On the dependence structure of order statistics

Given a random sample from a continuous variable, it is observed that the copula linking any pair of order statistics is independent of the parent distribution. To compare the degree of association between two such pairs of ordered random variables, a notion of relative monotone regression dependence (or stochastic increasingness) is considered. Using this concept, it is proved that for i<j, the dependence of the jth order statistic on the ith order statistic decreases as i and j draw apart. This extends earlier results of Tukey (Ann. Math. Statist. 29 (1958) 588) and Kim and David (J. Statist. Plann. Inference 24 (1990) 363). The effect of the sample size on this type of dependence is also investigated, and an explicit expression is given for the population value of Kendall's coefficient of concordance between two arbitrary order statistics of a random sample.     

Nonparametric estimation of the dependence function for a multivariate extreme value distribution

Understanding and modeling dependence structures for multivariate extreme values are of interest in a number of application areas. One of the well-known approaches is to investigate the Pickands dependence function. In the bivariate setting, there exist several estimators for estimating the Pickands dependence function which assume known marginal distributions [J. Pickands, Multivariate extreme value distributions, Bull. Internat. Statist. Inst., 49 (1981) 859-878; P. Deheuvels, On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions, Statist. Probab. Lett. 12 (1991) 429-439; P. Hall, N. Tajvidi, Distribution and dependence-function estimation for bivariate extreme-value distributions, Bernoulli 6 (2000) 835-844; P. Capéraà, A.-L. Fougères, C. Genest, A nonparametric estimation procedure for bivariate extreme value copulas, Biometrika 84 (1997) 567-577]. In this paper, we generalize the bivariate results to p-variate multivariate extreme value distributions with p?2. We demonstrate that the proposed estimators are consistent and asymptotically normal as well as have excellent small sample behavior.     

Analysis of two-sample truncated data using generalized logistic models

Parallel to Cox's [JRSS B34 (1972) 187-230] proportional hazards model, generalized logistic models have been discussed by Anderson [Bull. Int. Statist. Inst. 48 (1979) 35-53] and others. The essential assumption is that the two densities ratio has a known parametric form. A nice property of this model is that it naturally relates to the logistic regression model for categorical data. In astronomic, demographic, epidemiological, and other studies the variable of interest is often truncated by an associated variable. This paper studies generalized logistic models for the two-sample truncated data problem, where the two lifetime densities ratio is assumed to have the form exp{α+φ(x;β)}. Here φ is a known function of x and β, and the baseline density is unspecified. We develop a semiparametric maximum likelihood method for the case where the two samples have a common truncation distribution. It is shown that inferences for β do not depend the nonparametric components. We also derive an iterative algorithm to maximize the semiparametric likelihood for the general case where different truncation distributions are allowed. We further discuss how to check goodness of fit of the generalized logistic model. The developed methods are illustrated and evaluated using both simulated and real data.     

Empirical likelihood analysis of the Buckley-James estimator

The censored linear regression model, also referred to as the accelerated failure time (AFT) model when the logarithm of the survival time is used as the response variable, is widely seen as an alternative to the popular Cox model when the assumption of proportional hazards is questionable. Buckley and James [Linear regression with censored data, Biometrika 66 (1979) 429-436] extended the least squares estimator to the semiparametric censored linear regression model in which the error distribution is completely unspecified. The Buckley-James estimator performs well in many simulation studies and examples. The direct interpretation of the AFT model is also more attractive than the Cox model, as Cox has pointed out, in practical situations. However, the application of the Buckley-James estimation was limited in practice mainly due to its illusive variance. In this paper, we use the empirical likelihood method to derive a new test and confidence interval based on the Buckley-James estimator of the regression coefficient. A standard chi-square distribution is used to calculate the P-value and the confidence interval. The proposed empirical likelihood method does not involve variance estimation. It also shows much better small sample performance than some existing methods in our simulation studies.     

