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Likelihood ratio processes under nonstandard settings
Abstract
В статье показывается свойство локальной асимптотической нормальности (LAN) для криволинейных нормальных семейств и систем одновременных уравнений. Кроме того, показано, что односторонние случайные модели ANOVA не имеют свойства локальной асимптотической нормальности. Рассматриваются два случая, когда дисперсия случайного эффекта лежит внутри и на границе пространства параметров. В первом случае логарифм отношения правдоподобия сходится к 0. Во втором случае логарифм отношения правдоподобия имеет нетипичные предельные распределения, которые зависят от нормировок, отвечающих контигуальным гипотезам. Порядки этих нормировок, соответствующие дисперсиям случайных эффектов и возмущений, могут быть равными или больше единицы соответственно, а порядок, соответствующий общему среднему, может быть равен или больше половины. Следовательно, мы не можем использовать обычную оптимальную теорию, основанную на свойстве локальной асимптотической нормальности. Между тем с помощью классической схемы Неймана-Пирсона показано, что тест, основанный на логарифме отношения правдоподобия, асимптотически самый мощный.
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This paper studies a class of models where full identification is not necessarily assumed. We term such models partially identified. It is argued that partially identified systems are of practical importance since empirical investigators frequently proceed under conditions that are best described as apparent identification. One objective of the paper is to explore the properties of conventional statistical procedures in the context of identification failure. Our analysis concentrates on two major types of partially identified model: the classic simultaneous equations model under rank condition failures; and time series spurious regressions. Both types serve to illustrate the extensions that are needed to conventional asymptotic theory if the theory is to accommodate partially identified systems. In many of the cases studied, the limit distributions fall within the class of compound normal distributions. They are simply represented as covariance matrix or scalar mixtures of normals. This includes time series spurious regressions, where representations in terms of functionals of vector Brownian motion are more conventional in recent research following earlier work by the author.
Locally asymptotically most stringent tests for autoregressive against diagonal bilinear time series models are derived. A (restricted) local asymptotic normality property is therefore established for bilinear processes in the vicinity of linear autoregressive ones. The behaviour of the bispectrum under local alternatives of bilinear dependence shows the danger of misinterpreting skewness or kurtosis effects for nonlinearities. The proposed test statistic is a generalization of the Gaussian Lagrange multiplier statistic considered by P. Saikkonen and R. Luukkonen [Scand. J. Stat. 15, No. 1, 55-68 (1988; Zbl 0649.62087)], and is expressed as a closed-form function of the estimated residual spectrum and bispectrum. Its local power is explicitly provided. The local power of the Lagrange multiplier test follows as a particular case.
This paper constructs a two-country (Home and Foreign) general equilibrium model of Schumpeterian growth without scale effects. The scale effects property is removed by introducing two distinct specifications in the knowledge production function: the permanent effect on growth (PEG) specification, which allows policy effects on long-run growth; and the temporary effects on growth (TEG) specification, which generates semi-endogenous long-run economic growth. In the present model, the direction of the effect of the size of innovations on the pattern of trade and Home’s relative wage depends on the way in which the scale effects property is removed. Under the PEG specification, changes in the size of innovations increase Home’s comparative advantage and its relative wage, while under the TEG specification, an increase in the size of innovations increases Home’s relative wage but with an ambiguous effect on its comparative advantage.
A method is given for estimating the coefficients of a single equation in a complete system of linear stochastic equations (see expression (2.1)), provided that a number of the coefficients of the selected equation are known to be zero. Under the assumption of the knowledge of all variables in the system and the assumption that the disturbances in the equations of the system are normally distributed, point estimates are derived from the regressions of the jointly dependent variables on the predetermined variables (Theorem 1). The vector of the estimates of the coefficients of the jointly dependent variables is the characteristic vector of a matrix involving the regression coefficients and the estimate of the covariance matrix of the residuals from the regression functions. The vector corresponding to the smallest characteristic root is taken. An efficient method of computing these estimates is given in section 7. The asymptotic theory of these estimates is given in a following paper [2]. When the predetermined variables can be considered as fixed, confidence regions for the coefficients can be obtained on the basis of small sample theory (Theorem 3). A statistical test for the hypothesis of over-identification of the single equation can be based on the characteristic root associated with the vector of point estimates (Theorem 2) or on the expression for the small sample confidence region (Theorem 4). This hypothesis is equivalent to the hypothesis that the coefficients assumed to be zero actually are zero. The asymptotic distribution of the criterion is shown in a following paper [2] to be that of .
