High precision implementation of Steck's recursion method for use in goodness-of-fit tests

Classical continuous goodness-of-fit (GOF) testing is employed for examining whether the data come from an assumed parametric model. In many cases, GOF tests assume a uniform null distribution and examine extreme values of the order statistics of the samples. Many of these statistics can be expressed by a function of the order statistics and the p-values amount to a joint probability statement based on the uniform order statistics. In this paper, we utilize Steck''s recursion method and propose two high precision computing algorithms to compute the p-values for these GOF statistics. The numerical difficulties in implementing Steck''s method are discussed and compared with solutions provided in high precision libraries.     

Sample quantiles and additive statistics: Information,sufficiency, estimation

The order of the increase in the Fisher information measure contained in a finite number k of additive statistics or sample quantiles, constructed from a sample of size n, as n → ∞, is investigated. It is shown that the Fisher information in additive statistics increases asymptotically in a manner linear with respect to n, if 2 + δ moments of additive statistics exist for some δ > 0. If this condition does not hold, the order of increase in this information is non-linear and the information may even decrease. The problem of asymptotic sufficiency of sample quantiles is investigated and some linear analogues of maximum likelihood equations are constructed.     

Heteroskedasticity in the Tobit model

The paper deals with parameter estimation and the testing of individual parameters in heteroskedastic Tobit models. The statistical properties of semiparametric and maximum likelihood estimators are evaluated. Correspondingt-test statistics are compared. Results from a Monte Carlo experiment indicate that the semiparametric estimator performs relatively better than the maximum likelihood estimator. The associatedt-test statistics appear to perform better than the corresponding maximum likelihood test statistics. *** DIRECT SUPPORT *** A06GP002 00008     

On the Markov property of order statistics

The order statistics from a sample of size n≥3 from a discrete distribution form a Markov chain if and only if the parent distribution is supported by one or two points. More generally, a necessary and sufficient condition for the order statistics to form a Markov chain for (n≥3) is that there does not exist any atom x₀ of the parent distribution F satisfying F(x₀-)>0 and F(x₀)<1. To derive this result a formula for the joint distribution of order statistics is proved, which is of an interest on its own. Many exponential characterizations implicitly assume the Markov property. The corresponding putative geometric characterizations cannot then be reasonably expected to obtain. Some illustrative geometric characterizations are discussed.     

Comparing waiting times in a multi-stage model: A log-rank approach

A proper log-rank test for comparing two waiting (i.e. sojourn, gap) times under right censored data has been absent in the survival literature. The classical log-rank test provides a biased comparison even under independent right censoring since the censoring induced on the time since state entry depends on the entry time unless the hazards are semi-Markov. We develop test statistics for comparing K waiting time distributions from a multi-stage model in which censoring and waiting times may be dependent upon the transition history in the multi-stage model. To account for such dependent censoring, the proposed test statistics utilize an inverse probability of censoring weighted (IPCW) approach previously employed to define estimators for the cumulative hazard and survival function for waiting times in multi-stage models. We develop the test statistics as analogues to K-sample log-rank statistics for failure time data, and weak convergence to a Gaussian limit is demonstrated. A simulation study demonstrates the appropriateness of the test statistics in designs that violate typical independence assumptions for multi-stage models, under which naive test statistics for failure time data perform poorly, and illustrates the superiority of the test under proportional hazards alternatives to a Mann–Whitney type test. We apply the test statistics to an existing data set of burn patients.     

On the ABLUE of the normal mean from a censored sample

The asymptotically best linear unbiased estimate (ABLUE) of the normal mean is discussed. The estimate is based on k selected order statistics chosen from a singly or doubly censored large sample of size n(>k). The coefficients, the asymptotic relative efficiency of the estimate, and the optimum spacing of k real numbers between 0 and 1 which determines the optimum ranks of order statistics, are provided. A comparison between the ABLUE and the iterated maximum likelihood estimate is made.     

Recurrence relations for moments of progressively censored order statistics from logistic distribution with applications to inference

In this paper, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a logistic distribution. The use of these relations in a systematic manner allows us to compute all the means, variances and covariances of progressively Type-II right censored order statistics from the logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R₁,…,R_m). The results established here generalize the corresponding results for the usual order statistics due to [Shah, 1966] and [Shah, 1970]. These moments are then utilized to derive best linear unbiased estimators of the location and scale parameters of the logistic distribution. A comparison of these estimators with the maximum likelihood estimations is then made. The best linear unbiased predictors of censored failure times are briefly discussed. Finally, an illustrative example is presented.     

Measures of lack of fit from tests of chi-squared type

If X² is the Pearson chi-squared statistic for testing fit, then $\text{X}{}^2\text{n}$ has long been considered an associated measure of the degree of lack of fit. Here we consider two classes of statistics of chi-squared type, each having X² as a member. The first is a class of directed divergence statistics discussed by Cressie and Read, the second consists of nonnegative definite quadratic forms in the standardized cell frequencies. We investigate the large sample behavior of $\text{T}\text{n}$, where T is any of these statistics. A number of auxiliary results on the Cressie-Read statistics are also obtained. The measures are illustrated by application to data from classical physics compiled by Stigler.     

A family of tests for exponentiality against IFR alternatives

In this paper we consider the problem of testing exponentiality against IFR alternatives. A measure of deviation from exponentiality is developed and a class of test statistics are constructed on the basis of this measure. It is shown that the test statistic is an L-statistic. The asymptotic as well as the exact distributions of the test statistics are obtained and the test statistics are proved to be consistent. The Pitman efficiency has also been studied.     

