7,8]. ...
This paper develops a method to obtain multivariate kernel functions for density estimation problems in which the density function is defined on compact support. If domain-specific knowledge requires certain conditions to be satisfied at the boundary of the support of an unknown density, the proposed method incorporates the information contained in the boundary conditions into the kernel density estimators. The proposed method provides an exact kernel function that satisfies the boundary conditions, even for small samples. Existing methods primarily deal with a one-sided boundary in a one-dimensional problem. We consider density in a two-sided interval and extend it to a multi-dimensional problem.
... Since such an approach can give a kernel with the smallest AMSE criterion at least among the considered class, it would be superior to other kernel design methods that are not based on any discussion of the goodness of resulting higher-order kernels in kernel classes of the same order, such as jackknife (Schucany and Sommers, 1977;Wand and Schucany, 1990), as well as those that optimize alternative criteria other than the AMSE criterion, such as the minimumvariance kernels (Eddy, 1980;Müller, 1984). The existing researches (Gasser and Müller, 1979;Gasser et al., 1985;Müller, 1989, 1991) can be interpreted as following this approach, and they have focused on the relationship between the order of the kernel and the number of sign changes. More concretely, on the basis of the observation that any -th order univariate kernel should change its sign at least ( − 2) times on R, they considered kernel optimization among the class of -th order kernels with exactly ( − 2) sign changes (which we call the minimum-sign-change condition). ...
... In this paper, we also take the same approach as the one by (Gasser and Müller, 1979;Gasser et al., 1985;Müller, 1989, 1991): under the moment condition (17), one can show that changes its sign at least ( 2 −1) times on R ≥0 ; see Lemma B.2. Then, we consider the following modified problem to which the minimum-sign-change condition (P2-3) is added. ...
... In the same way as that by (Gasser and Müller, 1979;Gasser et al., 1985), we can show when = 2, 4 that this problem is solvable and the solution is provided as follows. ...
Kernel-based modal statistical methods include mode estimation, regression, and clustering. Estimation accuracy of these methods depends on the kernel used as well as the bandwidth. We study effect of the selection of the kernel function to the estimation accuracy of these methods. In particular, we theoretically show a (multivariate) optimal kernel that minimizes its analytically-obtained asymptotic error criterion when using an optimal bandwidth, among a certain kernel class defined via the number of its sign changes.
... Although kernel regression with a local bandwidth (Herrmann, 1997) gives relatively good results compared to the other methods frequently used for environmental data smoothing (Čampulová, 2018), in the outlier detection functions from the envoutliers package, the user may also choose kernel smoothing based on a global bandwidth (Gasser et al., 1991). For both local and global bandwidths the optimal kernels (Gasser et al., 1985) of the order k = 2 or k = 4 are used, respectively. ...
... • kernel.order a nonnegative integer giving the order of the optimal kernel (Gasser et al., 1985) used for smoothing. Possible options are kernel.order ...
... The function computes the estimate of kernel regression function using a local or global data-adaptive plug-in algorithm and optimal kernels (Gasser et al., 1985). The output of the method is a list with the following elements: ...
Environmental data often include outliers that may significantly affect further modelling and data analysis. Although a number of outlier detection methods have been proposed, their use is usually complicated by the assumption of the distribution or model of the analyzed data. However, environmental variables are quite often influenced by many different factors and their distribution is difficult to estimate. The envoutliers package has been developed to provide users with a choice of recently presented, semi-parametric outlier detection methods that do not impose requirements on the distribution of the original data. This paper briefly describes the methodology as well as its implementation in the package. The application is illustrated on real data examples.
... This variable smoothing allows asymmetric kernel estimators to behave better than traditional kernel estimators (see, e.g., Rosenblatt [4], Parzen [5]) near the boundary of the support in terms of their bias. Since the variable smoothing is integrated directly in the parametrization of the kernel function, asymmetric kernel estimators are also usually simpler to implement than boundary kernel methods (see, e.g., Gasser and Müller [6], Rice [7], Gasser et al. [8], Müller [9], Zhang and Karunamuni [10,11]). For these two reasons, asymmetric kernel estimators are, by now, well-known solutions to the boundary bias problem from which traditional kernel estimators suffer. ...
... For each of the eight target distributions (i = 1, 2, . . . , 8), each of the ten estimators (j = 1, 2, . . . , 10), each sample size (n = 256, 1000), and each sample (k = 1, 2, . . . ...
... , M, for the eight target distributions (i = 1, 2, . . . , 8), the ten estimators (j = 1, 2, . . . , 10) and the two sample sizes (n = 256, 1000). ...
In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimators were introduced for the estimation of cumulative distribution functions (c.d.f.s) supported on the half-line [0, ∞). They were the first authors to use asymmetric kernels in the context of c.d.f. estimation. Their estimators were shown to perform better numerically than traditional methods such as the basic kernel method and the boundary modified version from Tenreiro (2013). In the present paper, we complement their study by introducing five new asymmetric kernel c.d.f. estimators, namely the Gamma, inverse Gamma, lognormal, inverse Gaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For these five new estimators, we prove the asymptotic normality and we find asymptotic expressions for the following quantities: bias, variance, mean squared error and mean integrated squared error. A numerical study then compares the performance of the five new c.d.f. estimators against traditional methods and the Birnbaum-Saunders and Weibull kernel c.d.f. estimators from Mombeni et al. (2019). By using the same experimental design, we show that the lognormal and Birnbaum-Saunders kernel c.d.f. estimators perform the best overall, while the other asymmetric kernel estimators are sometimes better but always at least competitive against the boundary kernel method.
... This variable smoothing allows asymmetric kernel estimators to behave better than traditional kernel estimators (see, e.g., Rosenblatt [4], Parzen [5]) near the boundary of the support in terms of their bias. Since the variable smoothing is integrated directly in the parametrization of the kernel function, asymmetric kernel estimators are also usually simpler to implement than boundary kernel methods (see, e.g., Gasser and Müller [6], Rice [7], Gasser et al. [8], Müller [9], Zhang and Karunamuni [10,11]). For these two reasons, asymmetric kernel estimators are, by now, well-known solutions to the boundary bias problem from which traditional kernel estimators suffer. ...
... For each of the eight target distributions (i = 1, 2, . . . , 8), each of the ten estimators (j = 1, 2, . . . , 10), each sample size (n = 256, 1000), and each sample (k = 1, 2, . . . ...
... , M, for the eight target distributions (i = 1, 2, . . . , 8), the ten estimators (j = 1, 2, . . . , 10) and the two sample sizes (n = 256, 1000). ...