Semi-parametric estimation of partially linear single-index models

One of the most difficult problems in applications of semi-parametric partially linear single-index models (PLSIM) is the choice of pilot estimators and complexity parameters which may result in radically different estimators. Pilot estimators are often assumed to be root-n consistent, although they are not given in a constructible way. Complexity parameters, such as a smoothing bandwidth are constrained to a certain speed, which is rarely determinable in practical situations.In this paper, efficient, constructible and practicable estimators of PLSIMs are designed with applications to time series. The proposed technique answers two questions from Carroll et al. [Generalized partially linear single-index models, J. Amer. Statist. Assoc. 92 (1997) 477-489]: no root-n pilot estimator for the single-index part of the model is needed and complexity parameters can be selected at the optimal smoothing rate. The asymptotic distribution is derived and the corresponding algorithm is easily implemented. Examples from real data sets (credit-scoring and environmental statistics) illustrate the technique and the proposed methodology of minimum average variance estimation (MAVE).     

Comparison of Bartlett-type adjusted tests in the multiparameter case

Considering some Bartlett-type adjusted tests for a simple hypothesis about a multidimensional parameter, this paper clarifies similarities and dissimilarities with the one-parameter case developed in the 1990s, where a major emphasis is put on the issue posed by Rao and Mukerjee [C.R. Rao, R. Mukerjee, Comparison of Bartlett-type adjustments for the efficient score statistic, J. Statist. Plann. Inference 46 (1995) 137-146] on the power under a sequence of local alternatives. Not surprisingly, there is an infinite number of adjustments which extend Chandra-Mukerjee and Taniguchi approaches to the multiparameter case. Revisiting their ideas, this paper presents four specific cases (type K, K=0,1,2,3) and gives a sufficient condition under which our generalized adjustment for each case is uniquely determined, where type 0 is a counterpart of Chandra and Mukerjee’s original proposal for Rao’s test statistic, whereas the latter three types are introduced as double adjustments related to the Cordeiro and Ferrari approach. If the adjustment of type 1 is made instead of type K, K=0,2,3, it is shown that Chandra and Mukerjee’s approach is equivalent to Taniguchi’s approach in terms of the third-order local power. The same is partially true for type 0, depending on the model under consideration. However, the adjustments of type K, K=2,3, reveal, in general, the non-equivalence of these two approaches in terms of the third-order local power.     

On the optimality of a class of designs with three concurrences

In the present paper we consider a class of unequally replicated designs having concurrence range 2 and spectrum of the form μ₁(μ₂)^v−3μ₃. Now, Jacroux’s [Some sufficient conditions for the type I optimality of block designs, J. Statist. Plann. Inference 11 (1985) 385-396] Proposition 2.4 says that a design with spectrum of the above form, if satisfies some further conditions, is type 1 optimal. Unfortunately, this proposition does not apply to our designs since they have a poor status regarding E-optimality. Yet we are able to prove the A-optimality (in the general class) of these designs using majorisation technique. A method of construction of an infinite series of our A-optimal designs has also been given.The first and only known infinite series of examples of designs satisfying Jacroux’s conditions appears to be the first one in Section 4.1 of Morgan and Srivastav [On the Type-1 optimality of nearly balanced incomplete block designs with small concurrence range, Statist. Sinica 10 (2000) 1091-1116] - hitherto referred to as [MS]. In this paper, we use majorisation technique to prove stronger optimality properties of the above mentioned designs of [MS] as well as to present simpler proof of another optimality result in [MS].     

High breakdown estimators for principal components: the projection-pursuit approach revisited

Li and Chen (J. Amer. Statist. Assoc. 80 (1985) 759) proposed a method for principal components using projection-pursuit techniques. In classical principal components one searches for directions with maximal variance, and their approach consists of replacing this variance by a robust scale measure. Li and Chen showed that this estimator is consistent, qualitative robust and inherits the breakdown point of the robust scale estimator. We complete their study by deriving the influence function of the estimators for the eigenvectors, eigenvalues and the associated dispersion matrix. Corresponding Gaussian efficiencies are presented as well. Asymptotic normality of the estimators has been treated in a paper of Cui et al. (Biometrika 90 (2003) 953), complementing the results of this paper. Furthermore, a simple explicit version of the projection-pursuit based estimator is proposed and shown to be fast to compute, orthogonally equivariant, and having the maximal finite-sample breakdown point property. We will illustrate the method with a real data example.     

