Pitman Asymptotic Efficiency Research Articles

Statistical Insights: Analyzing Shock Models, Reliability Operations and Testing Exponentiality for NBRUmgf Class of Life Distributions

This study deals with a novel category of life distributions called the new better than renewal used in moment generating function (NBRUmgf) class. It delves into the connections between this form of aging and established aging categories, along with its relevance in a shock model. Additionally, it explores how this aging concept remains consistent through certain reliability operations which is consider a very important tool in reliability engineering. The research computes Pitman's asymptotic efficiencies for this testing procedure, comparing them with alternative methods. Furthermore, the study furnishes a comprehensive table of percentiles for the test statistic associated with this proposed method. Furthermore, in order to showcase the relevance of the study's conclusions, we utilize distinct real-world datasets, which illustrate the effectiveness of our test across various types of real data.

A New Class of Life Distributions Based on Moment Inequalities

In this paper, a new class of life distribution is named new better than renewal used in moment generating function (NBRUmgf). A new test for exponentiality versus NBRUmgf based on moment inequalities is established. Pitman's asymptotic efficiencies, powers and critical values of the new test are calculated to assess the performance of the test. The right censored data is handled also. Finally some applications are applied to the new test.

A note on Asymptotical Efficiency of the Goodness of Fit Tests Based on disjoint k-spacings Statistic

In this paper Pitman's asymptotic efficiencies (AE) as well as Kallenberg's intermediate AE of the goodness-of-fit tests based on higher-order non-overlapping spacings is considered. We study log statistic as well as entropy type statistic based on k-spacings when k may tend to infinity as n approaches infinity. It certainly compliments the available results for fixed k and provides more general result. We show that both types of statistics based on higher ordered spacings have higher efficiencies in Pitman's sense compared to their counterparts based on simple spacings. It is also shown that the Kallenberg's intermediate AE of such test coincides with its Pitman's AE, the power of the tests are also discussed.

Moments Inequalities for NBRUL Distributions with Hypotheses Testing Applications

In this paper, moment inequalities for the new better than renwal used in Laplace transform order ( NBRUL ) class of ageing distributions are derived. This inequalities demonstrate that if the mean life is finite, then all higher order moments exist. A new test for exponentiality versus NBRUL can be constructed using thes inequalities. Pitman's asymptotic efficiencies and critical values of the proposed test are calculated and tabulated. The powers of this test are estimated for some famously alternatives distributions in reliability such as Linear failure rate,Weibull and gamma distributions. Finally, examples in different areas are used as a practical applications of the proposed test.

On Testing Exponentiality Against UBA Class of Life Distributions Based On Laplace Transform

The problem of testing various classes of life distributions have been considered in the literature during the last decades. In this paper, we consider a new test statistic for testing exponentiality against used better than age (UBA) class of life distributions based on Laplace transform. This proposed test is presented for complete and right censored data. Furthermore, Pitman asymptotic efficiencies are calculated to assess the performance of our test. Selected critical values are tabulated. Some numerical simulations of power estimates are calculated. Proposed test is presented also in multivariate form. Finally, the test is applied to some real data.

Comparison of relative density of two random geometric digraph families in testing spatial clustering

We compare the performance of relative densities of two parameterized random geometric digraph families called proximity catch digraphs (PCDs) in testing bivariate spatial patterns. These PCD families are proportional edge (PE) and central similarity (CS) PCDs and are defined with proximity regions based on relative positions of data points from two classes. The relative densities of these PCDs were previously used as statistics for testing segregation and association patterns against complete spatial randomness. The relative density of a digraph, D, with n vertices (i.e., with order n) represents the ratio of the number of arcs in D to the number of arcs in the complete symmetric digraph of the same order. When scaled properly, the relative density of a PCD is a U-statistic; hence, it has asymptotic normality by the standard central limit theory of U-statistics. The PE- and CS-PCDs are defined with an expansion parameter that determines the size or measure of the associated proximity regions. In this article, we extend the distribution of the relative density of CS-PCDs for expansion parameter being larger than one, and compare finite sample performance of the tests by Monte Carlo simulations and asymptotic performance by Pitman asymptotic efficiency. We find the optimal expansion parameters of the PCDs for testing each alternative in finite samples and in the limit as the sample size tending to infinity. As a result of our comparisons, we demonstrate that in terms of empirical power (i.e., for finite samples) relative density of CS-PCD has better performance (which occurs for expansion parameter values larger than one) for the segregation alternative, while relative density of PE-PCD has better performance for the association alternative. The methods are also illustrated in a real-life data set from plant ecology.

