Pitman Asymptotic Relative Efficiency Research Articles

A new non-parametric hypothesis testing with reliability analysis applications to model some real data

Due to the essential necessities of current technology and contemporary sciences, a methodical approach to investigating every occurrence or action occurring in the world was required. As a result, it makes sense that such an approach would be required in the investigation of the reliability of technological systems and products. In actuality, there are instances when system parts gradually deteriorate over time t, at which point the system needs to be updated by replacing the spare parts. Although renewal tries to improve system usability, it is unable to restore the system to its prior state. We discovered that an entirely novel domain of reliability known as classes of life distributions has emerged as a result of the growth of criteria for determining the success or failure of dependability over the previous few decades. Moreover, the other nonparametric tests of life distribution lack test efficiency and have low test power. On the other hand, based on that approach, the non-parametric statistical test utilized in this study evaluates different types of treatment by examining failure behavior in the survival data that was collected. As a result, in this ivestigation, we have developed a novel test of life distribution in this study that considers the power and effectiveness of the test. In order to compare data belonging to the harmonic new better than used in expectation (HNBUE) class with data that follow an exponential distribution, this article suggests nonparametric tests for both censored and uncensored data. The higher percentile points of the test statistics are collected, and the suggested test's normality is verified. The Pitman asymptotic relative efficiency (PARE) and powers of proposed test are calculated a few well-known alternative asymmetric models. Three asymmetric real-world data sets are examined to demonstrate the paper's findings.

Asymptotic Relative Efficiency of the Kendall and Spearman Correlation Statistics

A necessary and sufficient condition for Pitman's asymptotic relative efficiency of the Kendall and Spearman correlation statistics for the independence test to be $1$ is given, in terms of certain smoothness and nondegeneracy properties of the model. Corresponding easy-to-use and broadly applicable sufficient conditions are obtained. These conditions hold for most well-known models of dependence.

Two Sample Statistical Test for Location Parameters

A class of distribution-free tests for the homogeneity of location parameters is proposed and compared with different competitors in terms of Pitman asymptotic relative efficiency. A numerical example is provided and a simulation study is made to check the performance of the tests.

Robust testing procedures for scale differences in paired data

Several statistical tests have been proposed in the literature to assess the difference between two scale parameters in paired data. If the location parameters of each sample are different, many of these test statistics are adjusted using the estimated location parameters. Then, the performance of the test statistics depends on the selection of the location estimators. In this study, the asymptotically distribution-free test based on the U-statistic, which does not depend on two location parameters, is constructed. The asymptotic normality and the asymptotic power of the proposed test are examined. Furthermore, the proposed test is compared with the existing tests in terms of Pitman's asymptotic relative efficiency. To improve the Type I error rate in the small sample size, the robust test procedure based on the percentile bootstrap is also proposed. Our simulation studies show that the Type I error rate of the proposed test is more controlled and robust than existing tests in many situations. Further, the empirical power of the proposed test is shown to be high for heavy-tailed or asymmetric distributions.

A moment inequality for decreasing mean time to failure distributions with hypothesis testing application

In this paper, we have derived a moment inequality for the Decreasing (Increasing) Mean Time to Failure (D(I)MTTF) life distributions. A new nonparametric test for testing exponentiality against DMTTF (IMTTF) alternatives is developed based on this moment inequality. Asymptotic property of the test statistic is derived under the null and alternative hypothesis. A comparative study has been carried out in terms of Pitman Asymptotic Relative Efficiency (PARE). The performance of the test is investigated by means of a simulation study. Utilizing the inverse probability censoring weighted scheme (IPCW) approach, we have also extended our test procedure to the case of right-censored data. Finally, application to real-life data sets has been carried out.

A goodness of fit test for mean time to failure function in age replacement

Mean time to failure (MTTF) function plays an important role in connection with age replacement policy. In this paper, a goodness of fit test is proposed for MTTF function utilizing a weighted integral as a measure of deviation. The test statistic is shown to be asymptotically normal and the consistency of the test is also established. The proposed test is compared with some parametric tests of exponentiality in the sense of Pitman Asymptotic Relative Efficiency (PARE). The performance of the test is assessed by means of a simulation study. Finally, the test is applied to a real-life data set for illustrative purposes and the results are shown to be in accordance with those obtained by applying the one-sample Kolmogorov-Smirnov test.

