Nonparametric Methods for Checking the Validity of Prior Order Information

A large number of statistical procedures have been proposed in the literature to explicitly utilize available information about the ordering of treatment effects at increasing treatment levels. These procedures are generally more efficient than those ignoring the order information. However, when the assumed order information is incorrect, order restricted procedures are inferior and, strictly speaking, invalid. Just as any statistical model needs to be validated by data, order information to be used in a statistical analysis should also be justified by data first. A common statistical format for checking the validity of order information is to test the null hypothesis of the ordering representing the order information. Parametric tests for ordered null hypotheses have been extensively studied in the literature. These tests are not suitable for data with nonnormal or unknown underlying distributions. The objective of this study is to develop a general distribution-free testing theory for ordered null hypotheses based on rank order statistics and score generating functions. Sufficient and necessary conditions for the consistency of the proposed general tests are rigorously established.     

Tests of Parameters of Several Gamma Distributions with Inequality Restrictions

Gamma distribution is one of the most used methods of modeling lifetime data. However, testing homogeneity of parameters of m 3 gamma distributions against order restrictions is almost non-existent in the current literature. We propose two methods to this end: one uses quadratic forms involving ratios of cumulants as test statistic and the other is a stepwise procedure which uses Fisher's method of combining p-values when shape parameters are equal but unknown. Both procedures allow use of arbitrary sample sizes of m populations. Test of the inequality restrictions as a null hypothesis against unrestricted alternatives is also considered. A Monte Carlo study of power at various alternatives shows that both methods are competitive when they are applicable.     

Inequalities concerning the expected selection differentials

In many situations two populations are compared on the basis of subsets of the available data. If this is done using the same fraction of best records, then the expectations of the arithmetic means of these fractions are strictly ordered in magnitude by the ordering of the sample sizes. The results are illustrated with the special cases of the uniform and negative exponential distributions, for which further inequalities are derived.     

多项分布总体的相等对增凸序问题的多样本检验

概率分布间随机序在实践中已经得到了广泛的应用,而且似然比检验是用以检验涉及随机序问题的最普遍的检验方法．但是,关于多个多项式总体间的增凸序约束的统计推断问题并没有得到充分发展．多样本的增凸序对无约束的检验问题已被研究．然而,多总体的相等性对增凸序的假设检验问题似乎更有研究意义．并且分布的相等对随机序的假设检验问题往往是统计学家最为普遍地考虑．对多样本的情况,本文考虑了分布的相等对增凸序的假设检验问题,并且获得似然比检验统计量的零渐近分布,它是一组x~2分布随机变量的加权和,即■~2分布．     

On approximation of the level probabilities for testing ordered parallel regression lines

Approximations for the level probabilities in testing order-restricted hypotheses are examined in this paper for generalized linear models with common slope. It is showed, in particular, that the asymptotic null distribution of the likelihood ratio statistic is a mixture of chi-squared distributions for the cases of simple order and simple tree order. Under a balanced structure, the asymptotic null distribution reduces to the well-known chi-bar-squared distribution. The use of the equal-weights level probabilities is also investigated. This approximation seems to be satisfactory when the sample sizes and the levels of the covariate are not too different among the strata.     

Nonparametric Bayesian Modeling for Stochastic Order

In comparing two populations, sometimes a model incorporating stochastic order is desired. Customarily, such modeling is done parametrically. The objective of this paper is to formulate nonparametric (possibly semiparametric) stochastic order specifications providing richer, more flexible modeling. We adopt a fully Bayesian approach using Dirichlet process mixing. An attractive feature of the Bayesian approach is that full inference is available regarding the population distributions. Prior information can conveniently be incorporated. Also, prior stochastic order is preserved to the posterior analysis. Apart from the two sample setting, the approach handles the matched pairs problem, the k-sample slippage problem, ordered ANOVA and ordered regression models. We illustrate by comparing two rather small samples, one of diabetic men, the other of diabetic women. Measurements are of androstenedione levels. Males are anticipated to produce levels which will tend to be higher than those of females.     

Zur Konstruktion von Hahn-Ternärkörpern als Ternärkörper formaler Potenzreihen

We endow the set of all formal power series with coefficients in a ternary field and exponents in a totally ordered loop with a ternary field operation such that a uniform valuation is given by the natural ultrametric distance. Any ordering of the coefficient ternary field can be extended to an ordering of the Hahn ternary field which is compatible with the given valuation.     