In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimators were introduced for the estimation of cumulative distribution functions (c.d.f.s) supported on the half-line $[0,\infty)$. They were the first authors to use asymmetric kernels in the context of c.d.f. estimation. Their estimators were shown to perform better numerically than traditional methods such as the basic kernel method and the boundary modified version from Tenreiro (2013). In the present paper, we complement their study by introducing five new asymmetric kernel c.d.f. estimators, namely the Gamma, inverse Gamma, lognormal, inverse Gaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For these five new estimators, we prove the asymptotic normality and we find asymptotic expressions for the following quantities: bias, variance, mean squared error and mean integrated squared error. A numerical study then compares the performance of the five new c.d.f. estimators against traditional methods and the Birnbaum-Saunders and Weibull kernel c.d.f. estimators from Mombeni et al. (2019). By using the same experimental design, we show that the lognormal and Birnbaum-Saunders kernel c.d.f. estimators perform the best overall, while the other asymmetric kernel estimators are sometimes better but always at least competitive against the boundary kernel method.
... The order p of a kernel is even when K is chosen symmetric. The kernel order has a direct connection to the best AMISE, namely O N − p 2p+1 , of the KDE estimator (Gasser et al. 1985, Silverman 1986). This suggests that high-order kernels should asymptotically perform better (though see Silverman (1986) and Marron and Wand (1992) on the usefulness of such kernels on moderate sample sizes). ...
... Examples of fourth-order kernels include K (u) = 9 8 1 − 5 3 u 2 1{u ≤ |1|} (Bartlett, 1963), and K (u) = 15 32 3 − 10u 2 + 7u 4 1{|u| ≤ 1} (Gasser et al., 1985) which, being polynomial kernels, are compatible with fast recursion (see subsection 3.2). ...
The problem of computing empirical cumulative distribution functions (ECDF) efficiently on large, multivariate datasets, is revisited. Computing an ECDF at one evaluation point requires O(N) operations on a dataset composed of N data points. Therefore, a direct evaluation of ECDFs at N evaluation points requires a quadratic O(N2) operations, which is prohibitive for large-scale problems. Two fast and exact methods are proposed and compared. The first one is based on fast summation in lexicographical order, with a O(NlogN) complexity and requires the evaluation points to lie on a regular grid. The second one is based on the divide-and-conquer principle, with a O(Nlog(N)(d−1)∨1) complexity and requires the evaluation points to coincide with the input points. The two fast algorithms are described and detailed in the general d-dimensional case, and numerical experiments validate their speed and accuracy. Secondly, a direct connection between cumulative distribution functions and kernel density estimation (KDE) is established for a large class of kernels. This connection paves the way for fast exact algorithms for multivariate kernel density estimation and kernel regression. Numerical tests with the Laplacian kernel validate the speed and accuracy of the proposed algorithms. A broad range of large-scale multivariate density estimation, cumulative distribution estimation, survival function estimation and regression problems can benefit from the proposed numerical methods.
... F n (x) is called a kernel distribution estimator. In the literature of density estimation, a kernel satisfying (3) is called an order (0, 2) kernel, where "0" means that the purpose is to estimate the density function and "2" means that such kernel yields bias of order O(h 2 ), see Gasser and Műller (1979), Gasser, Műller, and Mammitzsch (1985), Műller (1991) and Karunamuni (1998, 2000) for more references on this topic. Since the purpose of this paper is to discuss the estimation of the distribution function, to distinguish K from k, we will call K defined by (2) a distribution kernel and k a density kernel. ...
... This is the boundary problem of the kernel distribution estimator. Note that the boundary problem in kernel density estimation has the non-consistency problem, in addition to the slower convergence problem of the bias (Gasser and Műller 1979;Gasser et al. 1985;Műller 1991;Zhang and Karunamuni 1998). The boundary problem in kernel distribution estimation is less severe than in kernel density estimation. ...
Estimation of distribution functions has many real-world applications. We study kernel estimation of a distribution function when the density function has compact support. We show that, for densities taking value zero at the endpoints of the support, the kernel distribution estimator does not need boundary correction. Otherwise, boundary correction is necessary. In this paper, we propose a boundary distribution kernel estimator which is free of boundary problem and provides non-negative and non-decreasing distribution estimates between zero and one. Extensive simulation results show that boundary distribution kernel estimator provides better distribution estimates than the existing boundary correction methods. For practical application of the proposed methods, a data-dependent method for choosing the bandwidth is also proposed.
... The reflection method, proposed by Schuster (1985) and further investigated by Cline and Hart (1991), involves extending the support of the density function symmetrically beyond the observed data range and incorporating reflected data points into the estimation process. Boundary kernel estimators, initially proposed by Gasser and Müller (1979) and further refined by Gasser, Müller and Mammitzsch (1985), Jones (1993), Müller (1991), Karunamuni (1998, 2000), assign higher weights to data points located near the boundaries, effectively giving greater emphasis to the boundary regions during estimation and improving accuracy. Local polynomial smoothers with variable bandwidths, introduced by Fan and Gijbels (1992) in the regression context, see also Cheng, Fan and Marron (1997), Fan and Gijbels (1996), fit low-degree polynomials to local neighborhoods of data points, adaptively adjusting the polynomial degree and bandwidth to mitigate boundary bias. ...
This paper presents a novel approach for pointwise estimation of multivariate density functions on known domains of arbitrary dimensions using nonparametric local polynomial estimators. Our method is highly flexible, as it applies to both simple domains, such as open connected sets, and more complicated domains that are not star-shaped around the point of estimation. This enables us to handle domains with sharp concavities, holes, and local pinches, such as polynomial sectors. Additionally, we introduce a data-driven selection rule based on the general ideas of Goldenshluger and Lepski. Our results demonstrate that the local polynomial estimators are minimax under a L2 risk across a wide range of Hölder-type functional classes. In the adaptive case, we provide oracle inequalities and explicitly determine the convergence rate of our statistical procedure. Simulations on polynomial sectors show that our oracle estimates outperform those of the most popular alternative method, found in the sparr package for the R software. Our statistical procedure is implemented in an online R package which is readily accessible.
