Improved estimators for the location of double exponential distribution

In this paper, attention is focused on estimation of the location parameter in the double exponential case using a weighted linear combination of the sample median and pairs of order statistics, with symmetric distance to both sides from the sample median. Minimizing with respect to weights and distances we get smaller asymptotic variance in the second order. If the number of pairs is taken as infinite and the distances as null we attain the least asymptotic variance in this class of estimators. The Pitman estimator is also noted. Similarly improved estimators are scanned over their probability of concentration to investigate its bound. Numerical comparison of the estimators is shown.     

Classification rules for two parameter exponential populations under order restrictions on parameters

In this paper, we suggest classification procedures of an observation into one of two exponential populations assuming a known ordering between population parameters. We propose classification rules when either location or scale parameters are ordered. Some of these classification rules under ordering are better than usual classification rules with respect to the expected probability of correct classification. We also derive likelihood ratio-based classification rules. Comparison of these classification rules has been done using Monte Carlo simulations.     

Robust and nonparametric subset selection procedures

Subset selection procedures based on ranks have been investigated by a number of authors previously. Their methods are based on ranking the samples from all the populations jointly. However, as was pointed out by Rizvi and Woodworth (1970), the procedures they proposed cannot control the probability of a correct selection over the entire parameter space. In this paper, we propose a subset selection procedure based on pairwise rather than joint ranking of the samples. It is shown that this procedure controls the probability of a correct selection over the entire parameter space. It is also shown that the Pitman efficiency of this nonparametric procedure relative to the multivariate t procedure of Gupta (1956, 1965) is the same as the Pitman efficiency of the Mann-Whitney-Wilcoxon test relative to the t-test.     

Stochastic equivalence of ranking methods

The asymptotic behavior of linear rank statistics for comparing the locations of two populations, where the observations are ranked jointly with other populations, is considered. Under certain conditions, the asymptotic behavior of these statistics does not depend on which other populations are included in the ranking. In particular, the difference of a pair of these statistics, with the same score function, but based on two different rankings, converges to zero in probability under Pitman alternatives and Chernoff-Savage conditions on the scores and underlying distributions.     

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.     

A CLASS OF NONPARAMETRIC TESTS FOR THE ONE-SAMPLE LOCATION PROBLEM

A class of “optimal”U-statistics type nonparametric test statistics is proposed for the one-sample location problem by considering a kernel depending on a constant a and all possible (distinct) subsamples of size two from a sample of n independent and identically distributed observations. The “optimal” choice of a is determined by the underlying distribution. The proposed class includes the Sign and the modified Wilcoxon signed-rank statistics as special cases. It is shown that any “optimal” member of the class performs better in terms of Pitman efficiency relative to the Sign and Wilcoxon-signed rank statistics. The effect of deviation of chosen a from the “optimal” a on Pitman efficiency is also examined. A Hodges-Lehmann type point estimator of the location parameter corresponding to the proposed “optimal” test-statistics is also defined and studied in this paper.     

TESTS FOR DETECTING MORE NBU-NESS AT A SPECIFIC AGE

This paper proposes a class of non‐parametric test procedures for testing the null hypothesis that two distributions, F and G, are equal versus the alternative hypothesis that F is ‘more NBU (new better than used) at specified age t₀’ than G. Using Hoeffding's two‐sample U‐statistic theorem, it establishes the asymptotic normality of the test statistics and produces a class of asymptotically distribution‐free tests. Pitman asymptotic efficacies of the proposed tests are calculated with respect to the location and shape parameters. A numerical example is provided for illustrative purposes.     

Relative efficiencies of goodness-of-fit procedures with truncated data

Efficiency properties of the Kolmogorov-Smirnov, Kuiper, Cramér-von Mises, and Anderson-Darling statistics with truncated data are investigated. For these statistics, the approximate slopes are determined, and the equivalence of ratios of limiting approximate slopes to limiting Pitman efficiencies is established. Efficiencies for certain parametric families of distributions incorporating both location and scale alternatives are evaluated as the amount of truncation increases.     

Bounds on efficiencies for some two-sample nonparametric statistics

Bounds on the Pitman efficiency for two-sample scale and loca-tion statistics are presented along with densities for which these bounds are sharp.     

One-and two-sample sign tests for ranked set sample selective designs

Statistical inference based on a ranked set sample depends very much on the location of the quantified observations. A selective design which determines the location of the quantified observations in a ranked set sample is introduced. The paper investigates the effects of selective designs on one and two sample sign test statistics. The Pitman efficiencies of one- and two sample sign tests are calculated for selective designs and compared with ranked set samples of the same size. If the design quantifies observations at the center points, then the proposed procedure is superior to a ranked set sample of the same size in the sense of Pitman efficiency. Some practical problems are addressed for the two-sample sign test.     