In this paper we introduce a class of linear serial rank statistics for the problem of testing white noise against alternatives of ARMA serial dependence. The asymptotic normality of the proposed statistics is established, both under the null as well as alternative hypotheses, using LeCam's notion of contiguity. The efficiency properties of the proposed statistics are investigated, and an explicit formulation of the asymptotically most efficient score-generating functions is provided. Finally, we study the asymptotic relative efficiency of the proposed procedures with respect to their normal theory counterparts based on sample autocorrelations.
The local asymptotic normality property is established for a regression model with fractional ARIMA() errors. This result allows for solving, in an asymptotically optimal way, a variety of inference problems in the long-memory context: hypothesis testing, discriminant analysis, rank-based testing, locally asymptotically minimax andadaptive estimation, etc. The problem of testing linear constraints on the parameters, the discriminant analysis problem, and the construction of locally asymptotically minimax adaptive estimators are treated in some detail.
We consider the problem of testing uniformity on high-dimensional unit spheres.We are primarily interested in nonnull issues.We show that rotationally symmetric alternatives lead to two Local Asymptotic Normality (LAN) structures. The first one is for fixed modal location θ and allows to derive locally asymptotically most powerful tests under specified θ. The second one, that addresses the Fisher-von Mises-Langevin (FvML) case, relates to the unspecified-θ problem and shows that the high-dimensional Rayleigh test is locally asymptotically most powerful invariant. Under mild assumptions, we derive the asymptotic nonnull distribution of this test, which allows to extend away from the FvML case the asymptotic powers obtained there from Le Cam's third lemma. Throughout, we allow the dimension p to go to infinity in an arbitrary way as a function of the sample size n. Some of our results also strengthen the local optimality properties of the Rayleigh test in low dimensions.We perform aMonte Carlo study to illustrate our asymptotic results. Finally, we treat an application related to testing for sphericity in high dimensions.
We revisit, in an original and challenging perspective, the problem of
testing the null hypothesis that the mode of a directional signal is equal to a
given value. Motivated by a real data example where the signal is weak, we
consider this problem under asymptotic scenarios for which the signal strength
goes to zero at an arbitrary rate . Both under the null and the
alternative, we focus on rotationally symmetric distributions. We show that,
while they are asymptotically equivalent under fixed signal strength, the
classical Wald and Watson tests exhibit very different (null and non-null)
behaviours when the signal becomes arbitrarily weak. To fully characterize how
challenging the problem is as a function of , we adopt a Le Cam,
convergence-of-statistical-experiments, point of view and show that the
resulting limiting experiments crucially depend on . In the light of
these results, the Watson test is shown to be adaptively rate-consistent and
essentially adaptively Le Cam optimal. Throughout, our theoretical findings are
illustrated via Monte-Carlo simulations. The practical relevance of our results
is also shown on a real data example.
In this paper asymptotic expansions are derived for the density functions of the TSLS and LIML estimates of coefficients in a simultaneous equation system when the sample size increases and the effect of the exogenous variables increases along the sample size. These approximations are used to compare the asymptotic moments of the TSLS and LIML estimates and the concentration of probability around the true value of the estimates.
The classical method of least-squares estimation of the coefficients α in the (matrix) equation y = Zα + e yields estimators α̂ = Ay = + Ae. This method, however, employs only one of a class of transformation matrices, A, which yield this result; namely, the special case where A = (Z′Z)-1Z′. As is well known, the consistency of the estimators, α̂, requires that all of the variables whose sample values are represented as elements of the matrix Z be asymptotically uncorrelated with the error terms, e. In recent years some rather elaborate methods of obtaining consistent and otherwise optimal estimators of the coefficients α have been developed. In this paper we present a straightforward generalization of classical linear estimation which leads to estimates of α which possess optimal properties equivalent to those of existing limited-information single-equation estimators, and which is pedagogically simpler and less expensive to apply.3
This Tract presents an elaboration of the notion of 'contiguity', which is a concept of 'nearness' of sequences of probability measures. It provides a powerful mathematical tool for establishing certain theoretical results with applications in statistics, particularly in large sample theory problems, where it simplifies derivations and points the way to important results. The potential of this concept has so far only been touched upon in the existing literature, and this book provides the first systematic discussion of it. Alternative characterizations of contiguity are first described and related to more familiar mathematical ideas of a similar nature. A number of general theorems are formulated and proved. These results, which provide the means of obtaining asymptotic expansions and distributions of likelihood functions, are essential to the applications which follow.