... The reflection method, proposed by Schuster (1985) and further investigated by Cline and Hart (1991), involves extending the support of the density function symmetrically beyond the observed data range and incorporating reflected data points into the estimation process. Boundary kernel estimators, initially proposed by Gasser and Müller (1979) and further refined by Gasser, Müller and Mammitzsch (1985), Jones (1993), Müller (1991), Karunamuni (1998, 2000), assign higher weights to data points located near the boundaries, effectively giving greater emphasis to the boundary regions during estimation and improving accuracy. Local polynomial smoothers with variable bandwidths, introduced by Fan and Gijbels (1992) in the regression context, see also Cheng, Fan and Marron (1997), Fan and Gijbels (1996), fit low-degree polynomials to local neighborhoods of data points, adaptively adjusting the polynomial degree and bandwidth to mitigate boundary bias. ...
This paper presents a novel approach for pointwise estimation of multivariate density functions on known domains of arbitrary dimensions using nonparametric local polynomial estimators. Our method is highly flexible, as it applies to both simple domains, such as open connected sets, and more complicated domains that are not star-shaped around the point of estimation. This enables us to handle domains with sharp concavities, holes, and local pinches, such as polynomial sectors. Additionally, we introduce a data-driven selection rule based on the general ideas of Goldenshluger and Lepski. Our results demonstrate that the local polynomial estimators are minimax under a L2 risk across a wide range of Hölder-type functional classes. In the adaptive case, we provide oracle inequalities and explicitly determine the convergence rate of our statistical procedure. Simulations on polynomial sectors show that our oracle estimates outperform those of the most popular alternative method, found in the sparr package for the R software. Our statistical procedure is implemented in an online R package which is readily accessible.
... [1] p. 8. The idea of choosing a kernel of order q bigger (or equal) than r in order to ensure the Bias( f n,h (x)) to be O(h r ) dates back to the early 1960s in work of [2,3]; recent references on higher-order kernels include the following: [4][5][6][7][8][9][10]. Note that since r is typically unknown and can be arbitrarily large, it is possible to use kernels of infinite order that achieve the minimal bias condition Bias( f n,h (x)) = O(h r ) for any r; Ref. [11] gives many properties of kernels of infinite order. ...
The properties of non-parametric kernel estimators for probability density function from two special classes are investigated. Each class is parametrized with distribution smoothness parameter. One of the classes was introduced by Rosenblatt, another one is introduced in this paper. For the case of the known smoothness parameter, the rates of mean square convergence of optimal (on the bandwidth) density estimators are found. For the case of unknown smoothness parameter, the estimation procedure of the parameter is developed and almost surely convergency is proved. The convergence rates in the almost sure sense of these estimators are obtained. Adaptive estimators of densities from the given class on the basis of the constructed smoothness parameter estimators are presented. It is shown in examples how parameters of the adaptive density estimation procedures can be chosen. Non-asymptotic and asymptotic properties of these estimators are investigated. Specifically, the upper bounds for the mean square error of the adaptive density estimators for a fixed sample size are found and their strong consistency is proved. The convergence of these estimators in the almost sure sense is established. Simulation results illustrate the realization of the asymptotic behavior when the sample size grows large.
... In fact, when the regression function has bounded support, kernel estimates often overspill the boundaries and are consequently biased at and near these edges. To overcome this problem, many works are devoted to reducing the effects, we can list Gasser and Müller [8], Gasser et al. [9], Granovsky and Müller [11] and Müller [22] discuss boundary kernel methods. Djojosugito and Speckman [5] investigated boundary bias reduction based on a finite-dimensional projection in a Hilbert space. ...
If a regression function has a bounded support, the kernel estimates often exceed the boundaries and are therefore biased on and near these limits. In this paper, we focus on mitigating this boundary problem. We apply Bernstein polynomials and the Robbins-Monro algorithm to construct a non-recursive and recursive regression estimator. We study the asymptotic properties of these estimators, and we compare them with those of the Nadaraya-Watson estimator and the generalized Révész estimator introduced by [21]. In addition, through some simulation studies, we show that our non-recursive estimator has the lowest integrated root mean square error (ISE) in most of the considered cases. Finally, using a set of real data, we demonstrate how our non-recursive and recursive regression estimators can lead to very satisfactory estimates, especially near the boundaries.
... The models and corresponding scatter plots were obtained using the SPSS software. We applied the Epanechnikov kernel function due to its robustness (Gasser et al. 1985), adopting an alpha value of 0.40, algorithm to determine the number of clusters that best characterized the sample. The resulting dendrogram is displayed in Fig. 3, clearly indicating 3 clusters. ...
In this paper, we examine the impact of adopting Industry 4.0 (I4.0) base technologies on the development of seven learning dimensions used as proxies for organization learning capabilities. We conducted a grounded theory approach in which 129 practitioners from different manufacturing companies were surveyed, and their responses analyzed through multivariate techniques. Findings allowed us to raise a theoretical framework for explaining learning development in organizations undergoing I4.0 adoption. We identified three clusters of adopters: (i) beginners, (ii) in-transition, and (iii) advanced. We found an ascending learning trend in clusters (i) and (iii) and a stationary learning pattern in (ii). Our study advances the understanding of how learning capabilities are affected as organizations advance I4.0 adoption. Our findings also gauge expectations regarding the effects of I4.0 base technologies' adoption on learning capabilities, helping academics and practitioners anticipate potential issues and develop countermeasures accordingly.
... The Nadaraya-Watson kernel estimator is the most notable example of external approaches where data is first smoothed, before the support at observed targets is computed. In contrast, methods such as the Priestley-Chao estimator [20] and the Gasser-Muller estimators [21] are called internal approaches, where one first modifies the empirical function (i.e. focusing on unbiasedness) and kernel smoothing is carried out later. ...
We introduce the underlying concepts which give rise to some of the commonly used machine learning methods, excluding deep-learning machines and neural networks. We point to their advantages, limitations and potential use in various areas of pho-tonics. The main methods covered include parametric and non-parametric regression and classification techniques, kernel-based methods and support vector machines, decision trees, probabilistic models, Bayesian graphs, mixture models, Gaussian processes, message passing methods and visual informatics.
... 1 The estimator does su↵er from boundary bias though and in particular b g(0), b g(1) will not be consistent without modification. A standard way to correct for boundary bias is to use boundary kernels that adapt to the estimation point as they approach the boundary, Gasser et al. (1985). An alternative method is local linear kernel regression, which does not require an explicit boundary correction, Fan and Gijbels (1996). ...
