Probability plots and order statistics of the standard extreme value distribution

A comparison between the ordinary least-squares estimator and the weighted least-squares estimator when the data set arises from the standard extreme value distribution is provided. Probability plot of the extreme value distribution is applied. A goodness-of-fit test of the standard extreme value distribution is introduced. The percentage points of the test statistic are investigated. The results of power study for the test statistic under various alternatives show that in most situations the proposed test statistic serves as well as do competing alternatives.     

RAPTT: An Exact Two-Sample Test in High Dimensions Using Random Projections

In high dimensions, the classical Hotelling’s T² test tends to have low power or becomes undefined due to singularity of the sample covariance matrix. In this article, this problem is overcome by projecting the data matrix onto lower dimensional subspaces through multiplication by random matrices. We propose RAPTT (RAndom Projection T²-Test), an exact test for equality of means of two normal populations based on projected lower dimensional data. RAPTT does not require any constraints on the dimension of the data or the sample size. A simulation study indicates that in high dimensions the power of this test is often greater than that of competing tests. The advantages of RAPTT are illustrated on a high-dimensional gene expression dataset involving the discrimination of tumor and normal colon tissues.     

A Partial Empirical Likelihood Based Score Test Under a Semiparametric Finite Mixture Model

We propose a score statistic to test the null hypothesis that the two-component density functions are equal under a semiparametric finite mixture model. The proposed score test is based on a partial empirical likelihood function under an I-sample semiparametric model. The proposed score statistic has an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to a partial empirical likelihood ratio test and a Wald test. We present some results on a simulation study.     

Permutations of the Walsh system that preserve Lebesgue constants

A class of permutations of the Walsh-Paley system that preserve the lebesgue constants and theL ^p -norms of the Dirichlet kernel is distinguished. Thus it is proved, in particular, that Fine’s estimates and calculations of the Lebesgue constants for the Walsh—Paley system hold for the Walsh systems in the enumerations of Walsh and Kaczmarz. A third algorithm for calculating the Lebesgue constants, which is different from those obtained by Fine and which also makes it possible to calculate theL ^p -norms of the Dirichlet kernels, is proposed. It is shown that not all permutations of the Walsh system even within the blocks preserve the Lebesgue constants. The distinguished class of permutations includes theTW-systems of Schipp, which are not, in general, permutations within the blocks. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 36–48, July, 2000.     

Restricted Symmetric Permutations

Pattern-avoiding involutions, which have received much enumerative attention, are pattern-avoiding permutations which are invariant under the natural action of a certain subgroup of D ₈, the symmetry group of a square. Three other nontrivial subgroups of D ₈ also have invariant permutations under this action. For each of these subgroups, we enumerate the set of permutations which are invariant under the action of the subgroup and which also avoid a given set of forbidden patterns. The sets of forbidden patterns we consider include all subsets of S ₃. For each subgroup we also give a bijection between the invariant permutations and certain symmetric signed permutations. Received September 14, 2006     

VC-Dimension of Sets of Permutations

VC-dimension of a set of permutations to be the maximal k such that there exist distinct that appear in A in all possible linear orders, that is, every linear order of is equivalent to the standard order of for at least one permutation . In other words, the VC-dimension of A is the maximal k such that for some the restriction of A to contains all possible linear orders. This is analogous to the VC-dimension of a set of strings. Our main result is that there exists a universal constant C such that any set of permutations with VC-dimension 2 is of size . This is analogous to Sauer's lemma for the case of VC-dimension 2. One corollary of our main result is that any acyclic set of linear orders of is of size , (a set A of linear orders on is called acyclic if no 3 elements appear in A in all 3 orders (i,j,k), (k,i,j) and (j,k,i)). The size of the largest acyclic set of linear orders has interested researchers for many years because it is the largest number of linear orders of n alternatives such that the following is always satisfied: if each one of a set of voters chooses one of these orders as his preference then the majority relation between each two alternatives is transitive. Received August 6, 1998     

On summing permutations and some statistical properties

A closed form expression is obtained for the sum of all permutations of n objects taken r at a time. The average and variance of the permutations are derived and are shown to be proportional to the average and variance of the objects themselves. The proportionality constant is a function of only r, n and the base b and is independent of the actual objects considered. Previous results aimed at determining the sum of permutations are shown to be very specific cases of the current development.     