Some asymptotic results for hypothesis testing in multivariate failure time data

This paper presents a class of generalized Wald, generalized score and generalized likelihood ratio statistics for hypothesis testing and model selection for multivariate failure time data. These statistics are based on a marginal hazard model with a common baseline hazard function. The large sample distributions of these statistics are examined. It is shown that the proposed test statistics follow asymptotically a weighted sum of independent χ₁² distributions.     

Asymptotic linearity of serial and nonserial multivariate signed rank statistics

Asymptotic linearity plays a key role in estimation and testing in the presence of nuisance parameters. This property is established, in the very general context of a multivariate general linear model with elliptical VARMA errors, for the serial and nonserial multivariate rank statistics considered in Hallin and Paindaveine (Ann. Statist. 30 (2002a) 1103; Bernoulli 8 (2002b) 787 Ann. Statist. 32 (2004), to appear) and Oja and Paindaveine (J. Statist. Plann. Inference (2004), to appear). These statistics, which are multivariate versions of classical signed rank statistics, involve (i) multivariate signs based either on (pseudo-)Mahalanobis residuals, or on a modified version (absolute interdirections) of Randles's interdirections, and (ii) a concept of ranks based either on (pseudo-)Mahalanobis distances or on lift-interdirections.     

Two remarks on order statistics

Let X₍₁₎,…,X_(n) be the order statistics of n iid distributed random variables. We prove that (X_(i)) have a certain Markov property for general distributions and secondly that the order statistics have monotone conditional regression dependence. Both properties are well known in the case of continuous distributions.     

Relations for m-generalized order statistics via an Opial-type inequality

An Opial-type inequality is applied to obtain relations for expectations of functions of m-generalized order statistics (m-gOSs), their distribution functions, as well as moment-generating functions. Respective inequalities for common order statistics and record values are contained as particular cases.     

Nonparametric model checking for k-out-of-n systems

It is an important problem in reliability analysis to decide whether for a given k-out-of-n system the static or the sequential k-out-of-n model is appropriate. Often components are redundantly added to a system to protect against failure of the system. If the failure of any component of the system induces a higher rate of failure of the remaining components due to increased load, the sequential k-out-of-n model is appropriate. The increase of the failure rate of the remaining components after a failure of some component implies that the effects of the component redundancy are diminished. On the other hand, if all the components have the same failure distribution and whenever a failure occurs, the remaining components are not affected, the static k-out-of-n model is adequate. In this paper, we consider nonparametric hypothesis tests to make a decision between these two models. We analyze test statistics based on the profile score process as well as test statistics based on a multivariate intensity ratio and derive their asymptotic distribution. Finally, we compare the different test statistics.     

Some results for type I censored sampling from geometric distributions

This paper gives the likelihood function for a Type I censored sample from the geometric distribution with parameter p, and the maximum likelihood estimator for p. Exact and asymptotic sampling distributions of joint sufficient statistics for p are derived. Such distributional results make it possible to develop tests or confidence intervals based on discrete censored data, which are not available now in the literature. Neyman-Pearson tests for p are developed. Examples are given to illustrate these results.     

Test of symmetry based on copula function

We introduce some new nonparametric statistical tests of symmetry. The limiting behaviors of the proposed statistics are established under the null hypothesis. Emphasis is placed on explanation of the strong approximation methodology. The asymptotic validity of a re-sampling method to compute p-values is also established. A simulation study for some considered statistics is performed.     

New tests based on Rubin's empirical distribution function

In this paper we derive some new tests for goodness-of-fit based on Rubin's empirical distribution function (EDF). Substituting Rubin's EDF for the classical EDF in the Kolmogorov–Smirnov, Cramér–von Mises, Anderson–Darling statistics, since Rubin's EDF for a given sample is a randomized distribution function, randomized statistics are derived, of which the qth quantile and the expectation are chosen as test statistics. We show that the new tests are consistent under simple hypothesis. Several power comparisons are also performed to show that the new tests are generally more powerful than the classical ones.     

Asymptotic comparison of the critical values of step-down and step-up multiple comparison procedures

We consider the problem of comparing step-down and step-up multiple test procedures for testing n hypotheses when independent p-values or independent test statistics are available. The defining critical values of these procedures for independent test statistics are asymptotically equal, which yields a theoretical argument for the numerical observation that the step-up procedure is mostly more powerful than the step-down procedure. The main aim of this paper is to quantify the differences between the critical values more precisely. As a by-product we also obtain more information about the gain when we consider two subsequent steps of these procedures. Moreover, we investigate how liberal the step-up procedure becomes when the step-up critical values are replaced by their step-down counterparts or by more refined approximate values. The results for independent p-values are the basis for obtaining corresponding results when independent real-valued test statistics are at hand. It turns out that the differences of step-down and step-up critical values as well as the differences between subsequent steps tend to zero for many distributions, except for heavy-tailed distributions. The Cauchy distribution yields an example where the critical values of both procedures are nearly linearly increasing in n.     

Rank procedures for a large number of treatments

In this paper, we prove a novel result about the asymptotic distribution of a class of rank statistics that can be used in a situation where the number of replications is limited, whereas the number of treatments goes to infinity (large k, small n case).The results can be applied to, e.g., data from agricultural screening trials where usually the numbers of factor levels are large, but there are only few replications per factor level.     

Foreword (and Preface)

We present new algorithms for computing the exact distributions and p-values of quadratic t-sample distribution-free statistics of Kruskal–Wallis type. These algorithms are presented in terms of generating functions. We show that our algorithm also works for cases with ties and that it is much faster than existing algorithms. Moreover, we show how to use the results for the Kruskal–Wallis type statistics to compute the exact null distribution of the Chacko–Shorack statistic.