We introduce a new class of semiparametric dynamic autoregressive models for the Amihud illiquidity measure, which captures both the long-run trend in the illiquidity series with a nonparametric component and the short-run dynamics with an autoregressive component. We develop a GMM estimator based on conditional moment restrictions and an efficient semiparametric ML estimator based on an i.i.d. assumption. We derive large sample properties for our estimators. We further develop a methodology to detect the occurrence of permanent and transitory breaks in the illiquidity process. Finally, we demonstrate the model performance and its empirical relevance on two applications. First, we study the impact of stock splits on the illiquidity dynamics of the five largest US technology company stocks. Second, we investigate how the different components of the illiquidity process obtained from our model relate to the stock market risk premium using data on the S&P 500 stock market index.
... Otherwise, the Bias of F n is of order o(h 2 ) at interior instead is of order o(h) near the right boundary points this is the boundary problem of the kernel distribution estimator. In order to correct this problem, many methods have been proposed for kernel estimation in regression and density function estimation, among them, reflection of data [14], pseudo-data method [2] and also the boundary kernel method [3]. However, methods in kernel distribution function estimation are relatively few, this is due to the extra information F (0) = 0 and F (1) = 1. ...
In this paper, two kernel cumulative distribution function estimators are introduced and investigated in order to improve the boundary effects, we will restrict our attention to the right boundary. The first estimator uses a self-elimination between modify theoretical Bias term and the classical kernel estimator itself. The basic technique of construction the second estimator is kind of a generalized reflection method involving reflection a transformation of the observed data. The theoretical properties of our estimators turned out that the Bias has been reduced to the second power of the bandwidth, simulation studies and two real data applications were carried out to check these phenomena and are conducted that the proposed estimators are better than the existing boundary correction methods.
... 14 Our baseline implementation of the KPSS test uses the Bartlett kernel and a bandwidth 8(n/10) 1/4 , as studied by Hobijn, Franses & Ooms (2004). In addition to the results in Table 1, we have experimented with larger bandwidths, which, as shown by, e.g., Lee & Schmidt (1996) and Marmol (1998), make the KPSS test more robust against a fractional alternatives as well as considered the Epanechnikov (1969) and the Gasser, Müller & Mammitzsch (1985) optimal fourthorder kernel, also studied by Dew-Becker (2017). The results are qualitatively similar and left out for brevity. ...
... The standard normal distribution density function is often used as the kernel [62,63]: ...
Estimation of probability density functions (pdf) is considered an essential part of statistical modelling. Heteroskedasticity and outliers are the problems that make data analysis harder. The Cauchy mixture model helps us to cover both of them. This paper studies five different significant types of non-parametric multivariate density estimation techniques algorithmically and empirically. At the same time, we do not make assumptions about the origin of data from any known parametric families of distribution. The method of the inversion formula is made when the cluster of noise is involved in the general mixture model. The effectiveness of the method is demonstrated through a simulation study. The relationship between the accuracy of evaluation and complicated multidimensional Cauchy mixture models (CMM) is analyzed using the Monte Carlo method. For larger dimensions (d ~ 5) and small samples (n ~ 50), the adaptive kernel method is recommended. If the sample is n ~ 100, it is recommended to use a modified inversion formula (MIDE). It is better for larger samples with overlapping distributions to use a semi-parametric kernel estimation and more isolated distribution-modified inversion methods. For the mean absolute percentage error, it is recommended to use a semi-parametric kernel estimation when the sample has overlapping distributions. In the smaller dimensions (d = 2) and a sample is with overlapping distributions, it is recommended to use the semi-parametric kernel method (SKDE) and for isolated distributions, it is recommended to use modified inversion formula (MIDE). The inversion formula algorithm shows that with noise clusters, the results of the inversion formula improved significantly.
... There are three alternative proposals to achieve consistency at boundary of a kernel estimator of a copula. Boundary kernel methods are the most common techniques proposed in the context of kernel regression and density estimation (see [17,18]), the main difficulty with the use of this type of kernel being that it does not integrate one which, in practice, could be inconvenient. Chen and Huang [11] proposed a kernel estimator of copulas with linear boundary correction, the weakness of their method is that for many common families of copulas (e.g., Clayton, Gumbel, Gaussian and Student's t) the bias at some of the corners of the unit square is only of order O(b n ), versus the O(b 2 n ) that is reached in the central values of the domain, where O(·) is the asymptotic order operator. ...
A copula is a multivariate cumulative distribution function with marginal distributions Uniform(0,1). For this reason, a classical kernel estimator does not work and this estimator needs to be corrected at boundaries, which increases the difficulty of the estimation and, in practice, the bias boundary correction might not provide the desired improvement. A quantile transformation of marginals is a way to improve the classical kernel approach. This paper shows a Beta quantile transformation to be optimal and analyses a kernel estimator based on this transformation. Furthermore, the basic properties that allow the new estimator to be used for inference on extreme value copulas are tested. The results of a simulation study show how the new nonparametric estimator improves alternative kernel estimators of copulas. We illustrate our proposal with a financial risk data analysis.
... To obtain consistent estimates of β, we use kernel smoothing. In particular, we use different kernel statistics to estimate the above model as kernel smoothing methods can be useful for estimation in time varying events (Gasser et al., 1985;Fan et al., 1997). Furthermore, as X(t) ...
Early detection of a disease risk plays a vital role in successful treatment of the disease. Many chronic diseases, e.g., stroke, can be treated satisfactorily if they can be detected early. Traditionally, people evaluate their health conditions by comparing the current readings of their medical risk factors with some threshold values, and if irregular readings are triggered out, further medical tests are conducted to find the causes of the disease. A limitation of the traditional disease detection methods is the usage of only current time point data while ignoring the historical data. To use the history of the process for early detection of the disease using statistical process control, we suggest the use of a double exponential weighted moving average control chart to monitor risk factors sequentially. For the estimation of disease risk factors, different kernel functions are used. In particular, we use Epanechnikov, triangular, tricube, biweight, Gaussian, triweight, and cosine kernels. To evaluate the performance of different kernel functions, we conducted simulations as well as a real data set is used. The real data set is about 1055 stroke patients, out of which 27 individuals have suffered from a stroke attack at least once during the study time and the remaining 1028 people did not experience stroke. Numerical results show that the suggested method performs well in detection of early disease risk. Furthermore, it is observed that the biweight kernel function performs better than the other kernel functions for online disease risk monitoring. It is also noticed that for small smoothing parameters of the DEWMA chart, the Epanechinkov and cosine kernels perform better whereas tricube and biweight for the large smoothing parameters.
... The phenomenon is known as boundary effect, which is typical for local constant estimators. On the practical side, boundary correction methods, e.g., boundary kernel (Gasser et al., 1985), may be applied to improve the performance at boundary points, which is out of the scope of this paper. ...
