The idea of basing tests on the sample distribution function is a natural one. The Kolmogorov-Smirnov tests are of this nature. Blum, Kiefer & Rosenblatt (1961) made use of this approach to construct distribution free tests of independence. In this paper this method is further applied to the common two-sample problems of location and dispersion, the k-sample problem of location, and to the problem of dependence in bivariate samples. The rationale of the method is the following. Write the departure from the null hypothesis (against which the test must be sensitive) in terms of the true distribution functions and regard the mean value of this departure as the parameter of interest. The sample estimate of this parameter, expressed in terms of sample distribution functions, is then proposed as test statistic. This statistic is generally only dependent on the ranks of the samples and is consequently distribution free. Specifically this approach leads to Wilcoxon's two-sample test, a k-sample extension of Wilcoxon's test which is slightly different from the Kruskal-Wallis (1952) extension, the test of Ansari & Bradley (1960) for differences in dispersion, which is also a special case of a procedure proposed by Barton & David (1958), and to a test of dependence in bivariate samples which comes out to be a linear function of the rank correlation coefficients of Spearman and Kendall. The alternative k-sample extension of Wilcoxon's test has the same asymptotic relative efficiency properties as the Kruskal-Wallis test; it is however consistent against a slightly wider class than the latter. As is to be expected, the alternative test of independence behaves for large samples in much the same way as the tests of Spearman and Kendall.
The penalised least squares approach with smoothly clipped absolute deviation penalty has been consistently demonstrated to be an attractive regression shrinkage and selection method. It not only automatically and consistently selects the important variables, but also produces estimators which are as efficient as the oracle estimator. However, these attractive features depend on appropriately choosing the tuning parameter. We show that the commonly used the generalised crossvalidation cannot select the tuning parameter satisfactorily, with a nonignorable overfitting effect in the resulting model. In addition, we propose a bic tuning parameter selector, which is shown to be able to identify the true model consistently. Simulation studies are presented to support theoretical findings, and an empirical example is given to illustrate its use in the Female Labor Supply data.
The estimation of parameters in an absorbing Markov chain has been discussed by Gani (1956), Bhat & Gani (1960), Bhat (1961) and possibly by other authors. Asymptotic theory for the distribution of maximum-likelihood estimators applies if a large number of independent replicates of the chain is available. These replicates, however, could be considered as occurring sequentially in time, so that the chain has implicitly been altered in such a way that the absorbing state has been replaced by an ‘instantaneous return’ state; once it is reached, a new replicate is commenced starting from the original initial state.
In this paper, we concern ourselves with the asymptotic theory of maximum-likelihood estimators from a single realization of truly absorbing chains. It is clear that if the number of states is kept fixed there can be no asymptotic theory, since with probability one, an absorbing state will be reached after a finite number of transitions, and no further information can be obtained by continuing observation.
In § 2, we show by means of two simple examples that asymptotic theory may, at least in some cases, be available if the number of non-absorbing states is large. In the remaining sections, forming the main part of the paper, we discuss a more complex example, a population genetic model of Moran (1958a). We do not prove that the conjectured asymptotic theory holds for the estimation of the parameter, but produce numerical evidence from simulation studies to support this conjecture.
Further research is needed to clarify the general problem of inference in absorbing Markov chains. The problem can be made to depend on the theory of positively regular chains but the latter is itself incomplete for this purpose.
This paper extends the induced smoothing procedure of Brown & Wang (2006) for the semiparametric accelerated failure time model to the case of clustered failure time data. The resulting procedure permits fast and accurate computation of regression parameter estimates and standard errors using simple and widely available numerical methods, such as the Newton-Raphson algorithm. The regression parameter estimates are shown to be strongly consistent and asymptotically normal; in addition, we prove that the asymptotic distribution of the smoothed estimator coincides with that obtained without the use of smoothing. This establishes a key claim of Brown & Wang (2006) for the case of independent failure time data and also extends such results to the case of clustered data. Simulation results show that these smoothed estimates perform as well as those obtained using the best available methods at a fraction of the computational cost.
A bivariate correlated Poisson model for the number of accidents sustained by a set of individuals exposed to a risk situation in two successive periods of time is considered. It is shown that a selection of individuals free from accidents in the first period reduces the average number of accidents to be expected in the next period, provided that the accident proneness varies from individual to individual with an arbitrary non-degenerate distribution.
