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.     

Two-sample Pitman closeness comparison under progressive Type-II censoring

In this paper, we consider two problems concerning two independent progressively Type-II censored samples. We first consider the Pitman closeness (PC) of order statistics from two independent progressively censored samples to a specific population quantile. We then consider the point prediction of a future progressively censored order statistic and discuss the determination of the closest progressively censored order statistic from the current sample according to the simultaneous closeness probabilities. For both these problems, explicit expressions are derived for the pertinent PC probabilities, and then special cases are given as examples. For various censoring schemes, we also present numerical results for the standard uniform, standard exponential, and standard normal distributions. Finally, a distribution-free result for the median is obtained.     

Finding the best treatment under heavy censoring and hidden bias

Summary.  We analyse male survival duration after hospitalization following an acute myocardial infarction with a large ( N =11024) Finnish data set to find the best performing hospital district (and to disseminate its treatment protocol). This is a multiple-treatment problem with 21 treatments (i.e. 21 hospital districts). The task of choosing the best treatment is difficult owing to heavy right censoring (73%), which makes the usual location measures (the mean and median) unidentified; instead, only lower quantiles are identified. There is also a sample selection issue that only those who made it to a hospital alive are observed (54%); this becomes a problem if we wish to know their potential survival duration after hospitalization, if they had survived to a hospital contrary to the fact. The data set is limited in its covariates—only age is available—but includes the distance to the hospital, which plays an interesting role. Given that only age and distance are observed, it is likely that there are unobserved confounders. To account for them, a sensitivity analysis is conducted following pair matching. All estimators employed point to a clear winner and the sensitivity analysis indicates that the finding is fairly robust.     

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.     

Average rank correlation statistics in the presence of ties

The problem of n judges ranking r objects is considered in the situation where ties are permitted. Asymtotic distributions under the null hypothesis of complete randomness in the rankings are derived for the test statistics of average rank correlations between all pairs of ranking where the rank correlations are measured either by Spearman rho or Kendall tau. The relative efficeincies of these average rank correlatins are derived using approximate Bahadur slope and limiting pitman efficiency, and in both cases the Kendall statistic is shown to be more efficient. Some interpretatins of these and related results are also given.     

Two sample nonparametric tests based on subsamples

The paper introduces a general class of nonparametric tests for the two-sample location problem based on subsamples. Includ- ed in this class is the Mann-Whitney (or the Wilcoxon rank sum) test. General formulas for the Pitman efficacy for different methods of subsampling are derived. A small sample power simu- lation compares the performance of members of this class     

On the relation between Hodges-Lehmann efficiency and Pitman efficiency

The relation between Hodges–Lehmann efficiency and Pitman efficiency is studied in the context of testing one-sided hypotheses about a real-valued parameter. It is first shown that for tests based on sums of independently and identically distributed observations local Hodges-Lehmann efficiency is equivalent to Pitman efficiency. Then, it is proved that this equivalence also carries over to tests based on two broad classes of M -estimators for the location problem. In all cases considered explicit formulas of the Hodges-Lehmann efficacies are obtained.     

Pitman closeness,monotonicity and consistency of best linear unbiased and invariant estimators for exponential distribution under Type II censoring

Comparisons of best linear unbiased estimators with some other prominent estimators have been carried out over the last 50 years since the ground breaking work of Lloyd [E.H. Lloyd, Least squares estimation of location and scale parameters using order statistics, Biometrika 39 (1952), pp. 88–95]. These comparisons have been made under many different criteria across different parametric families of distributions. A noteworthy one is by Nagaraja [H.N. Nagaraja, Comparison of estimators and predictors from two-parameter exponential distribution, Sankhyā Ser. B 48 (1986), pp. 10–18], who made a comparison of best linear unbiased (BLUE) and best linear invariant (BLIE) estimators in the case of exponential distribution. In this paper, continuing along the same lines by assuming a Type II right censored sample from a scaled-exponential distribution, we first compare BLUE and BLIE of the exponential mean parameter in terms of Pitman closeness (nearness) criterion. We show that the BLUE is always Pitman closer than the BLIE. Next, we introduce the notions of Pitman monotonicity and Pitman consistency, and then establish that both BLUE and BLIE possess these two properties.     

The Bahadur representation for sample quantiles under strongly mixing sequence

Sun [2006, The Bahadur representation for sample quantiles under weak dependence. Statist. Probab. Lett. 76, 1238-1244] established the Bahadur representation for sample quantiles under the strongly mixing sequence. But there are some problems in the proofs of the main results. In the paper, we further investigate the Bahadur representation for sample quantiles under strongly mixing sequence and get better bound than that in Sun (2006).     

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.     

Large sample properties of nonparametric copula estimators under bivariate censoring

A new nonparametric estimator is proposed for the copula function of a bivariate survival function for data subject to random right-censoring. We consider two censoring models: univariate and copula censoring. We show strong consistency and we obtain an i.i.d. representation for the copula estimator. In a simulation study we compare the new estimator to the one of Gribkova and Lopez [Nonparametric copula estimation under bivariate censoring; doi:10.1111/sjos.12144].     