Principal component analysis (PCA) is a versatile tool to reduce the dimensionality which has wide applications in statistics and machine learning community. It is particularly useful to model data in high-dimensional scenarios where the number of variables $p$ is comparable to, or much larger than the sample size $n$. Despite extensive literature on this topic, researches have focused on modeling static principal eigenvectors or subspaces, which is unsuitable for stochastic processes that are dynamic in nature. To characterize the change in the whole course of high-dimensional data collection, we propose a unified framework to estimate dynamic principal subspaces spanned by leading eigenvectors of covariance matrices. In the proposed framework, we formulate an optimization problem by combining the kernel smoothing and regularization penalty together with the orthogonality constraint, which can be effectively solved by the proximal gradient method for manifold optimization. We show that our method is suitable for high-dimensional data observed under both common and irregular designs. In addition, theoretical properties of the estimators are investigated under $l_q (0 \leq q \leq 1)$ sparsity. Extensive experiments demonstrate the effectiveness of the proposed method in both simulated and real data examples.
... We also suppress the region of integration A, and write short-handed ∫ K 2 v,ps (u)du instead of ∫ A K 2 v,ps (u; A)du. The kernel K v,ps (u) is an equivalent kernel as defined by Gasser et al. (1985). For several kernels K satisfying Assumption (A9), the expressions for K v,ps (u) for different value of v and p s are tabulated in Fan and Gijbels (1996) (pp. ...
Quantile regression is an important tool in data analysis. Linear regression, or more generally, parametric quantile regression imposes often too restrictive assumptions. Nonparametric regression avoids making distributional assumptions, but might have the disadvantage of not exploiting distributional modelling elements that might be brought in. A semiparametric approach towards estimating conditional quantile curves is proposed. It is based on a recently studied large family of asymmetric densities of which the location parameter is a quantile (and not a mean). Passing to conditional densities and exploiting local likelihood techniques in a multiparameter functional setting then leads to a semiparametric estimation procedure. For the local maximum likelihood estimators the asymptotic distributional properties are established, and it is discussed how to assess finite sample bias and variance. Due to the appealing semiparametric framework, one can discuss in detail the bandwidth selection issue, and provide several practical bandwidth selectors. The practical use of the semiparametric method is illustrated in the analysis of maximum winds speeds of hurricanes in the North Atlantic region, and of bone density data. A simulation study includes a comparison with nonparametric local linear quantile regression as well as an investigation of robustness against miss-specifying the parametric model part.
... In signal approximation, a noisy signal is usually approximated by a piecewise constant function. A variety of denoising methods have been developed, including lowess (Cleveland (1979)), kernel estimators (Gasser, Müller and Mammitzsch (1985); Müller and Stadtmüller (1987)), penalized smoothing splines (Ruppert, Wand and Carroll (2009)), Markov random field (Geman and Geman (1984)), and wavelets (Donoho and Johnstone (1994); Chang, Yu and Vetterli (2000)). To encourage the underlying sparse or blocky structure of y, we set d j = e j , for j = 1, 2, . . . ...
... To compare kernel estimates, the kernels must have the same support (Härdle 1991). Gasser et al. (1985) suggest comparing kernels over the common interval ;1 1 ]. Applying that suggestion, our kernel programs are listed in the following where varname is the input variable, bandwidth is a scalar that specifies the half-width of each bin, and density and midpoint are new variables that will contain the density estimates and bin midpoints, respectively. ...
Stata Techical Bulletin STB-26
Kernel density estimators are important tools for exploring and analyzing data distributions (see the references in Salgado-
Ugarte et al. 1993). However, one drawback of these procedures is the large number of calculations required to compute them.
As a consequence, it can be time consuming to compute kernel density estimators even for moderate sample sizes and when using
fast processors. Scott (1985) suggested an alternative procedure to overcome this problem: the Averaged S hifted Histogram
(ASH). Subsequently, H¨ardle and Scott (1988) developed a more general framework called WARPing (weighted averaging of
rounded points).
This insert, based mainly on some chapters from the books by H¨ardle (1991) and Scott (1992), briefly introduces the ASH
and WARPing procedures for density estimation and presents some ado-files and Turbo Pascal programs for their calculation.
... Zhang and Karunamuni [185] showed that the performance of the Beta kernel estimator is very similar to that of the reflection estimator of Schuster [155], which does not have the boundary problem only for densities exhibiting a shoulder condition at the endpoints of the support. For densities not exhibiting a shoulder condition, they showed that the performance of the Beta kernel estimator at the boundary was inferior to that of the well-known boundary kernel estimator, see, e.g., [55,56,183,184] and references therein. ...
We study theoretically, for the first time, the Dirichlet kernel estimator introduced by Aitchison and Lauder (1985) for the estimation of multivariate densities supported on the d-dimensional simplex. The simplex is an important case as it is the natural domain of compositional data and has been neglected in the literature on asymmetric kernels. The Dirichlet kernel estimator, which generalizes the (non-modified) unidimensional Beta kernel estimator from Chen (1999), is free of boundary bias and non-negative everywhere on the simplex. We show that it achieves the optimal convergence rate O(n−4/(d+4)) for the mean squared error and the mean integrated squared error, we prove its asymptotic normality and uniform strong consistency, and we also find an asymptotic expression for the mean integrated absolute error. To illustrate the Dirichlet kernel method and its favorable boundary properties, we present a case study on minerals processing.
... The Gaussian kernel consistent with the distribution of normal φ(x) [9], [19] selection: ...
The problem of nonparametric estimation of probability density function is considered. The performance of kernel estimators based on various common kernels and a new kernel K (see (14)) with both fixed and adaptive smoothing bandwidth is compared in terms of the symmetric mean absolute percentage error using the Monte Carlo method. The kernel K is everywhere positive but has lighter tails than the Gaussian density. Gaussian mixture models from a collection introduced by Marron and Wand (1992) are taken for Monte Carlo simulations. The adaptive kernel method outperforms the smoothing with a fixed bandwidth in the majority of models. The kernel K shows better performance for Gaussian mixtures with considerably overlapping components and multiple peaks (double claw distribution).
... Zhang and Karunamuni [185] showed that the performance of the Beta kernel estimator is very similar to that of the reflection estimator of Schuster [155], which does not have the boundary problem only for densities exhibiting a shoulder condition at the endpoints of the support. For densities not exhibiting a shoulder condition, they showed that the performance of the Beta kernel estimator at the boundary was inferior to that of the well-known boundary kernel estimator, see, e.g., [55,56,183,184] and references therein. ...