Testing exponentiality against a class of alternatives which includes the RNBUE distributions based on the empirical laplace transform

A scale-free test for exponentiality is proposed which is consistent within an extended set of models, including, but not limited to, the “renewal new better than used in expectation” (RNBUE) class of life distributions. The limiting null distribution of the test statistics is derived, and the approximate local Bahadur efficiency is calculated for several families of alternatives. Finite-sample properties of the proposed procedures are investigated via simulation. Bibliography: 24 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2006, pp. 63–77.     

Classifying efficient alternatives in SMAA using cross confidence factors

Stochastic multicriteria acceptability analysis (SMAA) is a family of methods for aiding multicriteria group decision making. These methods are based on exploring the weight space in order to describe the preferences that make each alternative the most preferred one. The main results of the analysis are rank acceptability indices, central weight vectors and confidence factors for different alternatives. The rank acceptability indices describe the variety of different preferences resulting in a certain rank for an alternative; the central weight vectors represent the typical preferences favouring each alternative; and the confidence factors measure whether the criteria data are sufficiently accurate for making an informed decision.In some cases, when the problem involves a large number of efficient alternatives, the analysis may fail to discriminate between them. This situation is revealed by low confidence factors. In this paper we develop cross confidence factors, which are based on computing confidence factors for alternatives using each other’s central weight vectors. The cross confidence factors can be used for classifying efficient alternatives into sets of similar and competing alternatives. These sets are related to the concept of reference sets in Data Envelopment Analysis (DEA), but generalized for stochastic models. Forming these sets is useful when trying to identify one or more most preferred alternatives, or suitable compromise alternatives. The reference sets can also be used for evaluating whether criteria need to be measured more accurately, and at which alternatives the measurements should be focused. This may cause considerable savings in measurement costs. We demonstrate the use of the cross confidence factors and reference sets using a real-life example.     

Empirical likelihood ratio tests for multivariate regression models

This paper proposes some diagnostic tools for checking the adequacy of multivariate regression models including classical regression and time series autoregression. In statistical inference, the empirical likelihood ratio method has been well known to be a powerful tool for constructing test and confidence region. For model checking, however, the naive empirical likelihood (EL) based tests are not of Wilks’ phenomenon. Hence, we make use of bias correction to construct the EL-based score tests and derive a nonparametric version of Wilks’ theorem. Moreover, by the advantages of both the EL and score test method, the EL-based score tests share many desirable features as follows: They are self-scale invariant and can detect the alternatives that converge to the null at rate n ^−1/2, the possibly fastest rate for lack-of-fit testing; they involve weight functions, which provides us with the flexibility to choose scores for improving power performance, especially under directional alternatives. Furthermore, when the alternatives are not directional, we construct asymptotically distribution-free maximin tests for a large class of possible alternatives. A simulation study is carried out and an application for a real dataset is analyzed.      

Challenge Versus Exchange in Collective Decision Making: A Comparison of Two Simulation Models Based on Simulated Data

In collective decision making, actors can use different influence strategies to get their way. Differences in influence strategies may, or may not, be connected to differences in collective outcomes. This research studies two influence strategies: the exchange strategy and the challenge strategy. In the existing literature, these strategies are analyzed and compared using simulation models in which actor behavior regarding influence attempts based on one of the strategies is modeled explicitly. Until now, these models have been tested only empirically on limited data sets. However, a theoretical test is necessary to gain more precise insights in the effect of characteristics of collective decision making situations on the collective outcomes. In the present research, computer simulations are used in a structured comparison of two competing models (the iterative exchange model and challenge model). The analyses show that the outcomes of both models are captured for a large part in the actor characteristics on the issues. Besides this, the expected directions of challenges and exchanges play a major part in explaining the outcomes of the models. This research shows that the use of simulated data allows a structured search of the input space, which led to new insights into the iterative exchange model and challenge model, and therefore in the exchange strategy and the challenge strategy.     