On constructions for two dimensional balanced sampling plan excluding contiguous units with block size four

Balanced sampling plans excluding contiguous units (BSEC) were first introduced in 1988 by Hedayat, Rao and Stufken [A.S. Hedayat, C.R. Rao, J. Stufken, Sampling plans excluding contiguous units, J. Statist. Plann. Inference 19 (1988) 159-170]. These designs can be used for survey sampling when the contiguous units provide similar information. In this paper, we show some recursive constructions for two dimensional BSECs with block size four, and give the existence of some infinite classes.     

Multivariate nonparametric tests in a randomized complete block design

In this paper multivariate extensions of the Friedman and Page tests for the comparison of several treatments are introduced. Related unadjusted and adjusted treatment effect estimates for the multivariate response variable are also found and their properties discussed. The test statistics and estimates are analogous to the traditional univariate methods. In test constructions, the univariate ranks are replaced by multivariate spatial ranks (J. Nonparam. Statist. 5 (1995) 201). Asymptotic theory is developed to provide approximations for the limiting distributions of the test statistics and estimates. Limiting efficiencies of the tests and treatment effect estimates are found in the multivariate normal and t distribution cases. The tests are rotation invariant only, but affine invariant versions can be easily constructed. The theory is illustrated by an example.     

Empirical likelihood inference for linear transformation models

Empirical likelihood inference is developed for censored survival data under the linear transformation models, which generalize Cox's [Regression models and life tables (with Discussion), J. Roy. Statist. Soc. Ser. B 34 (1972) 187-220] proportional hazards model. We show that the limiting distribution of the empirical likelihood ratio is a weighted sum of standard chi-squared distribution. Empirical likelihood ratio tests for the regression parameters with and without covariate adjustments are also derived. Simulation studies suggest that the empirical likelihood ratio tests are more accurate (under the null hypothesis) and powerful (under the alternative hypothesis) than the normal approximation based tests of Chen et al. [Semiparametric of transformation models with censored data, Biometrika 89 (2002) 659-668] when the model is different from the proportional hazards model and the proportion of censoring is high.     

Estimation, prediction and the Stein phenomenon under divergence loss

We consider two problems: (1) estimate a normal mean under a general divergence loss introduced in [S. Amari, Differential geometry of curved exponential families — curvatures and information loss, Ann. Statist. 10 (1982) 357-387] and [N. Cressie, T.R.C. Read, Multinomial goodness-of-fit tests, J. Roy. Statist. Soc. Ser. B. 46 (1984) 440-464] and (2) find a predictive density of a new observation drawn independently of observations sampled from a normal distribution with the same mean but possibly with a different variance under the same loss. The general divergence loss includes as special cases both the Kullback-Leibler and Bhattacharyya-Hellinger losses. The sample mean, which is a Bayes estimator of the population mean under this loss and the improper uniform prior, is shown to be minimax in any arbitrary dimension. A counterpart of this result for predictive density is also proved in any arbitrary dimension. The admissibility of these rules holds in one dimension, and we conjecture that the result is true in two dimensions as well. However, the general Baranchick [A.J. Baranchick, a family of minimax estimators of the mean of a multivariate normal distribution, Ann. Math. Statist. 41 (1970) 642-645] class of estimators, which includes the James-Stein estimator and the Strawderman [W.E. Strawderman, Proper Bayes minimax estimators of the multivariate normal mean, Ann. Math. Statist. 42 (1971) 385-388] class of estimators, dominates the sample mean in three or higher dimensions for the estimation problem. An analogous class of predictive densities is defined and any member of this class is shown to dominate the predictive density corresponding to a uniform prior in three or higher dimensions. For the prediction problem, in the special case of Kullback-Leibler loss, our results complement to a certain extent some of the recent important work of Komaki [F. Komaki, A shrinkage predictive distribution for multivariate normal observations, Biometrika 88 (2001) 859-864] and George, Liang and Xu [E.I. George, F. Liang, X. Xu, Improved minimax predictive densities under Kullbak-Leibler loss, Ann. Statist. 34 (2006) 78-92]. While our proposed approach produces a general class of predictive densities (not necessarily Bayes, but not excluding Bayes predictors) dominating the predictive density under a uniform prior. We show also that various modifications of the James-Stein estimator continue to dominate the sample mean, and by the duality of estimation and predictive density results which we will show, similar results continue to hold for the prediction problem as well.     