We study theoretically, for the first time, the Dirichlet kernel estimator introduced by Aitchison and Lauder (1985) for the estimation of multivariate densities supported on the d-dimensional simplex. The simplex is an important case as it is the natural domain of compositional data and has been neglected in the literature on asymmetric kernels. The Dirichlet kernel estimator, which generalizes the (non-modified) unidimensional Beta kernel estimator from Chen (1999), is free of boundary bias and non-negative everywhere on the simplex. We show that it achieves the optimal convergence rate O(n−4/(d+4)) for the mean squared error and the mean integrated squared error, we prove its asymptotic normality and uniform strong consistency, and we also find an asymptotic expression for the mean integrated absolute error. To illustrate the Dirichlet kernel method and its favorable boundary properties, we present a case study on minerals processing.
In this paper, we propose a general framework to select tuning parameters for the nonparametric derivative estimation. The new framework broadens the scope of the previously proposed generalized
criterion by replacing the empirical derivative with any other linear nonparametric smoother. We provide the theoretical support of the proposed derivative estimation in a random design and justify it through simulation studies. The practical application of the proposed framework is demonstrated in the study of the age effect on hippocampal gray matter volume in healthy adults from the IXI dataset and the study of the effect of age and body mass index on blood pressure from the Pima Indians dataset.
In this paper, we propose a new frequency domain test for pairwise time reversibility at any specific couple of quantiles of two-dimensional marginal distribution. The proposed test is applicable to a very broad class of time series, regardless of the existence of moments and Markovian properties. By varying the couple of quantiles, the test can detect any violation of pairwise time reversibility. Our approach is based on an estimator of the L2-distance between the imaginary part of copula spectral density kernel and its value under the null hypothesis. We show that the limiting distribution of the proposed test statistic is normal and investigate the finite sample performance by means of a simulation study. We illustrate the use of the proposed test by applying it to stock price data.
The Sharpe ratio function is a commonly used risk/return measure in financial econometrics. To estimate this function, most existing methods take a two-step procedure that first estimates the mean and volatility functions separately and then applies the plug-in method. In this paper, we propose a direct method via local maximum likelihood to simultaneously estimate the Sharpe ratio function and the negative log-volatility function as well as their derivatives. We establish the joint limiting distribution of the proposed estimators, and moreover extend the proposed method to estimate the multivariate Sharpe ratio function. We also evaluate the numerical performance of the proposed estimators through simulation studies, and compare them with existing methods. Finally, we apply the proposed method to the three-month US Treasury bill data and that captures a well-known covariate-dependent effect on the Sharpe ratio.
p>Nonparametric regression provides an intuitive estimate of a regression function or conditional expectation without the restrictions imposed by parametric models. This is a particularly useful property since the rigidity of such parametric models is not always desirable. The application of nonparametric regression, in the univariate setting, is investigated in the context of predicting a finite population total. We propose, instead of parametric estimators of the finite population total, nonparametric regression estimators obtained by smoothing the data and interpolating the smooth to predict nonsample values. It is shown how such estimation can be more robust and efficient than inference tied to parametric regression models. The nonparametric regression estimators considered are classified as operational and model-based . They require the selection of a smoothing parameter which controls the smoothness of the resultant curve. Methods of choosing the smoothing parameter are discussed.
One important property that some of the estimators are shown to possess is `total preservation'. Suppose ^y = Sy , where S os a smoother matrix and ^y and y are vectors of estimated and observed y respectively. Then an estimator is said to be `total-preserving' if 1 <sup>T</sup>^y = 1 <sup>T</sup> Sy = 1 <sup>T</sup>y. It is shown how, under repeated sampling, `total preserving' nonparametric regression estimators are design-unbiased or approximately design-unbiased and how they remain more efficient than standard parametric methods, for a suitable choice of the smoothing parameter.</p
p>This thesis is concerned with the search for robust efficient procedures for estimating the regression coefficient in the marginal distribution of the survey variables for data collected from complex surveys. Parametric model based procedures which take into account the structure of the population by assuming a parametric model are found to be very sensitive to the violations of these underlying parametric model assumptions. However these procedures perform very well when the parametric model assumptions hold. Randomization based estimators, which takes into account the population structure through the selection probabilities, are found to be robust unconditionally but their conditional and efficiency properties may be poor in some circumstances. We propose nonparametric model based procedures which do not make any explicit assumptions about the distribution describing the population structure. One nonparametric procedure, namely the Nadaraya-Watson kernel estimator of the regression coefficient is the most efficient and robust when the structure of the population is unknown. However the estimator is biased when the population is linear and homoscedastic. The validity of the theoretical results was assessed in a series of simulation studies based on a variety of stratified sampling schemes.</p
The problem of nonparametric estimation of a univariate density with rth continuous derivative on compact support is addressed (r≥2). If the density function has compact support and is non-zero at either boundary, regular kernel estimator will be completely biased at such boundary. Although several correction methods were proposed to improve the bias at the boundary to h2 in the last decades, this paper initiates a way to further improve bias to higher order (hr) for interior area of density function support, while remaining the order of bias h2 at boundary. We will first review flat-top kernel estimator and flat-top series estimator, then propose the Transformed Flat-top Series estimator. The theoretical analysis is supplemented with simulation results as well as real data applications.
The local polynomial modeling of the Kaplan–Meier estimate for random designs under the right censored data setting is investigated in detail. Two classes of boundary aware estimates are developed: estimates of the distribution function and its derivatives of any arbitrary order and estimates of integrated distribution function derivative products. Their statistical properties are quantified analytically and their implementation is facilitated by the development of corresponding data driven plug-in bandwidth selectors. The asymptotic rate of convergence of the plug–in rule for the estimates of the distribution function and its derivatives is quantified analytically. Numerical evidence is also provided on its finite sample performance. A real life data analysis illustrates how the methodological advances proposed herein help to generate additional insights in comparison to existing methods.
In the era of big data, there are many data sets recorded in equal intervals of time. To model the change rate of such data, one often constructs a nonparametric regression model and then estimates the first derivative of the mean function. Along this direction, we propose a symmetric two-sided local constant regression for interior points, an asymmetric two-sided local polynomial regression for boundary points, and a one-sided local linear forecasting model for outside points. Specifically, under the framework of locally weighted least squares regression, we derive the asymptotic bias and variance of the proposed estimators, as well as establish their asymptotic normality. Moreover, to reduce the estimation bias for highly-oscillatory functions, we propose debiased estimators based on high-order polynomials and derive their corresponding kernel functions. A data-driven two-step procedure for simultaneous selection of the model and tuning parameters is also proposed. Finally, the usefulness of our proposed estimators is demonstrated by simulation studies and two real data examples.
n this article we make a comparison between three different nonparametric density estimators .The first estimator is called fixed kernel which uses a fixed bandwidth.