TESTING EXPONENTIALITY AGAINST HNBUET CLASS BASED ON GOODNESS OF FITAPPROACH

Based on goodness of fit approach a new test for testing exponentiality against harmonic new better than used in expectation upper tail (HNBUET) is proposed. The percentiles of this test are calculated and tabulated for sample size . For some commonly used life distributions in reliability such as linear failure rate, Makeham and Weibull distributions, the Pitman asymptotic efficiency (PAE) and the power of proposed test are calculated. Finally the proposed test is applied to two sets of real data.

On Greenwood goodness-of-fit test

Abstracts The Greenwood test is based on the sum of squares of disjoint spacings. For the goodness-of-fit problem within the class of symmetric tests, the Greenwood test is known to be optimal in terms of the Pitman asymptotic efficiency (AE), whereas it is much inferior to the log-spacings test in terms of the Bahadur AE. In this paper we extend these properties of Greenwood test showing that it remains optimal in terms of the weak intermediate and intermediate AE, but it is much inferior to those tests satisfying the Cramer condition in terms of the strong intermediate AE.

Testing for NBUL using goodness of fit approach with applications

Most of the approaches suggested during the last decades for solving life testing problems are markedly different from those used in the related but wider area of goodness of fit problems. In this paper, it is demonstrate that applying the goodness of fit approach makes sense also for solving life testing problems and that result in simpler procedures that are asymptotically equivalent or better than standard ones. They may have superior finite sample behavior. A U-statistic is obtained for testing exponentiality against the new better than used in the Laplace transform order aging class (NBUL). The pitman asymptotic efficiency of (NBUL) is studied. The power and the critical values of the proposed statistic are calculated. Real examples are given to elucidate the use of the proposed test statistic in the reliability analysis. Finally, the problem with the right censored data is also handled with some applications.

On testing exponentiality against NBARFR life distribution

This paper considers testing exponentiality against new better than average renewal failure rate (NBARFR) alternatives. The percentiles of this test statistic are tabulated for simple sizes 5(1)50. Pitman's asymptotic efficiencies relative to the tests of the new better than used (NBU), new better than used failure rate (NBUFR) and new better than used renewal failure rate (NBURFR) (Amhmed, 1994; Hendi et al., 2000 and Mahmoud and Abdul Alim, 2002). The powers of this test are also calculated for some used life distributions. The problem when right-censored data is available is handled. Pratical applications of our tests in the medical sciences are present.

A new family of random graphs for testing spatial segregation

AbstractThe authors discuss a graph‐based approach for testing spatial point patterns. This approach falls under the category of data‐random graphs, which have been introduced and used for statistical pattern recognition in recent years. The authors address specifically the problem of testing complete spatial randomness against spatial patterns of segregation or association between two or more classes of points on the plane. To this end, they use a particular type of parameterized random digraph called a proximity catch digraph (PCD) which is based on relative positions of the data points from various classes. The statistic employed is the relative density of the PCD, which is a U‐statistic when scaled properly. The authors derive the limiting distribution of the relative density, using the standard asymptotic theory of U‐statistics. They evaluate the finite‐sample performance of their test statistic by Monte Carlo simulations and assess its asymptotic performance via Pitman's asymptotic efficiency, thereby yielding the optimal parameters for testing. They further stress that their methodology remains valid for data in higher dimensions.

A weighted multivariate sign test for cluster-correlated data

We consider the multivariate location problem with cluster-correlated data. A family of multivariate weighted sign tests is introduced for which observations from different clusters can receive different weights. Under weak assumptions, the test statistic is asymptotically distributed as a chi-squared random variable as the number of clusters goes to infinity. The asymptotic distribution of the test statistic is also given for a local alternative model under multivariate normality. Optimal weights maximizing Pitman asymptotic efficiency are provided. These weights depend on the cluster sizes and on the intracluster correlation. Several approaches for estimating these weights are presented. Using Pitman asymptotic efficiency, we show that appropriate weighting can increase substantially the efficiency compared to a test that gives the same weight to each cluster. A multivariate weighted t-test is also introduced. The finite-sample performance of the weighted sign test is explored through a simulation study which shows that the proposed approach is very competitive. A real data example illustrates the practical application of the methodology.