On a further generalization of the savage test

A statistic based on the frequencies within the k+1 intervals specified by k arbitrary quantiles is proposed for a LMP test against Lehmann alternatives generalizing the Savage test for the two-sample problem. The maximum efficiency relative to the Savage test for optimally chosen k quantiles is also provided for k=l(2)l5. The asymptotic normality of the statistic follows from the asymptotic multinomial distribution of the frequencies in the classes determined by the k quantiles.     

Shrinkage testimators for the shape parameter of weibull distributin under type ii censoring

In this paper we propose some shrinkage testimators for the shape parameter of the Weibull distribution when censored samples are available and study their properties. Comparison of the testimators with Singh and Bhatkulikar (1978) and with the usual estimator, interms of mean squared error are made. It is shown that the proposed testimators     

The bahadur efficiency of paired versus joint ranking procedures for pa1rwtse multiple comparisons

The performance of paired versus joint ranking procedures for pairwise multiple comparisons is investigated using approxiirete Bahadur efficiency, When the populations to be compared are widely separated, or-when the. data arise from a shift model with an underlying unimodal density, the paired ranking procedure is found to be better for comparing two adjacent populations while the joint ranking procedure is more efficient for comparing the two most distant populations     

Comparison of two weibull distributions under random censoring

The two-sample problem for comparing Weibull scale parameters is studied for randomly censored data. Three different test statistics are considered and their asymptotic properties are established under a sequence of local alternatives, It is shown that both the test statistic based on the mlefs (maximum likelihood estimators) and the likelihood ratio test are asymptotically optimum. The third statistic based only on the number of failures is not, Asymptotic relative efficiency of this statistic is obtained and its numerical values are computed for uniform and Weibull censoring, Effects of uniform random censoring on the censoring level of the experiment are illus¬trated, A direct proof for the joint asymptotic normality of the mlefs of the shape and the scale parameters is also given     

On the asymptotic normality of the kernel estimators of the density function and its derivatives under censoring

In this paper, we study asymptotic normality of the kernel estimators of the density function and its derivatives as well as the mode in the randomly right censorship model. The mode estimator is defined as the random variable that maximizes the kernel density estimator. Our results are stated under some suitable conditions upon the kernel function, the smoothing parameter and both distributions functions that appear in this model. Here, the Kaplan–Meier estimator of the distribution function is used to build the estimates. We carry out a simulation study which shows how good the normality works.     

Optimal sample size allocation for multi-level stress testing with Weibull regression under Type-II censoring

We discuss the optimal allocation problem in a multi-level stress test with Type-II censoring and Weibull (extreme value) regression model. We derive the maximum-likelihood estimators and their asymptotic variance–covariance matrix through the Fisher information. Four optimality criteria are used to discuss the optimal allocation problem. Optimal allocation of units, both exactly for small sample sizes and asymptotically for large sample sizes, for two- and four-stress-level situations are determined numerically. Conclusions and discussions are provided based on the numerical studies.     

Optimal sample size allocation for accelerated life test under progressive type-II censoring with competing risks

In this article, we study the optimization problem of sample size allocation when the competing risks data are from a progressive type-II censoring in a constant-stress accelerated life test with multiple levels. The failure times of the individual causes are assumed to be statistically independent and exponentially distributed with different parameters. We obtain the estimates of the unknown parameters through a maximum likelihood method, and also derive the Fisher information matrix. We propose three optimization criteria and two search scenarios to obtain the sample size allocation at each stress level. Some numerical results are studied to illustrate the usage of the proposed methods.     

Some new results on univariate and multivariate permutation tests for ordinal categorical variables under restricted alternatives

In several sciences, especially when dealing with performance evaluation, complex testing problems may arise due in particular to the presence of multidimensional categorical data. In such cases the application of nonparametric methods can represent a reasonable approach. In this paper, we consider the problem of testing whether a “treatment” is stochastically larger than a “control” when univariate and multivariate ordinal categorical data are present. We propose a solution based on the nonparametric combination of dependent permutation tests (Pesarin in Multivariate permutation test with application to biostatistics. Wiley, Chichester, 2001), on variable transformation, and on tests on moments. The solution requires the transformation of categorical response variables into numeric variables and the breaking up of the original problem’s hypotheses into partial sub-hypotheses regarding the moments of the transformed variables. This type of problem is considered to be almost impossible to analyze within likelihood ratio tests, especially in the multivariate case (Wang in J Am Stat Assoc 91:1676–1683, 1996). A comparative simulation study is also presented along with an application example.     

Some results on maximum likelihood estimators of parameters of exponential distribution under type I progressive censoring with changing failure rates

In this paper the study of relative bias (RB), exact variance and mean square error (MSE) of the maximum likelihood estimators of the exponential distribution under type I progressive censoring with changing failure rates is considered. A minimum mean square error (MMSE) estimator for the parameter at each stage is proposed. The numerical evalution of their relative performance is made for selected values of n and p. Further results concerning group-censoring, total expected waiting time and optimal spacings of the times of censoring are derived and results obtained by Kendell and Anderson (1971) are deduced as special cases.