Testing multivariate normality in incomplete data of small sample size

In longitudinal studies with small samples and incomplete data, multivariate normal-based models continue to be a powerful tool for analysis. This has included a broad scope of biomedical studies. Testing the assumption of multivariate normality (MVN) is critical. Although many methods are available for testing normality in complete data with large samples, a few deal with the testing in small samples. For example, Liang et al. (J. Statist. Planning and Inference 86 (2000) 129) propose a projection procedure for testing MVN for complete-data with small samples where the sample sizes may be close to the dimension. To our knowledge, no statistical methods for testing MVN in incomplete data with small samples are yet available. This article develops a test procedure in such a setting using multiple imputations and the projection test. To utilize the incomplete data structure in multiple imputation, we adopt a noniterative inverse Bayes formulae (IBF) sampling procedure instead of the iterative Gibbs sampling to generate iid samples. Simulations are performed for both complete and incomplete data when the sample size is less than the dimension. The method is illustrated with a real study on an anticancer drug.     

A test for elliptical symmetry

This paper presents a statistic for testing the hypothesis of elliptical symmetry. The statistic also provides a specialized test of multivariate normality. We obtain the asymptotic distribution of this statistic under the null hypothesis of multivariate normality, and give a bootstrapping procedure for approximating the null distribution of the statistic under an arbitrary elliptically symmetric distribution. We present simulation results to examine the accuracy of the asymptotic distribution and the performance of the bootstrapping procedure. Finally, for selected alternatives, we compare the power of our test statistic with that of recently proposed tests for elliptical symmetry given by Manzotti et al. [A statistic for testing the null hypothesis of elliptical symmetry, J. Multivariate Anal. 81 (2002) 274-285] and Schott [Testing for elliptical symmetry in covariance-matrix-based analyses, Statist. Probab. Lett. 60 (2002) 395-404], and with that of the well known tests for multivariate normality of Mardia [Measures of multivariate skewness and kurtosis with applications, Biometrika 57 (1970) 519-530] and Baringhaus and Henze [A consistent test for multivariate normality based on the empirical characteristic function, Metrika 35 (1988) 339-348].     

The hyper-Dirichlet process and its discrete approximations: The butterfly model

The aim of this paper is the study of some random probability distributions, called hyper-Dirichlet processes. In the simplest situation considered in the paper these distributions charge the product of three sample spaces, with the property that the first and the last component are independent conditional to the middle one. The law of the marginals on the first two and on the last two components are specified to be Dirichlet processes with the same marginal parameter measure on the common second component. The joint law is then obtained as the hyper-Markov combination, introduced in [A.P. Dawid, S.L. Lauritzen, Hyper-Markov laws in the statistical analysis of decomposable graphical models, Ann. Statist. 21 (3) (1993) 1272-1317], of these two Dirichlet processes. The processes constructed in this way in fact are in fact generalizations of the hyper-Dirichlet laws on contingency tables considered in the above paper. Our main result is the convergence to the hyper-Dirichlet process of the sequence of hyper-Dirichlet laws associated to finer and finer “discretizations” of the two parameter measures, which is proved by means of a suitable coupling construction.     