We propose a graphical measure, the generalized negative predictive function, to quantify the predictive accuracy of covariates
for survival time or recurrent event times. This new measure characterizes the event-free probabilities over time conditional
on a thresholded linear combination of covariates and has direct clinical utility. We show that this function is maximized
at the set of covariates truly related to event times and thus can be used to compare the predictive accuracy of different
sets of covariates. We construct nonparametric estimators for this function under right censoring and prove that the proposed
estimators, upon proper normalization, converge weakly to zero-mean Gaussian processes. To bypass the estimation of complex
density functions involved in the asymptotic variances, we adopt the bootstrap approach and establish its validity. Simulation
studies demonstrate that the proposed methods perform well in practical situations. Two clinical studies are presented.
Recent scientific and technological innovations have produced an abundance of potential markers that are being investigated for their use in disease screening and diagnosis. In evaluating these markers, it is often necessary to account for covariates associated with the marker of interest. Covariates may include subject characteristics, expertise of the test operator, test procedures or aspects of specimen handling. In this paper, we propose the covariate-adjusted receiver operating characteristic curve, a measure of covariate-adjusted classification accuracy. Nonparametric and semiparametric estimators are proposed, asymptotic distribution theory is provided and finite sample performance is investigated. For illustration we characterize the age-adjusted discriminatory accuracy of prostate-specific antigen as a biomarker for prostate cancer.
Directed acyclic graphs are commonly used to represent causal relationships among random variables in graphical models. Applications of these models arise in the study of physical and biological systems where directed edges between nodes represent the influence of components of the system on each other. Estimation of directed graphs from observational data is computationally NP-hard. In addition, directed graphs with the same structure may be indistinguishable based on observations alone. When the nodes exhibit a natural ordering, the problem of estimating directed graphs reduces to the problem of estimating the structure of the network. In this paper, we propose an efficient penalized likelihood method for estimation of the adjacency matrix of directed acyclic graphs, when variables inherit a natural ordering. We study variable selection consistency of lasso and adaptive lasso penalties in high-dimensional sparse settings, and propose an error-based choice for selecting the tuning parameter. We show that although the lasso is only variable selection consistent under stringent conditions, the adaptive lasso can consistently estimate the true graph under the usual regularity assumptions.
We give a definition of a bounded edge within the causal directed acyclic graph framework. A bounded edge generalizes the
notion of a signed edge and is defined in terms of bounds on a ratio of survivor probabilities. We derive rules concerning
the propagation of bounds. Bounds on causal effects in the presence of unmeasured confounding are also derived using bounds
related to specific edges on a graph. We illustrate the theory developed by an example concerning estimating the effect of
antihistamine treatment on asthma in the presence of unmeasured confounding.
Several sparseness penalties have been suggested for delivery of good predictive performance in automatic variable selection
within the framework of regularization. All assume that the true model is sparse. We propose a penalty, a convex combination
of the L1- and L∞-norms, that adapts to a variety of situations including sparseness and nonsparseness, grouping and nongrouping. The proposed
penalty performs grouping and adaptive regularization. In addition, we introduce a novel homotopy algorithm utilizing subgradients
for developing regularization solution surfaces involving multiple regularizers. This permits efficient computation and adaptive
tuning. Numerical experiments are conducted using simulation. In simulated and real examples, the proposed penalty compares
well against popular alternatives.
We propose a semiparametric additive rate model for modelling recurrent events in the presence of a terminal event. The dependence between recurrent events and terminal event is nonparametric. A general transformation model is used to model the terminal event. We construct an estimating equation for parameter estimation and derive the asymptotic distributions of the proposed estimators. Simulation studies demonstrate that the proposed inference procedure performs well in realistic settings. Application to a medical study is presented.
Results are given concerning inferences that can be drawn about interaction when binary exposures are subject to certain forms
of independent nondifferential misclassification. Tests for interaction, using the misclassified exposures, are valid provided
the probability of misclassification satisfies certain bounds. Results are given for additive statistical interactions, for
causal interactions corresponding to synergism in the sufficient cause framework and for so-called compositional epistasis.
Both two-way and three-way interactions are considered. The results require only that the probability of misclassification
be no larger than 1/2 or 1/4, depending on the test. For additive statistical interaction, a method to correct estimates and
confidence intervals for misclassification is described. The consequences for power of interaction tests under exposure misclassification
are explored through simulations.