Testing of Location Parameters Against Restricted Alternatives with Optimal Choice of Weights

Testing of equality of populations is one of the fundamental problems in Science. In this paper, we propose a new class of nonparametric tests for testing of location parameters of the distribution of populations against restricted alternatives. The asymptotic distribution of the proposed test statistics is derived. In order to attain maximum efficiency of the proposed tests, a general optimal choice of weights of the proposed class of test statistics is established. Pitman asymptotic relative efficiency and simulation study is carried out of the proposed class of tests. An illustrative example is provided to see the execution of the proposed test.

On Tests for Special Two-Sample Location Problem Based on Subsample Minimum and Median

Shetty and Umarani(2005) considered a special type of two-sample location problem which has some potential applications, particularly in comparing the performances of two packing machines. In this paper, a new class of test statistics is proposed which is based on sub-sample minima of first sample and sub-sample median of second sample. The performance of members of the new class is evaluated in terms of Pitman asymptotic relative efficiency in comparision with Shetty and Umarani(2005). It can be concluded that the performance of the new proposed class of tests is better for medium and heavy tailed distributions such as Normal, Logistic, Laplace and Cauchy. For light tailed distributions the performance of new testis better in specific cases. It is proved that the test is consistent for one-sided alternatives.

An equivalent formula for Pitman’s asymptotic relative efficiency under weakened assumptions

The present article is concerned with proving a well-known and very useful equivalent formula for the Pitman asymptotic relative efficiency A.R.E between rank tests. Furthermore, we also weaken the assumptions needed on the score generating function by using techniques arising from functional analysis.

Nonparametric multiple sample scale testing using U-statistics

Testing of equality of scale parameters is one of the fundamental problems in Statistics and is useful in many fields such as agriculture, engineering, experimental methods and dose level testing. In this paper, we propose a new class of nonparametric tests, to test whether the scale parameters of all the populations are equal, or there is an umbrella ordering in the scale parameters. The proposed test is based on weighted linear combination of two-sample U-statistics. The optimal choice of weights is determined so that proposed tests have maximum efficiency. The comparison of proposed test with respect to existing tests is made in terms of Pitman asymptotic relative efficiency. A numerical example is provided for illustrative purpose. Power of the proposed test is also assessed using Monte Carlo simulation study.

A new test for exponentiality against HNBUE alternatives

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.

Asymptotic Relative Efficiencies of the Score and Robust Tests in Genetic Association Studies

Introduction:The score statisticZ(θ)and the maximin efficient robust test statisticZMERTare commonly used in genetic association study, but according to our knowledge there is no formal comparison of them.Methods:In this report, we compare the asymptotic behavior ofZ(θ)andZMERT, by computing their Asymptotic Relative Efficiencies (AREs) relative to each other. Four commonly used ARE measures, the Pitman ARE, Chernoff ARE, Hodges-Lehmann ARE and the Bahadur ARE are considered. Some modifications of these methods are made to simplify the computations. We found that the Chernoff, Hodges-Lehmann and Bahadur AREs are suitable for our setting.Results and Conclusion:Based on our study, the efficiencies of the two test statistic varies for different criterion used, and for different parameter values under the same criterion, so each test has its advantages and dis-advantages according to the criterion used and the parameters involved, which are described in the context. Numerical examples are given to illustrate the use of the two statistics in genetic association study.

A General Class of Tests for Testing Homogeneity of Location Parameters against Umbrella Alternatives

In this paper, a general class of non-parametric tests for testing homogeneity of location parameter against umbrella alternatives is proposed. Testing for umbrella alternatives has many applications in the field of biology, medicine, botany, dose level testing, engineering, economics, psychology, zoology. As an example, the effectiveness of a drug is likely to increase with increase of dose up to a certain level and then its effect begins to decrease. The proposed test is based on linear combination of two-sample U-statistics. The null distribution of the test statistics is developed. We compare the test with some other competing tests in terms of Pitman asymptotic relative efficiency. To see execution of the test, a numerical example is provided. Simulation study is carried out to assess the power of proposed class of tests.