The second estimator is called variable kernel, which uses variable bandwidth, then the third estimator, refers to a new version of nonparametric density estimator (this estimator is a hybrid of the above estimators.
This comparison depends on using different kernels, two of these functions are proposed in this article .To make this comparison suitable, we use standardization techniques, which are called canonical kernel.
Simulation is used to find kernel function based on the standardization techniques (canonical kernel).
Missing covariate data arise frequently in biomedical studies. In this article, we propose a class of weighted estimating equations for the additive hazard regression model when some of the covariates are missing at random. Time-specific and subject-specific weights are incorporated into the formulation of weighted estimating equations. Unified results are established for estimating selection probabilities that cover both parametric and non-parametric modeling schemes. The resulting estimators have closed forms and are shown to be consistent and asymptotically normal. Simulation studies indicate that the proposed estimators perform well for practical settings. An application to a mouse leukemia study is illustrated.
Modern scientific instruments readily record various dynamical phenomena at high frequency and for extended durations. Spanning timescales across several orders of magnitude, such “high-throughput” (HTP) data are routinely analyzed with parametric models in the frequency domain. However, the large size of HTP datasets can render maximum likelihood estimation prohibitively expensive. Moreover, HTP recording devices are operated by extensive electronic circuitry, producing periodic noise to which parameter estimates are highly sensitive. This article proposes to address these issues with a two-stage approach. Preliminary parameter estimates are first obtained by a periodogram variance-stabilizing procedure, for which data compression greatly reduces computational costs with minimal impact to statistical efficiency. Next, a novel test with false discovery rate control eliminates most periodic outliers, to which the second-stage estimator becomes more robust. Extensive simulations and experimental results indicate that for a widely-used model in HTP data analysis, a substantial reduction in mean squared error can be expected by applying our methodology.
Introduction: Estimation of a probability density function is an important area of nonparametric statistical inference that has received much attention in recent decades. The kernel method is widely used in nonparametric estimation of the probability density function of an absolutely continuous distribution with support on the whole real line. However, for a distribution with support on a subset of the real line, the kernel density estimator with fixed symmetric kernels encounters bias at the boundaries of the support, which is known as the boundary bias issue. This is due to smoothing data near the boundary points by the fixed symmetric kernel that leads to allocating probability density to outside of the distribution’s support (see Silverman, 1986).
There are many applications, such as reliability, insurance and life testing, dealing with non-negative data and estimating the probability density function of distributions with support on the non-negative real line is the object of interest. Using the kernel estimator with fixed symmetric kernels in these cases results in the boundary bias issue at the origin. A number of methods have been proposed to avoid the boundary bias issue at the origin. A simple remedy is to replace symmetric kernels by asymmetric kernels which never assign density to negative values. The Gamma kernels proposed by Chen (2000) are the effective asymmetric kernels to estimate the probability density function of distributions on the non-negative real line.
Orthogonal series estimators form another class of nonparametric probability density estimators, which go back to Cencov (1964). In this approach, as reviewed in Efromovich (2010), the target probability function is expanded in terms of a sequence of orthogonal basis functions. After selecting a suitable sequence of orthogonal basis functions, the observed data are used to estimated the coefficients of the expansion in order to obtain the orthogonal series density estimator. Similar to kernel estimators, under some mild conditions the orthogonal series estimators have appealing large sample properties. Moreover, the boundary issue can be avoided by using orthogonal density estimators with suitable basis functions.
Although small sample properties of asymmetric kernel estimators with the Gamma kernels and orthogonal series estimators are well-studied separately, but to the best of our knowledge, there have been no reports of comparing their performance in estimating the probability density function of distributions on the non-negative real line. In this paper, a simulation study is conducted to compare the small-sample performance of the Gamma kernel estimators and orthogonal series estimators for a set of distributions on the positive real line.
Material and methods: Following Malec and Schienle (2014), we consider six parameter settings for the generalized F distribution to obtain probability density functions with different shapes, near-origin behaviors and tail decays (Figure 2). Based on 5000 simulations from any of these density functions with sample sizes , we estimate the target density function using the type I and II Gamma kernel estimators and the orthogonal series estimators with Hermite and Laguerre basis functions and compute the mean integrated squared error (MISE).
The bandwidth parameter in the Gamma kernel estimators and the cutoff and smoothing parameters in the orthogonal series estimators are significantly affect the performance of the estimators. We use optimal bandwidths for the Gamma kernel estimators and optimal cutoff and smoothing parameters of the orthogonal series estimators to avoid variations due to uncertainty of tuning parameters. To obtain the optimal tuning parameters for each target density, we compute and minimize the MISE with respect to the tuning parameters based on additional 5000 simulations from the true density function.
Results and discussion: For each density function, the optimal tuning parameters and the MISE’s of the estimators are reported (Table 2). As expected from the large sample properties, increasing the sample size improves the performance of all estimators. The performances of estimators vary from cases to cases and there no considered estimator is the best in all cases. In all cases except one, the type II Gamma kernel estimator is superior to the type I Gamma kernel estimator, which is in agreement with Chen (2000) suggestion of preferring type II to type I Gamma kernel estimator. However, in one case the type I Gamma kernel estimator is better than all other estimators. In cases where the shape and near-origin behavior of the target density is similar to the Hermit or Laguerre basis functions, the corresponding orthogonal series estimator outperforms all the other competing estimators.
Conclusion
The following conclusions were drawn from this research to choose among the considered estimators.
If the basis functions of the orthogonal series estimator are chosen to have similar shape and near-origin behavior to the target density function, then the corresponding orthogonal series estimator can outperform the Gamma kernel estimators.
If there is no prior knowledge about the shape and near-origin behavior of the target density function and the sample size is relatively large (n=400), then the type II Gamma kernel estimator can outperform the orthogonal series estimators.
