Vasant P. Bhapkar

Conditioning on ancillary statistics and loss of information in the presence of nuisance parameters

Abstract The results by Godambe (1980, 1984) and Liang (1983) concerning ancillarity of statistic T for parameter of interest θ1 in the presence of parameter θ2, and the loss of information in using the conditional distribution for inference concerning θ1, are extended to the general multiparametric case of θ1 and θ2. Also, relationships are examined concerning the appropriate information matrices.

Loss of information in the presence of nuisance parameters and partial sufficiency

Abstract Loss of information concerning parameter of interest θ in the absence of any knowledge of nuisance parameters is considered when statistic S = S ( X ) is used in place of X , especially when S has marginal distribution depending only on parameter θ. Some definitions of partial sufficiency of S for θ are investigated for possible information loss with respect to Fisher information matrix generalizations.

Distribution of Q When Testing Equality of Matched Proportions

Abstract The asymptotic distribution of Cochrans Q statistic is obtained under the null hypothesis H 0 for comparing proportions in matched samples under the multinomial model. Simulation studies show that a Satterthwaithe-type approximation provides a better approximation to the limiting distribution than does the chi-square distribution. Also, if by the nature of the problem, H 0 implies the side condition H, which is necessary for Q to have a limiting chi-square distribution, then Q has the same Pitman efficiency as a Wald statistic that is known to have some optimal asymptotic properties and that has the χ2 distribution in the limit under H 0 whether or not H holds.

ON FISHER INFORMATION INEQUALITIES IN THE PRESENCE OF NUISANCE PARAMETERS

The existence of a generalized Fisher information matrix for a vector parameter of interest is established for the case where nuisance parameters are present under general conditions. A matrix inequality is established for the information in an estimating function for the vector parameter of interest. It is shown that this inequality leads to a sharper lower bound for the variance matrix of unbiased estimators, for any set of functionally independent functions of parameters of interest, than the lower bound provided by the Cramér-Rao inequality in terms of the full parameter.

Multiple comparisons of matched proportions

It is well known (see, e.g., Scheffe (1959)) that if confidence intervals are desired for several treatment comparisons of interest, especially after a preliminary test of significance, then the appropriate technique is to consider simultaneous confidence intervals with a certain joint confidence coefficient. Goodman (1964) derived such simultaneous confidence intervals for contrasts among several multinomial populations, each with the same number, say J, of classes. The special case involving simultaneous confidence intervals for contrasts among several binomial populations on the basis of independent samples follows simply by taking J=2. This paper now deals with the problem of construction of simultaneous confidence intervals among probabilities of ‘success’ on the basis of matched samples.

Completeness and optimality of marginal likelihood estimating equations

A counter-example shows that the proof of optimality of the marginal likelihood estimating function for parameter of interest, under the conditions assumed in Lloyd (1987), contains a gap and is, thus, invalid. The same comment applies to the generalized version of Lloyd’s Theorem given by Bhapkar and Srinivasan (1993). In the light of known results concerning Fisher information for parameter of interest and partial sufficiency of a suitable statistic, the counter-example reveals a similar gap in the proof of corollary 3.2 of Bhapkar (1991).

On nonparametric profile analysis of several multivariate samples

Abstract A unified development is offered for asymptotically distribution-free profile analysis of several multivariate samples. This includes as special cases procedures based on generalized U-statistics and also those based on linear rank statistics. Furthermore, it includes as special cases analysis of location profiles and also scalar profiles. Finally, asymptotic power and consistency properties are discussed for tests of hypotheses and subhypotheses of interest.

On the equivalence of some test criteria based on BAN estimators for the multivariate exponential family

Abstract For the general multivariate exponential family of distributions it is shown that Raos test criterion based on efficient scores is algebraically identical to the general chi-squared criterion based on maximum likelihood estimates and, similarly, that the Wald statistic is algebraically identical to the general minimum modified chi-squared statistic using linearization; these results are valid also for the multisample versions. Thus, these are extensions to the general exponential family of the findings due to Silvey (1970) and Bhapkar (1966), respectively, for the special case of the multinomial family. It is also shown that the general forms of the chi-squared and modified chi-squared criteria reduce to their respective well-known forms for the multivariate symmetric power series distribution. This finding is, thus, an extension of results noted by Ferguson (1958) and Clickner (1976) for the special case of the multinomial distribution.

NONPARAMETRIC TESTS FOR SCALAR PROFILE ANALYSIS OF SEVERAL MULTIVARIATE SAMPLES

SummaryIn an earlier paper [3] a nonparametric analogue of the hypothesis of parallelism of population profiles was formulated for several multivariate populations and asymptotically distribution-free tests were proposed. This formulation was primarily focussed on thelocation characteristics of the populations. We now extend the concept to consider the parallelism of populationscalar profiles. A class of asymptotically distribution-free tests based on generalizedU-statistics is studied as regards consistency and asymptotic power from scalar point of view.

On the optimality of estimators based on P-sufficient statistics

For estimation of functions involving only parameters of interest, in the presence of nuisance parameters, some optimality properties are established for partially sufficient (i.e. p-sufficient) statistics in two classes of regular probability models. The results are based on a characterization of regular unbiased estimating functions for parameters of interest in probability models for which a statistic exists such that its marginal distribution depends on unknown parameters only through the parameters of interest.