Jonckheere Type Test Procedure with Optimal Criterion Under Order Restrictions

SYNOPTIC ABSTRACTIn many practical applications, there is a need for testing of an hypothesis under order restrictions. We propose a general class of tests which generalizes the tests proposed by Jonckheere (1954), Kumar et al. (1994a, b), and Patil (2007). The asymptotic distribution of test statistic under null hypothesis, as well as under an equally spaced sequence of local alternatives, is derived. Pitman asymptotic relative efficiency comparisons with respect to competing tests are made to evaluate the performance of the proposed class of tests. An example is provided in Hundal (1969) data for implementing the proposed test procedure. Furthermore, a simulation study is carried out to estimate the power and level of significance of the proposed class of tests.

A Class of Nonparametric Tests for the Two-Sample Location Problem

The two-sample location problem is one of the fundamental problems encountered in Statistics. In many applications of Statistics, two-sample problems arise in such a way as to lead naturally to the formulations of the null hypothesis to the effect that the two samples come from identical populations. A class of nonparametric test statistics is proposed for two-sample location problem based on U-statistic with the kernel depending on a constant ’a’ when the underlying distribution is symmetric. The optimal choice of ’a’ for different underlying distributions is determined. An alternative expression for the class of test statistics is established. Pitman asymptotic relative efficiencies indicate that the proposed class of test statistics does well in comparison with many of the test statistics available in the literature. The small sample performance is also studied through Monte-Carlo Simulation technique

A CLASS OF DISTRIBUTION-FREE TESTS FOR INDEPENDENCE AGAINST POSITIVE QUADRANT DEPENDENCE

A class of distribution-free tests based on convex combination of two U-statistics is considered for testing independence against positive quadrant dependence. The class of tests proposed by Kochar and Gupta (1987) and Kendall’s test are members of the proposed class. The performance of the proposed class is evaluated in terms of Pitman asymptotic relative efficiency for Block- Basu (1974) model and Woodworth family of distributions. It has been observed that some members of the class perform better than the existing tests in the literature. Unbiasedness and consistency of the proposed class of tests have been established.

Testing Exponentiality Against UBAC Using Kernel Methods

In this paper, the problem of testing exponentiality against used better than aged in convex ordering classes of life distributions is investigated. For this property a nonparametric test is presented based on kernel method. The test is presented for complete and right censored data. Furthermore, Pitman's asymptotic relative efficiency (PARE) is discussed to assess the performance of the test with respect to other tests. Selected critical values are tabulated. Some numerical simulations on the power estimates are presented for proposed test. Finally, examples in medical sciences are used as practical applications for the proposed test.

On the Gini Mean Difference Test for Circular Data

We introduce a new test of isotropy or uniformity on the circle, based on the Gini mean difference of the sample arc-lengths and obtain both the exact and asymptotic distributionsunder the null hypothesis of circular uniformity. We also provide a table of upper percentile values of the exact distribution for small to moderate sample sizes. Illustrative examples in circular data analysis are also given. It is shown that a “generalized” Gini mean difference test has better asymptotic efficiency than a corresponding “generalized” Rao's test in the sense of Pitman asymptotic relative efficiency.

A class of nonparametric tests for testing homogeneity of variances against ordered alternatives based on subsample extrema

A class of nonparametric test statistics for several sample scale problem is proposed by considering extrema of subsamples of size c. The proposed class of statistics has the advantage of not requiring the several distribution functions to have a common median, but rather any common quantile of order α, 0 ≤ α ≤1, (not necessarily 1/2) which is assumed to be known. The proposed class of tests is compared with other known tests already existing in the literature in the Pitman asymptotic relative efficiency sense for different distributions. Some results of Monte Carlo power study are also presented.

New nonparametric tests for testing homogeneity of scale parameters against umbrella alternative

Three nonparametric tests for several sample scale problem against umbrella alternative, when the peak of the umbrella is known, are proposed. The proposed statistics have the advantage of not requiring the several distribution functions to have a common median, but rather any common quantile of order α,0≤α≤1, (not necessarily 1/2) which is assumed to be known. The proposed tests are compared with each other in the Pitman asymptotic relative efficiency sense for different distributions.