Guarded Weights of Evidence and Acceptability Profiles Based on Signs: II. Asymptotic Results

The concepts of guarded weights of evidence and acceptability profiles have been extended to the distribution-free setting in Dollinger, Kulinskaya & Staudte (1999). In that first of two parts the advantages of these concepts relative to traditional ones such as p -values and confidence intervals derived from hypothesis tests are emphasized for small samples. Here in Part II asymptotic expressions are found for guarded weights of evidence for hypothesesregarding the median of a symmetric distribution and related acceptability profiles for the median. It is also seen that for local alternatives the efficacy and Pitman asymptotic relative efficiency of the sign statistic for testing hypotheses carries over to the more general setting of guarded weights of evidence.     

A new test for exponentiality against HNBUE alternatives

Abstract

Two families of test statistics for testing the null hypothesis of exponentiality against Harmonic New Better than Used in Expectation (HNBUE) alternatives are proposed. Asymptotic distributions of the test statistics are derived under the null and alternative hypotheses and the consistency of the tests established. Comparison with competing tests are made in terms of Pitman Asymptotic Relative Efficiency (PARE). Simulation studies have been carried out to assess the performance of the tests. Finally, the test has been applied to three real life data sets described in Proschan, Susarla and Van Ryzin and Engelhardt, Bain and Wright.     

On tail-ordering and comparison of failure rates

In this paper, we have studied some implications between tail-ordering (also known as dispersive ordering) and failure rate ordering (also called TP₂ ordering) of two probability distribution functions. Based on independent random samples from these distributions, a class of distribution-free tests has been proposed for testing the null hypothesis that the two life distributions are identical against the alternative that one failure rate is uniformly smaller than the other. The tests have good efficiencies as compared to their competitors.     

Combining poisson means

The problem of estimating the Poisson mean is considered based on the two samples in the presence of uncertain prior information (not in the form of distribution) that two independent random samples taken from two possibly identical Poisson populations. The parameter of interest is λ₁ from population I. Three estimators, i.e. the unrestricted estimator, restricted estimator and preliminary test estimator are proposed. Their asymptotic mean squared errors are derived and compared; parameter regions have been found for which restricted and preliminary test estimators are always asymptotically more efficient than the classical estimator. The relative dominance picture of the estimators is presented. Maximum and minimum asymptotic efficiencies of the estimators relative to the classical estimator are tabulated. A max-min rule for the size of the preliminary test is also discussed. A Monte Carlo study is presented to compare the performance of the estimator with that of Kale and Bancroft (1967).     

Tests for exponentiality against new better than old in expectation and new better than some used in expectation alternatives

We present statistical procedures for testing exponentiality againt New Better than Old in Expectation (NBOE) and New Better than Some Used in Expectation (NBSUE) alternatives. The test statistics devised for the purpose are U-Statistics and hence asymptotically normally distributed. Pitman's asymptotic relative efficiency results have been obtained and Monte Carlo study presented to compare power of the test proposed with the other tests.     

Ordres stochastiques et efficacité asymptotique des L-estimateurs

The efficiency of an estimator depends heavily on the tails of the distribution of the observations. Several partial orders have been defined to compare probability distributions according to their tails. In this paper we show that the asymptotic relative efficiency of two L-estimators with monotone weight functions is isotonic with respect to the partial orders defined by van Zwet (1964) and Lawrence (1975). We also give results concerning trimmed means.     

A robust test for the homogeneity of scales

Many robust tests for the equality of variances have been proposed recently. Brown and Forsythe (1974) and Layard (1973) review some of the well-known procedures and compare them by simulation methods. Brown and Forsythe’s alternative formulation of Levene’s test statistic is found to be quite robust under certain nonnormal distributions. The performance of the methods, however, suffers in the presence of heavy tailed distributions such as the Cauchy distribution.

In this paper, we propose and study a simple robust test. The results obtained from the Monte Carlo study compare favorably with those of the existing procedures.     

Comparison of estimators of interclass and intraclass correlations from familial data

When familles have different numbers of offspring, Srivastava (1984) gave an alternative approach to deriving the maximum-likelihood estimators of inter- and intraclass correlations, which requires solving only one equation. Since the procedure is iterative and requires considerable computation, several alternative estimators have been proposed in the literature. In this paper, a comparison is made between the maximum-likelihood estimator and two alternative estimators proposed by Srivastava (1984). By obtaining the asymptotic normal distributions of the estimators, it is shown that one of the easily computable estimators is comparable to the maximum-likelihood estimator.     