The Gaussian maximum likelihood estimate is investigated for time series models that have locally a stationary behaviour (e.g. for time varying autoregressive models). The asymptotic properties are studied in the case where the fitted model is either correct or misspecified. For example the behaviour of the maximum likelihood estimate is explained in the case where a stationary model is fitted to a nonstationary process. As a general model selection criterion the AIC is considered. It can for example automatically select between stationary models, nonstationary models and deterministic trends.
This paper provides estimates of varieties × environments and plot error variances based on more than 1000 trials of varieties of oats, barley, wheat, perennial and Italian ryegrass, timothy, cocksfoot, potato, sugar beet, swedes, autumn kale, forage maize and field beans. Tables of critical differences and acceptance probabilities are presented to guide in planning future series of trials.(Received July 27 1983)
The concepts of the curved exponential family of distributions and ancillarity are applied to estimation problems of a single structural equation in a simultaneous equation model, and the effect of conditioning on ancillary statistics on the limited information maximum-likelihood (LIML) estimator is investigated. The asymptotic conditional covariance matrix of the LIML estimator conditioned on the second-order asymptotic maximal ancillary statistic is shown to be efficiently estimated by Liu and Breen's formula. The effect of conditioning on a second-order asymptotic ancillary statistic, i.e., the smallest characteristic root associated with the LIML estimation, is analyzed by means of an asymptotic expansion of the distribution as well as the exact distribution. The smallest root helps to give an intuitively appealing measure of precision of the LIML estimator.
Comparisons of estimators are made on the basis of their mean squared errors and their concentrations of probability computed by means of asymptotic expansions of their distributions when the disturbance variance tends to zero and alternatively when the sample size increases indefinitely. The estimators include k-class estimators (limited information maximum likelihood, two-stage least squares, and ordinary least squares) and linear combinations of them as well as modifications of the limited information maximum likelihood estimator and several Bayes' estimators. Many inequalities between the asymptotic mean squared errors and concentrations of probability are given. Among medianunbiasedestimators, the limited information maximum likelihood estimator dominates the median-unbiased fixed k-class estimator.
Let X
0, X
1,, X
nbe r.v.'s coming from a stochastic process whose finite dimensional distributions are of known functional form except that they involve a k-dimensional parameter. From the viewpoint of statistical inference, it is of interest to obtain the asymptotic distributions of the log-likelihood function and also of certain other r.v.'s closely associated with the likelihood function. The probability measures employed for this purpose depend, in general, on the sample size n. These problems are resolved provided the process satisfies some quite general regularity conditions. The results presented herein generalize previously obtained results for the case of Markovian processes, and also for i.n.n.i.d. r.v.'s. The concept of contiguity plays a key role in the various derivations.
This book is an introduction to the field of asymptotic statistics. The treatment is both practical and mathematically rigorous. In addition to most of the standard topics of an asymptotics course, including likelihood inference, M-estimation, the theory of asymptotic efficiency, U-statistics, and rank procedures, the book also presents recent research topics such as semiparametric models, the bootstrap, and empirical processes and their applications. The topics are organized from the central idea of approximation by limit experiments, which gives the book one of its unifying themes. This entails mainly the local approximation of the classical i.i.d. set up with smooth parameters by location experiments involving a single, normally distributed observation. Thus, even the standard subjects of asymptotic statistics are presented in a novel way. Suitable as a graduate or Master's level statistics text, this book will also give researchers an overview of research in asymptotic statistics.
For a class of locally stationary processes introduced by Dahlhaus, we derive the LAN theorem under non-Gaussianity and apply the results to asymptotically optimal estimation and testing problems. For a class F of statistics which includes important statistics, we derive the asymptotic distributions of statistics in F under contiguous alternatives of unknown parameter. Because the asymptotics depend on the non-Gaussianity of the process, we discuss the non-Gaussian robustness. An interesting feature of effect of non-Gaussianity is elucidated in terms of LAN. Furthermore, the LAN theorem is applied to adaptive estimation when the innovation density is unknown.
We consider families of real valued random variables which form a stationary autoregressive process with infinite order. Using a certain local parametrization we are able to establish that such models are locally asymptotically normal. This property admits (using well-known results) a characterization and a construction of local asymptotically minimax estimates for a finite dimensional subparameter of the model. Finally we will show that even adaptive estimation is possible.