This paper studies standard predictive regressions in economic systems governed by persistent vector autoregressive dynamics for the state variables. In particular, all – or a subset – of the variables may be fractionally integrated, which induces a spurious regression problem. We propose a new inference and testing procedure – the Local speCtruM (LCM) approach – for joint significance of the regressors, that is robust against the variables having different integration orders and remains valid regardless of whether predictors are significant and, if they are, whether they induce cointegration. Specifically, the LCM procedure is based on fractional filtering and band spectrum regression using a suitably selected set of frequency ordinates. Contrary to existing procedures, we establish a uniform Gaussian limit theory and a standard χ2-distributed test statistic. Using the LCM inference and testing techniques, we explore predictive regressions for the realized return variation. Standard least squares inference indicates that popular financial and macroeconomic variables convey valuable information about future return volatility. In contrast, we find no significant evidence using our robust LCM procedure. If anything, our tests support a reverse chain of causality, with rising financial volatility predating adverse innovations to key macroeconomic variables. Simulations are employed to illustrate the relevance of the theoretical arguments for finite-sample inference.
Kernel smoothers belong to the most popular nonparametric functional estimates. They provide a simple way of finding structure in data. Kernel smoothing can be very well applied on the regression model. In the context of kernel estimates of a regression function, the choice of a kernel from the different points o view can be investigated. The main idea of this paper is to present construction of the optimal kernel and edge optimal kernel by means of the Gegenbauer and Legendre polynomial.
In this paper we used non parametric density estimator (Kernel Estimator) to estimate probability density function for Image data of Hand Gesture and warping Hand Gesture and we see the curve for Kernel Estimator and combine the curve between Kernel Estimator and normal Distribution. Programs written using the language Matlab (R2009a) .
This textbook gives an overview of statistical methods that have been developed during the last years due to increasing computer use, including random number generators, Monte Carlo methods, Markov Chain Monte Carlo (MCMC) methods, Bootstrap, EM algorithms, SIMEX, variable selection, density estimators, kernel estimators, orthogonal and local polynomial estimators, wavelet estimators, splines, and model assessment. Computer Intensive Methods in Statistics is written for students at graduate level, but can also be used by practitioners.
Features
Presents the main ideas of computer-intensive statistical methods
Gives the algorithms for all the methods
Uses various plots and illustrations for explaining the main ideas
Features the theoretical backgrounds of the main methods.
Includes R codes for the methods and examples
In recent years, nonparametric curve estimates have been extensively explored in theoretical work. There has, however, been a certain lack of convincing applications, in particular involving comparisons with parametric techniques. The present investigation deals with the analysis of human height growth, where longitudinal measurements were collected for a sample of boys and a sample of girls. Evidence is presented that kernel estimates of acceleration and velocity of height, and of height itself, might offer advantages over a parametric fitting via functional models recently introduced. For the specific problem considered, both approaches are biased, but the parametric one shows qualitative and quantitative distortion which both are not easily predictable. Data-analytic problems involved with kernel estimation concern the choice of kernels, the choice of the smoothing parameter, and also whether the smoothing parameter should be chosen uniformly for all subjects or individually.
In this note we consider the problem of fitting a general functional relationship between two variables. We require only that the function to be fitted is, in some sense, “smooth”, and do not assume that it has a known mathematical form involving only a finite number of unknown parameters.
Estimation of the value of a density function at a point of continuity using a kernel-type estimator is discussed and improvements of the technique are presented. The generalized jackknife method is employed to reduce the asymptotic and small sample mean square error and bias of these estimators. The procedure presented has the flexibility to afford the user a choice between bias reduction, variance reduction, or both.
S ummary
We consider a nonparametric technique proposed by Priestley and Chao (1972) for estimating an unknown regression function. Conditions for strong convergence and asymptotic normality are discussed. Special consideration is given to the optimal choice of a weighting function.
There is a large class of problems in which the estimation of curves arises naturally (see [15], [34]). It is curious that
one of the earliest extensive investigations of this type involves the estimation of the spectral density function when sampling
from a stationary sequence ([1], [17], [27], [33]). Even though the simple histogram has been used for years, it was only
later that the simpler question of estimating a probability density function was dealt with at some length ([26], [25], [9]).
Because the final character of the usual results obtained in both problem areas is quite similar, and the arguments are much
more transparent in the case of the probability density function, we shall develop the results for the probability density
function first. Later some corresponding results for spectra will be given.
This paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression. We analyze a tapered Fourier series estimate and discuss the relationship of this estimate to a kernel estimate. We first consider a method based on an unbiased estimate of mean square error, and show that the bandwidth thus chosen is asymptotically optimal. Other methods are examined as well and are shown to be asymptotically equivalent. A small simulation shows, however, that for small or moderate sample size, the methods perform quite differently.
The statistical properties of a cubic smoothing spline and its derivative are analyzed. It is shown that unless unnatural boundary conditions hold, the integrated squared bias is dominated by local effects near the boundary. Similar effects are shown to occur in the regularized solution of a translation-kernel integral equation. These results are derived by developing a Fourier representation for a smoothing spline.
It is pointed out that several results due to Walter and Blum do not hold strictly as they are stated. We find expansions for the mean square errors and mean integrated square errors of trigonometric series estimates of densities, and use them to compare the efficiencies of the estimates.
Let $X_1, \cdots, X_n$ be independent observations with common density $f$. A kernel estimate of the mode is any value of $t$ which maximizes the kernel estimate of the density $f_n$. Conditions are given restricting the density, the kernel, and the bandwidth under which this estimate of the mode has an asymptotic normal distribution. By imposing sufficient restrictions, the rate at which the mean squared error of the estimator converges to zero can be decreased from $n^{-\frac{4}{7}}$ to $n^{-1+\varepsilon}$ for any positive $\varepsilon$. Also, by bounding the support of the kernel it is shown that for any particular bandwidth sequence the asymptotic mean squared error is minimized by a certain truncated polynomial kernel.
Based on a random sample from a population with (unknown) probability density f, this note exhibits a class of statistics f(p) for each fixed integer p [greater, double equals] 0. It is shown that f(p) are uniformly strongly consistent estimators of f(p), the pth order derivative of f, if and only if f(p) is bounded and uniformly continuous.
A note on kernels fulfilling certain moment conditions
- Mammitzsch
Kernel estimation of regression functions Smoothing Techniques for Curve Estimation.
- Th. Gasser
- H. G. Miiller
Optimal convergence properties of kernel estimates of derivatives of a destiny function Smoothing Techniques for Curve Estimation.
- H. G. Millier
- Th. Gasser
Some comments on the asymptotic behaviour of robust smoothers Smoothing Techniques for Curve Estimation.
- W. Stützle
- Y. Mittal
Optimal convergence properties of kernel estimates of derivatives of a destiny function
- Millier