DISTRIBUTION-FREE TESTS BASED ON SUB-SAMPLE EXTREMA FOR TESTING AGAINST POSITIVE DEPENDENCE

A class of distribution-free tests based on U-statistics has been proposed for testing the null hypothesis of independence against positive quadrant dependence. The tests are based on U-statistics and the Kendall's-tau test belongs to this class.     

Robust M-estimators of scale: Minimax bias versus maximal variance

In the location-scale estimation problem, we study robustness properties of M-estimators of the scale parameter under unknown ?-contamination of a fixed symmetric unimodal error distribution F₀. Within a general class of M-estimators, the estimator with minimax asymptotic bias is shown to lie within the subclass of α-interquantile ranges of the empirical distribution symmetrized about the sample median. Our main result is that as ? → 0, the limiting minimax asymptotic bias estimator is sometimes (e.g., when F_o is Cauchy), but not always, the median absolute deviation about the median. It is also shown that contamination in the neighbourhood of a discontinuity of the influence function of a minimax bias estimator can sometimes inflate the asymptotic variance beyond that achieved by placing all the ?-contamination at infinity. This effect is quantified by a new notion of asymptotic efficiency that takes into account the effect of infinitesimal contamination of the parametric model for the error distribution.     

Some recurrence relation-based estimators of the parameters of a generalized log-series distribution

Two families of closed form estimators are proposed for estimating the single parameter of the log-series distribution(LSD)and for estimating the two parameters of a generalization of the LSD distribution(GLSD)presented by Tripathi and Gupta(1985). These families are based on the recurrence relations obtained from these distributions, are of closed form, and have very high asymptotic relative effi¬ciencies. Some two-stage procedures are suggested.     

A-optimal designs for optimum mixture in an additive quadratic mixture model

In the analysis of experiments with mixtures, quadratic models have been widely used. The optimum designs for the estimation of optimum mixing proportions in a quadratic mixture model have been studied by Pal and Mandal [Optimum designs for optimum mixtures. Statist Probab Lett. 2006;76:1369–1379] and Mandal et al. [Optimum mixture designs: a pseudo-Bayesian approach. J Ind Soc Agric Stat. 2008;62(2):174–182; Optimum mixture designs under constraints on mixing components. Statist Appl. 2008;6(1&2) (New Series): 189–205], using a pseudo-Bayesian approach. In this paper, a similar approach has been employed to obtain the A-optimal designs for the estimation of optimum proportions in an additive quadratic mixture model, proposed by Darroch and Waller [Additivity and interaction in three-component experiments with mixture. Biometrika. 1985;72:153–163], when the number of components is 3, 4 and 5. It has been shown that the vertices of the simplex are necessarily the support points of the optimum design, and the other support points include barycentres of depth at most 2.     

A brief survey of nonparametric statistics

The area of non-parametrie statistics has briefly been surveyed. The survey is limited to univariate fixed-sample size problems     

Testing for No Effect by Cosine Series Methods

The properties of three lack-of-fit tests that are related to non-parametric cosine regression analysis are examined in the context of testing for a constant mean function. Analytic power comparisons of these tests vs a most powerful test are made using intermediate asymptotic relative efficiency. In particular, a data-driven test is produced which is asymptotically as efficient as the most powerful test over a class of alternatives. A small scale simulation experiment is conducted to ascertain the extent that the large sample comparisons are applicable to finite samples.     

Ordinary least squares estimation of simultaneous equation systems with trended data: further results

The paper gives sufficient conditions for the consistency and asymptotic normality of OLS in linear simultaneous equation systems with trend in some exogenous variables, extending the results of Krämer (1981, 1984) to more general types of trend. When con- sistent, OLS is also shown to have the same limiting distribution as any k-class estimator with a stochastically bounded k, and to produce a consistent estimate of the error variance in the equation.     