Testing renewal new better than used life distributions based on U-test

New test-statistic for testing exponentiality against renewal new better than used (RNBU) class of life distribution based on U-statistic is studied. Selected critical values are tabulated for sample size n = 5(5)50. The Pitman asymptotic efficiency (PAE) of the test is calculated and compared with, the (PAE) of the test for new renewal better than used (NRBU) class of life distribution [see M.A.W. Mahmoud, S.M. EL-arishy, L.S. Diab, A non-parametric test of new renewal better than used class of life distributions, in: Proceedings of the International Conference on Mathematics Trends and Developments, Cairo, Egypt, vol. 4, 2002, pp. 191–203]. The power of the test is estimated by simulation at 0.05 significant level. A real example is given to elucidate the use of the proposed test. The problem in case of right-censored data is also handled.

Multivariate Nonparametric Tests

Multivariate nonparametric statistical tests of hypotheses are described for the one-sample location problem, the several-sample location problem and the problem of testing independence between pairs of vectors. These methods are based on affine-invariant spatial sign and spatial rank vectors. They provide affine-invariant multivariate generalizations of the univariate sign test, signed-rank test, Wilcoxon rank sum test, Kruskal–Wallis test, and the Kendall and Spearman correlation tests. While the emphasis is on tests of hypotheses, certain references to associated affine-equivariant estimators are included. Pitman asymptotic efficiencies demonstrate the excellent performance of these methods, particularly in heavy-tailed population settings. Moreover, these methods are easy to compute for data in common dimensions.

Tests for the total time on test transform order

In this paper we consider two test statistics for testing the strict TTT transform order between two life distributions of interest. We give their asymptotic distributions and compare our tests with some other related tests in terms of Pitman's asymptotic efficiency. Also we present some results to show the performance and the asymptotic normality of our tests.

An affine‐invariant multivariate sign test for cluster correlated data

AbstractThe author presents a multivariate location model for cluster correlated observations. He proposes an affine‐invariant multivariate sign statistic for testing the value of the location parameter. His statistic is an adaptation of that proposed by Randles (2000). The author shows, under very mild conditions, that his test statistic is asymptotically distributed as a chi‐squared random variable under the null hypothesis. In particular, the test can be used for skewed populations. In the context of a general multivariate normal model, the author obtains values of his test's Pitman asymptotic efficiency relative to another test based on the overall average. He shows that there is an improvement in the relative performance of the new test as soon as intra‐cluster correlation is present Even in the univariate case, the new test can be very competitive for Gaussian data. Furthermore, the statistic is easy to compute, even for large dimensional data. The author shows through simulations that his test performs well compared to the average‐based test. He illustrates its use with real data.

On testing exponentiality against IFRA alternatives

In a recent paper, Ahmad [J. Statist. Plann. Inference 92 (2001) 121] proposes a family of tests for new better than used, new better than used in expectation and harmonic new better than used in expectation. His family of tests is based on some moment inequalities of these classes. In this paper we present a family of tests to test exponentiality against increasing failure rate average (IFRA) alternatives. Our proposed family is based on a moment inequality of the IFRA class. The results of a Pitman asymptotic relatives efficiency comparison indicate the superiority of our tests.

２×２表の条件付検定： 工藤・垂水検定の漸近検出力

Kudô and Tarumi (1978) proposed a conditional test for testing equivalence of two proportions in 2 × 2 tables by assuming the date collected by a negative binomial sampling. The test is undoubtedly a counterpart of the Fisher exact test, but no comparison has been made of the two tests. In this paper we obtain asymptotic power of the Kudô-Tarumi test and compare it with the Fisher exact test by means of the Pitman asymptotic efficiency. It will be shown that the Kudô-Tarumi test is the most powerful similar test for a risk ratio, whereas the Fisher exact test is, the most powerful similar test for a risk ratio, whereas the Fisher exact test is, as well known, the most powerful similar test for an odds ratio. It is also shown that the Kudô-Tarumi test has higher efficiency than the Fisher exact test in many cases.

SUB-SAMPLE MANN–WHITNEY–WILCOXON TEST

ABSTRACT This paper presents a class of sub-sample rank-sum statistics to test the stochastic ordering between two distributions. The proposed class includes as special case the min–max test of Öztürk.[1] It is shown that the asymptotic distribution of the test statistic is normal and its Pitman asymptotic efficiency is as good as or higher than the competitors in the class of sub-sample Mann–Whitney–Wilcoxon statistics.

A goodness of fit approach to testing mean residual times

The random residual lifetime is a random variable representing the remaining life of a system of age x⩾0. The mean residual life at age x, is the expected value of the remaining life beyond that age which is well known and extensively studied for both testing and estimation. In this article, a new approach is presented for testing constant versus decreasing residual life. The new test is much simpler to compute, asymptotically normal and performs better than previous tests in terms of power and Pitman asymptotic efficiencies for several alternatives.