Repeated significance tests based on likelihood ratio statistics

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     

Non-parametric tests for several sample location problem based on subsample extrema

In this paper non-parametric tests for homogeneity of several populations against locationtype alternatives are proposed. For this all possible subsamples of fixed size are drawn from each sample and their maxima and minima are computed One class of tests is obtained using these subsample minima whereas other class of tests involves use of sub sample maxima. Tests belonging t o these two classes have been compared with many of the presently available tests in terms of their Pitman asymptotic relative efficiency . Some of the members of these proposed classes of tests prove to robust in terms of efficiency.     

A class of k-sample distribution-free tests for location against ordered alternatives

This paper introduces a new class of distribution-free tests for testing the homogeneity of several location parameters against ordered alternatives. The proposed class of test statistics is based on a linear combination of two-sample U-statistics based on subsample extremes. The mean and variance of the test statistic are obtained under the null hypothesis as well as under the sequence of local alternatives. The optimal weights are also determined. It is shown via Pitman ARE comparisons that the proposed class of test statistics performs better than its competitor tests in case of heavy-tailed and long-tailed distributions     

Sequential and two-stage procedures for selecting the better exponential population covering the case of unknown and unequal scale parameters

The problems of selecting the larger location parameter of two exponential distributions are discussed. When the scale parameters are the same but unknown, we consider the procedure of Desu et al. (1977) in detail, and study some of its exact and asymptotic properties. We indicate how this procedure can be modified along the lines of Mukhopadhyay (1979, 1980) to achieve first-order asymptotic efficiency. We then propose a sequential procedure for this set-up and show that it is asymptotically second-order efficient according to Ghosh and Mukhopadhyay (1981). In case the scale parameters are completely unknown and unequal, we propose a two-stage procedure that guarantees the probability of correct selection to exceed the prescribed nominal level in the preference zone. We do not need any new tables to implement this particular procedure other than those in Krishnaiah and Armitage (1964), Gupta and Sobel (1962), Guttman and Milton (1969). We also propose a sequential method in this case and derive some of its asymptotic properties.     

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.     

On rules based on sample medians for selection of the largest location parameter

The problem of selection of a subset containing the largest of several location parameters is considered, and a Gupta-type selection rule based on sample medians is investigated for normal and double exponential populations. Numerical comparisons between rules based on medians and means of small samples are made for normal and contaminated normal populations, assuming the popula-tion means to be equally spaced. It appears that the rule based on sample means loses its superiority over the rule based on sample medians in case the samples are heavily contaminated. The asymptotic relative efficiency (ARE) of the medians procedure relative to the means procedure is also computed, assuming the normal means to be in a slippage configuration. The means proce-dure is found to be superior to the median procedure in the sense of ARE. As in the small sample case, the situation is reversed if the normal populations are highly contaminate.     

Estimators of percentiles based on absolute loss

Estimators of percentiles of location and scale parameter distributions are optimized based on Pitman closeness and absolute risk. A median unbiased (MU) estimator and a minimum risk (MR) estimator are shown to exist within a class of estimators, and to depend upon the medians of two completely specified distributions.     

A Generalization of Ahmad's Class of Mann–Whitney–Wilcoxon Statistics

In the spirit of the recent work of Ahmad (1996) this paper introduces another class of Mann–Whitney–Wilcoxon test statistics. The test statistic compares the r th and s th powers of the tail probabilities of the underlying probability distributions. The choice of r + s = 4 improves the Pitman efficiency for uniform, exponential, lognormal and normal distributions and keeps the same efficiency as the Mann–Whitney–Wilcoxon test for logistic and double exponential distributions. The two-sample test is modified for the one-sample problem with symmetric underlying distribution.     

SELECTION PROCEDURES FOR SCALE PARAMETERS USING TWO-SAMPLE U-STATISTICS

Let be k independent populations having the same known quantile of order p (0 p 1) and let F(x)=F(x/_i) be the absolutely continuous cumulative distribution function of the ith population indexed by the scale parameter ₁, i = 1,…, k. We propose subset selection procedures based on two-sample U-statistics for selecting a subset of k populations containing the one associated with the smallest scale parameter. These procedures are compared with the subset selection procedures based on two-sample linear rank statistics given by Gill & Mehta (1989) in the sense of Pitman asymptotic relative efficiency, with interesting results.     

Test for independence between time to failure and cause of failure in competing risks with k causes

In this paper, we develop a simple nonparametric test for testing the independence of time to failure and cause of failure in competing risks set up. We generalise the test to the situation where failure data is right censored. We obtain the asymptotic distribution of the test statistics for complete and censored data. The efficiency loss due to censoring is studied using Pitman efficiency. The performance of the proposed test is evaluated through simulations. Finally we illustrate our test procedure using three real data sets.