Showing papers by "Pranab Kumar Sen published in 1981"

The Cox Regression Model, Invariance Principles for Some Induced Quantile Processes and Some Repeated Significance Tests

Abstract: For the Cox regression model, the partial likelihood functions involve linear combinations of induced order statistics. Some invariance principles pertaining to such linear combinations of induced order statistics are studied and the theory is incorporated in the formulation of some repeated significance tests (for the hypothesis of no regression) based on these partial likelihoods.

On permutational central limit theorems for general multivariate linear rank statistics

Abstract: In nonparametric multivariate analysis, permutational central limit theorems (PCLT) play a vital role. In the context of multivariate multi sample rank order tests, Puri and Sen (1966, Theorem 4.1) considered a PCLT and this was later (Puri and Sen, 1969, Theorem 3.2) extended to general linear models. In either case, the proof provided by these authors is not correct (as will be explained in Section 2); nevertheless, the theorems remain valid under the stringent regularity conditions stated there. These theorems are multivariate generalizations of the classical (univariate) PCLT, which in its most general form is due to H?jek (1961). He was able to in corporate a powerful (quadratic mean) equivalence result for linear rank statistics and linear combinations of independent random variables (r.v.) which provide the desired result through the classical CLT. Puri and Sen (1971, Theorem 5.4.1) adapted this technique in the multivariate case; but, their result on the PCLT does not properly follow from the H?jek (1961) result, though the conclusion on the unconditional null distribution remains true. Conclusion on the PCLT in the multivariate case based on the moment convergence property (cf. Wald and Wolfowitz, 1944) is, of course, valid, but demands comparatively stringent regularity conditions. All these call for a re-examination of multivariate PCLT.

Some invariance principles for mized rank statistics and induced order statistics and some application

Abstract: The structural affinity of mixed rank statistics and linear combinations of functions of concomitants of order statistics (or induced order statistics) is examined here. Some weal as well as strong invariance principles for these statistics are studied. A variety of models (depend on the nature of stochastic dependence of the two variates) is considered and the regularity conditions are tailored for these diverse situations. Some possible applications of these results in some problems of sequential (statistical) inference are also considered.

Repeated significance tests based on likelihood ratio statistics

Abstract: For a specified decision rule, a general class of likelihood ratio based repeated significance tests is considered. An invariance principle for the likelihood ratio statistics is established and incorporated in the study of the asymptotic theory of the proposed tests. For comparing these tests with the conventional likelihood ratio tests, based solely on the target sample size, some Bahadur efficiency results are presented. The theory is then adapted in the study of some multiple comparison procedures

On asymptotic non-normal null distributions for locally most powerful rank test statistics

Abstract: For probability densities not necessarily possessing finite Fisher information, an extended class of locally most powerful rank tests ‘LMPRT’ is considered. This class of LMPRT is partitioned into three disjoint subsets, characterized respectively by ‘i’ being asymptotically most powerful for «Pitman-type sequences of’ local ‘contiguous’ alternatives «AMPLA» and having asymptotically a normal null distribution ‘ANND» of the test statistic, ‘ii’ not necessarily being AMPLA, but having the ANND property, and ‘iii’ not necessarily being AMPLA and having asymptotically a non-normal null distribution of the test statistic. The current study focuses on the existence of the asymptotic null distributions of all these LMPRT statistics. In addition, an alternative distribution is suggested for inferences based on the LMPRT statistics of case ‘iii’

Estimation of p{x > y} based on progressively truncated versions of the wilcqxon-mann-whitney statistics

Abstract: Stochastic processes related to some generalized U-statistics (with especial emphasis on the Wilcoxon-Mann-Whitney (WMW-) statistic), under progressive right censoring, are considered for prediction purposes Their weak convergence results are incorporated in the study of the asymptotic properties of the predictors The form of the estimable parameter as a function of the truncation point is studied for the WMW case for some typical distributions and the theoretical results are supplemented by simulated ones

An Invariance Principle for Progressively Truncated Likelihood Ratio Statistics.

Abstract: In the context of time-sequential tests based on likelihood ratio statistics, weak convergence (in the Skorokhod as well as the sup-norm metric) of progressively truncated likelihood ratio statistics (to appropriate Wiener processes) is established through the construction of a continuous time-parameter martingale process. The case of local (contiguous) alternative hypotheses is also treated. The results are extended to the multiparameter case as well.

On Invariance Principles for LMP Conditional test Statistics

Abstract: For a set of random variables, not necessarily independent, along with a characterization of locally most powerful (LMP) conditional tests, some weak as well as strong invariancc principles for tbe allied test statistics arc considered. In the light of asymptotic sufficiency of sub-sigma fields (generating tbe conditional tests). the closeness of LMP and LMP conditional test statistics is examined with reference to the invariancc principles for these statistics. these results include, as special cases, parallel results on various LMP rank test statistica. scattered in the literature.