Likelihood ratio tests of correlated multivariate samples

We develop methods to compare multiple multivariate normally distributed samples which may be correlated. The methods are new in the context that no assumption is made about the correlations among the samples. Three types of null hypotheses are considered: equality of mean vectors, homogeneity of covariance matrices, and equality of both mean vectors and covariance matrices. We demonstrate that the likelihood ratio test statistics have finite-sample distributions that are functions of two independent Wishart variables and dependent on the covariance matrix of the combined multiple populations. Asymptotic calculations show that the likelihood ratio test statistics converge in distribution to central Chi-squared distributions under the null hypotheses regardless of how the populations are correlated. Following these theoretical findings, we propose a resampling procedure for the implementation of the likelihood ratio tests in which no restrictive assumption is imposed on the structures of the covariance matrices. The empirical size and power of the test procedure are investigated for various sample sizes via simulations. Two examples are provided for illustration. The results show good performance of the methods in terms of test validity and power.     

The new test criterion for the homogeneity of parameters of several populations

Summary The new test criterion for testing the homogeneity of parameters of several populations is proposed and the test properties of it is discussed. The asymptotic expansions of the distributions of test criterion are discussed under (i) null hypothesis, (ii) fixed alternative hypothesis and (iii) local alternative hypothesis converging to the null hypothesis with appropriate rate of convergence as the sample size increases. As a particular case the asymptotic theory of a statistic for a homogeneity of variances of normal populations is also discussed and the exact moments of it under a null hypothesis can be used to obtain a percentage point by a Pearsonian curve fitting. This Institute of statistical Mathematics     

On the testing of marginal homogeneity with a one-sided alternative in the analysis of variance

We consider a two-factor experiment in which the factors have the same levels with a natural ordering among the levels. Likelihood ratio tests for testing equality of the main effects with a one-sided alternative and for testing the one-sided hypothesis as a null hypothesis are studied. Closed form expressions for the maximum likelihood estimates under the various hypotheses are obtained. The null hypothesis distributions for these test statistics are derived.The efforts of the first author were supported by the NSERC of Canada. The efforts of the second author were supported by the Office of Naval Research under Contract ONR N00014-80-C-0321. The efforts of the third author were supported by the Office of Naval Research under Contract ONR N00014-80-C-0322.     

On Some Tests of the Covariance Matrix Under General Conditions

We consider the problem of testing the hypothesis about the covariance matrix of random vectors under the assumptions that the underlying distributions are nonnormal and the sample size is moderate. The asymptotic expansions of the null distributions are obtained up to n ^−1/2. It is found that in most cases the null statistics are distributed as a mixture of independent chi-square random variables with degree of freedom one (up to n ^−1/2) and the coefficients of the mixtures are functions of the fourth cumulants of the original random variables. We also provide a general method to approximate such distributions based on a normalization transformation.     

有供应方限制的临时价格折扣与最优存贮模型

供应方提供临时价格折扣，一般都附带特别定货量限制。购买方可在所提供折扣价格条件下从有限种订货量中选择最小的订货量，但不一定是最优订货量。本文提出有供应方限制的临时价格折扣条件下的最优存贮模型和订货策略。并给出应用实例。     

A bartlett-type adjustment for the likelihood ratio statistic with an ordered alternative

For testing the equality of normal variances with an increasing alternative, under the null hypothesis the likelihood ratio test statistic is asymptotically distributed as a mixture of chi-squared distributions. In this paper a Bartlett-type adjustment is proposed to improve the approximation of the null distribution of the likelihood ratio test statistic with an ordered alternative.     

A Preference Order Dynamic Program for a Knapsack Problem with Stochastic Rewards

In this paper, we extend the knapsack problem to include more realistic situations by treating the rewards (or values) associated with each item included in the solution as random variables with distributions that are known (or may be estimated) rather than known integers, as in the usual formulation. We propose a dynamic programming solution methodology where the usual real-valued return function is replaced by a preference ordering on the distributions of returns from the items selected. In addition to extending previous solutions to the knapsack problem, we demonstrate the selection of a preference ordering criterion and illustrate the conditions required of the ordering to guarantee optimality of the procedure. A sample problem is shown to demonstrate the algorithm, and results of computational experience with 459 problems of varying sizes and parameters are presented.     