Profile Hellinger distance estimation

The successful application of the Hellinger distance approach to fully parametric models is well known. The corresponding optimal estimators, known as minimum Hellinger distance (MHD) estimators, are efficient and have excellent robustness properties [Beran R. Minimum Hellinger distance estimators for parametric models. Ann Statist. 1977;5:445–463]. This combination of efficiency and robustness makes MHD estimators appealing in practice. However, their application to semiparametric statistical models, which have a nuisance parameter (typically of infinite dimension), has not been fully studied. In this paper, we investigate a methodology to extend the MHD approach to general semiparametric models. We introduce the profile Hellinger distance and use it to construct a minimum profile Hellinger distance estimator of the finite-dimensional parameter of interest. This approach is analogous in some sense to the profile likelihood approach. We investigate the asymptotic properties such as the asymptotic normality, efficiency, and adaptivity of the proposed estimator. We also investigate its robustness properties. We present its small-sample properties using a Monte Carlo study.     

A thurstone-type model for paired comparisons with unequal numbers of repetitions

In paired comparison experiments t objects are ranked for any particular characteristic x by offering the objects in all possible pairs to a judge, each pair being repeated a certain number of times. The judge is to express his preference by giving a score 1 to the preferred object and a score 0 to the non-preferred object. A modification of Thurstone Model for analysis of data from such experiments has been given by Mosteller ‘1951a,b,c’. In this paper angular transformation is used to generalize Mosteller1s model in order to make the preference proportions independent and incidentally ensure homo-scedastlcity of variances and correlations and additivlty of scale in the subjective continuum for the stimuli." The model is extended to unequal numbers of repetitions of the pairs. Using the model two different types of treatment ratings are obtained along with the respective standard errors for moderately large numbers of repetitions, one setting the location parameter S, "0 and the other using the constraint S1 + S2 +…,+ St = 0.     

Statistical inference on asymptotic properties of two estimators for the partially linear single-index models

The outer product of gradients (OPG) estimation procedure based on least squares (LS) approach has been presented by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] to estimate the single-index parameter in partially linear single-index models (PLSIM). However, its asymptotic property has not been established yet and the efficiency of LS-based method can be significantly affected by outliers and heavy-tailed distributions. In this paper, we firstly derive the asymptotic property of OPG estimator developed by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] in theory, and a novel robust estimation procedure combining the ideas of OPG and local rank (LR) inference is further developed for PLSIM along with its theoretical property. Then, we theoretically derive the asymptotic relative efficiency (ARE) of the proposed LR-based procedure with respect to LS-based method, which is shown to possess an expression that is closely related to that of the signed-rank Wilcoxon test in comparison with the t-test. Moreover, we demonstrate that the new proposed estimator has a great efficiency gain across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error. Even in the worst case scenarios, the ARE owns a lower bound equalling to 0.864 for estimating the single-index parameter and a lower bound being 0.8896 for estimating the nonparametric function respectively, versus the LS-based estimators. Finally, some Monte Carlo simulations and a real data analysis are conducted to illustrate the finite sample performance of the estimators.     

On testing exponentiality against new better than used of specified age

R.M. Hollander, D.H. Park and F. Proschan [A class of life distributions for aging, J. Amer. Statist. Assoc. 81 (1986) 91–95] introduced the concept of the larger class of life distributions called new better than used of specified age. In practice, one might be interested in the new better than used behaviour at an unknown but estimable age t₀. Here, we investigate the testing of new better than used of specified age t₀ (NBU-t₀) alternatives. A class of test statistics for testing NBU-t₀ (t₀ is known) based on a U-statistic whose kernel depends on sub-sample minima is proposed. A member of the class of tests proposed by N. Ebrahimi and M. Habbibullah [Testing whether the survival distribution is new better than used of specified age, Biometrika 77 (1990) 212–215] for this problem belongs to the class of tests proposed here. The distributional properties of the class of test statistics are studied. The performances of a few members of the proposed class of tests are studied in terms of Pitman asymptotic relative efficiency. The Pitman ARE values show that the members of the class perform well in comparison with the N. Ebrahimi and M. Habbibullah [Testing whether the survival distribution is new better than used of specified age, Biometrika 77 (1990) 212–215] tests. The proposed class of tests is shown to be consistent for NBU-t₀ alternatives.     

Estimators based on ranks for arma models

In this paper we introduce a new family of robust estimators for ARMA models. These estimators are defined by replacing the residual sample autocovariances in the least squares equations by autocovariances based on ranks. The asymptotic normality of the proposed estimators is provided. The efficiency and robustness properties of these estimators are studied. An adequate choice of the score functions gives estimators which have high efficiency under normality and robustness in the presence of outliers. The score functions can also be chosen so that the resulting estimators are asymptotically as efficient as the maximum likelihood estimators for a given distribution.