Efficient estimators and LAN in canonical bivariate POT models

Bivariate generalized Pareto distributions (GPs) with uniform margins are introduced and elementary properties such as peaks-over-threshold (POT) stability are discussed. A unified parameterization with parameter ?∈[0,1] of the GPs is provided by their canonical parameterization. We derive efficient estimators of ? and of the dependence function of the GP in various models and establish local asymptotic normality (LAN) of the loglikelihood function of a 2×2 table sorting of the observations. From this result we can deduce that the estimator of ? suggested by Falk and Reiss (2001, Statist. Probab. Lett. 52, 233-242) is not efficient, whereas a modification actually is.     

Density testing in a contaminated sample

We study non-parametric tests for checking parametric hypotheses about a multivariate density f of independent identically distributed random vectors Z₁,Z₂,… which are observed under additional noise with density ψ. The tests we propose are an extension of the test due to Bickel and Rosenblatt [On some global measures of the deviations of density function estimates, Ann. Statist. 1 (1973) 1071-1095] and are based on a comparison of a nonparametric deconvolution estimator and the smoothed version of a parametric fit of the density f of the variables of interest Z_i. In an example the loss of efficiency is highlighted when the test is based on the convolved (but observable) density g=f*ψ instead on the initial density of interest f.     

On Maximum Likelihood Estimation for Gaussian Spatial Autoregression Models

The article presents a central limit theorem for the maximum likelihood estimator of a vector-valued parameter in a linear spatial stochastic difference equation with Gaussian white noise right side. The result is compared to the known limit theorems derived for the approximate likelihood e.g. by Whittle (1954, Biometrika, 41, 434-439), Guyon (1982, Biometrika, 69, 95-105) and Rosenblatt (1985, Stationary Sequences and Random Fields, Birkhäuser, Boston) and to the asymptotic properties of the quasi-likelihood studied by Heyde and Gay (1989, Stochastic Process. Appl., 31, 223-236; 1993, Stochastic Process. Appl., 45, 169-182). Application of the theory is demonstrated on several classes of models including the one considered by Niu (1995, J. Multivariate Anal., 55, 82-104).     

On the block thresholding wavelet estimators with censored data

We consider block thresholding wavelet-based density estimators with randomly right-censored data and investigate their asymptotic convergence rates. Unlike for the complete data case, the empirical wavelet coefficients are constructed through the Kaplan-Meier estimators of the distribution functions in the censored data case. On the basis of a result of Stute [W. Stute, The central limit theorem under random censorship, Ann. Statist. 23 (1995) 422-439] that approximates the Kaplan-Meier integrals as averages of i.i.d. random variables with a certain rate in probability, we can show that these wavelet empirical coefficients can be approximated by averages of i.i.d. random variables with a certain error rate in L². Therefore we can show that these estimators, based on block thresholding of empirical wavelet coefficients, achieve optimal convergence rates over a large range of Besov function classes , p≥2, q≥1 and nearly optimal convergence rates when 1≤p<2. We also show that these estimators achieve optimal convergence rates over a large class of functions that involve many irregularities of a wide variety of types, including chirp and Doppler functions, and jump discontinuities. Therefore, in the presence of random censoring, wavelet estimators still provide extensive adaptivity to many irregularities of large function classes. The performance of the estimators is tested via a modest simulation study.     

A weighted multivariate signed-rank test for cluster-correlated data

A weighted multivariate signed-rank test is introduced for an analysis of multivariate clustered data. Observations in different clusters may then get different weights. The test provides a robust and efficient alternative to normal theory based methods. Asymptotic theory is developed to find the approximate p-value as well as to calculate the limiting Pitman efficiency of the test. A conditionally distribution-free version of the test is also discussed. The finite-sample behavior of different versions of the test statistic is explored by simulations and the new test is compared to the unweighted and weighted versions of Hotelling’s T² test and the multivariate spatial sign test introduced in [D. Larocque, J. Nevalainen, H. Oja, A weighted multivariate sign test for cluster-correlated data, Biometrika 94 (2007) 267-283]. Finally, a real data example is used to illustrate the theory.