It is shown that the likelihood ratio of an autoregressive time series of finite order with a regression trend is asymptotically normal. This result is used to derive the power of a test for positive correlation of the residuals under local autoregressive alternatives. The test is based on the Durbin-Watson statistics.
Modulating the dynamics of a nonlinear autoregressive model with a radial basis function (RBF) of exogenous variables is known to reduce the prediction error. Here, RBF is a function that decays to zero exponentially if the deviation between the exogenous variables and a center location becomes large. This paper introduces a class of RBF-based multiplicatively modulated nonlinear autoregressive (mmNAR) models. First, we establish the local asymptotic normality (LAN) for vector conditional heteroscedastic autoregressive nonlinear (CHARN) models, which include the mmNAR and many other well-known time-series models as special cases. Asymptotic optimality for estimation and testing is described in terms of LAN properties. The mmNAR model indicates goodness-of-fit for surface electromyograms (EMG) using electrocorticograms (ECoG) as the exogenous variables. Concretely, it is found that the negative potential of the motor cortex forces change in the frequency of EMG, which is reasonable from a physiological point of view. The proposed mmNAR model fitting is both useful and efficient as a signal-processing technique for extracting information on the action potential, which is associated with the postsynaptic potential
We consider the estimation problem for the parameter of a stationary ARMA process, with independent and identically, but not necessary normally distributed errors. First we prove local asymptotic normality (LAN) for this model. Then we construct locally asymptotically minimax (LAM) estimators, which asymptotically achieve the smallest possible covariance matrix. Utilizing these, we finally obtain strongly adaptive estimators, by using usual kernel estimators for the score function , where f denotes the density of the error distribution. These estimates turn out to be asymptotically optimal in the LAM sense for a wide class of symmetric densities f.
For a class of locally stationary processes introduced by Dahlhaus, this paper discusses the problem of testing composite hypotheses. First, for the Gaussian likelihood ratio test (GLR), Wald test (W) and Lagrange multiplier test (LM), we derive the limiting distribution under a composite hypothesis in parametric form. It is shown that the distribution of GLR, W and LM tends to a K^2 distribution under the hypothesis. We also evaluate their local powers under a sequence of local alternatives, and discuss their asymptotic optimality. The results can be applied to testing for stationarity. Some examples are given. They illuminate the local power property via simulation. On the other hand, we provide a nonparametric LAN theorem. Based on this result, we obtain the limiting distribution of the GLR under both null and alternative hypotheses described in nonparametric form. Finally, the numerical studies are given. Copyright 2003 Blackwell Publishing Ltd.
We estimate the causal effect of mandatory participation in the military service on the involvement in criminal activities. We exploit the random assignment of young men to military service in Argentina through a draft lottery to identify this causal effect. Using a unique set of administrative data that includes draft eligibility, participation in the military service, and criminal records, we find that participation in the military service increases the likelihood of developing a criminal record in adulthood. The effects are not only significant for the cohorts that performed military service during war times, but also for those that provided service at peace times. We also find that military service has detrimental effects on future performance in the labor market.
The primary purpose of this paper is to review a very few results on some basic elements of large sample theory in a restricted structural framework, as described in detail in the recent book by LeCam and Yang (1990, Asymptotics in Statistics: Some Basic Concepts . New York: Springer), and to illustrate how the asymptotic inference problems associated with a wide variety of time series regression models fit into such a structural framework. The models illustrated include many linear time series models, including cointegrated models and autoregressive models with unit roots that are of wide current interest. The general treatment also includes nonlinear models, including what have become known as ARCH models. The possibility of replacing the density of the error variables of such models by an estimate of it (adaptive estimation) based on the observations is also considered.
Under the framework in which the asymptotic problems are treated, only the approximating structure of the likelihood ratios of the observations, together with auxiliary estimates of the parameters, will be required. Such approximating structures are available under quite general assumptions, such as that the Fisher information of the common density of the error variables is finite and nonsingular, and the more specific assumptions, such as Gaussianity, are not required. In addition, the construction and the form of inference procedures will not involve any additional complications in the non-Gaussian situations because the approximating quadratic structure actually will reduce the problems to the situations similar to those involved in the Gaussian cases.
Локальная асимптотическая нормальность для неодинаково распределенных наблюдений
- И. А. Ибрагимов, Р. З. Хасьминский
Application of variance components to horticultural problems with special reference to a parathion residue study
- R. H. Sharpe, C. H. Van Middelem