Multivariate process capability via Löwner ordering

Probability bounds can be derived for distributions whose covariance matrices are ordered with respect to Löwner partial ordering, a relation that is based on whether the difference between two matrices is positive definite. One example is Anderson’s Theorem. This paper develops a probability bound that follows from Anderson’s Theorem that is useful in the assessment of multivariate process capability. A statistical hypothesis test is also derived that allows one to test the null hypothesis that a given process is capable versus the alternative hypothesis that it is not capable on the basis of a sample of observed quality characteristic vectors from the process. It is argued that the proposed methodology is viable outside the multivariate normal model, where the p-value for the test can be computed using the bootstrap. The methods are demonstrated using example data, and the performance of the bootstrap approach is studied empirically using computer simulations.     

Testing with Exponentially Tilted Empirical Likelihood

Imposing restrictions without assuming underlying distributions to modelize complex realities is a valuable methodological tool. However, if a subset of restrictions were not correctly specified, the usual test-statistics for correctly specified models tend to reject erronously a simple null hypothesis. In this setting, we may say that the model suffers from misspecification. We study the behavior of empirical phi-divergence test-statistics, introduced in Balakrishnan et al. Statistics 49:951–977 (2015), by using the exponential tilted empirical likelihood estimators of Schennach Ann Stat 35:634–672 (2007), as a good compromise between the efficiency of the significance level for small sample sizes and the robustness under misspecification.     

Optimal training sample allocation and asymptotic expansions for error rates in discriminant analysis

The sample-based rule obtained from Bayes classification rule by replacing the unknown parameters by ML estimates from a stratified training sample is used for the classification of a random observationX into one ofL populations. The asymptotic expansions in terms of the inverses of the training sample sizes for cross-validation, apparent and plug-in error rates are found. These are used to compare estimation methods of the error rate for a wide range of regular distributions as probability models for considered populations. The optimal training sample allocation minimizing the asymptotic expected error regret is found in the cases of widely applicable, positively skewed distributions (Rayleigh and Maxwell distributions). These probability models for populations are often met in ecology and biology. The results indicate that equal training sample sizes for each populations sometimes are not optimal, even when prior probabilities of populations are equal.     

Restricted one way analysis of variance using the empirical likelihood ratio test

Empirical likelihood (EL) ratio tests are developed for testing for or against the hypothesis that k-population means μ₁,μ₂,…,μ_k are isotonic with respect to some quasi-order ? on {1,2,…,k}. The null asymptotic distributions are derived and are shown to be of chi-bar squared type. The asymptotic power of the proposed test for testing for equality of these means against the order restriction is derived under contiguous alternatives and a simulation study is carried out to investigate the finite sample behaviors of this test. In addition, an adjusted EL test is used to improve the small size performance of our test and an example is also discussed to illustrate the theoretical results.     

Sufficient conditions for stochastic equality of two distributions under some partial orders

We establish some conditions for stochastic equality of two nonnegative random variables which are ordered with respect to variability ordering or with respect to mean residual life ordering or with respect to second order stochastic ordering.     

Selecting the Best Population Using a Test for Equality Based on Minimal Wilcoxon Rank-sum Precedence Statistic

In this paper, we first give an overview of the precedence-type test procedures. Then we propose a nonparametric test based on early failures for the equality of two life-time distributions against two alternatives concerning the best population. This procedure utilizes the minimal Wilcoxon rank-sum precedence statistic (Ng and Balakrishnan, 2002, 2004) which can determine the difference between populations based on early (100q%) failures. Hence, this procedure can be useful in life-testing experiments in biological as well as industrial settings. After proposing the test procedure, we derive the exact null distribution of the test statistic in the two-sample case with equal or unequal sample sizes. We also present the exact probability of correct selection under the Lehmann alternative. Then, we generalize the test procedure to the k-sample situation. Critical values for some sample sizes are presented. Next, we examine the performance of this test procedure under a location-shift alternative through Monte Carlo simulations. Two examples are presented to illustrate our test procedure with selecting the best population as an objective.