The bias of three pseudo-random shuffles

Three schemes for shuffling a deck ofn cards are studied, each involving a random choice from [n] ⁿ . The shuffles favor some permutations over others sincen! does not dividen ⁿ . The probabilities that the shuffles lead to some simple permutations, for instance cycles left and right and the identity, are calculated. Some inequalities are obtained which lead to information about the least and most likely permutations. Numbers of combinatorial interest occur, notably the Catalan numbers and the Bell numbers.     

A Data Driven Smooth Test for Circular Uniformity

We propose a new omnibus test for uniformity on the circle. The new test is based upon the idea of data driven smooth tests as presented in Ledwina (1994, J. Amer. Statist. Assoc., 89, 1000–1005). Our simulations indicate that the test performs very well for multifarious alternatives. In particular, it seems to outperform other known omnibus tests when testing against multimodal alternatives. We also investigate asymptotic properties of our test and we prove that it is consistent against every departure from uniformity.     

Comparing Algorithms for Sorting with <Emphasis Type="Italic">t</Emphasis> Stacks in Series

We show that the left-greedy algorithm is a better algorithm than the right-greedy algorithm for sorting permutations using t stacks in series when t > 1. We also supply a method for constructing some permutations that can be sorted by t stacks in series and from this get a lower bound on the number of permutations of length n that are sortable by t stacks in series. Finally we show that the left-greedy algorithm is neither optimal nor defines a closed class of permutations for t > 2.AMS Subject Classification: 05A05, 68R05, 68W01.     

Testing for random effects in linear mixed models for longitudinal data under moment conditions

In this paper, we consider whether the random effect exists in linear mixed models （LMMs） when only moment conditions are assumed. Based on the estimators of parameters and their asymptotic properties, a Wald-type test is constructed. It is consistent against global alternatives and is sensitive to the local alternatives converging to the null hypothesis at parametric rates, a fastest possibly rate for goodness-of-fit testing. Moreover, a simulation study shows the performance of the test is good. The procedure also applies to a real data.     

Test for a specified signal when the noise covariance matrix is unknown

In the univariate case it is well known that the one sided t test is uniformly most powerful for the null hypothesis against all one sided alternatives. Such a property does not easily extend to the multivariate case. In this paper, a test derived for the hypothesis that the mean of a vector random variable is zero against specified alternatives, when the covariance matrix is unknown. This test depends on the given alternatives and is more powerful than Hotelling's T². The results are derived both for real and complex vector observations and under normal and spherical distributions. The properties of the proposed tests are investigated in detail when a single alternative is specified.     

Coherent random permutations with biased record statistics

We consider random permutations that are defined coherently for all values of n, and for each n have a probability distribution which is conditionally uniform given the set of upper and lower record values. Our central example is a two-parameter family of random permutations that are conditionally uniform given the counts of upper and lower records. This family may be seen as an interpolation between two versions of Ewens’ distribution. We discuss characterisations of the conditionally uniform permutations, their asymptotic properties, constructions and relations to random compositions.     

Local efficiency of a Cramér-von Mises test of independence

Deheuvels proposed a rank test of independence based on a Cramér-von Mises functional of the empirical copula process. Using a general result on the asymptotic distribution of this process under sequences of contiguous alternatives, the local power curve of Deheuvels’ test is computed in the bivariate case and compared to that of competing procedures based on linear rank statistics. The Gil-Pelaez inversion formula is used to make additional comparisons in terms of a natural extension of Pitman's measure of asymptotic relative efficiency.     

Some t-Homogeneous Sets of Permutations

Perpendicular Arrays are orderedcombinatorial structures, which recently have found applicationsin cryptography. A fundamental construction uses as ingredientscombinatorial designs and uniformly t-homogeneoussets of permutations. We study the latter type of objects. Thesemay also be viewed as generalizations of t-homogeneousgroups of permutations. Several construction techniques are given.Here we concentrate on the optimal case, where the number ofpermutations attains the lower bound. We obtain several new optimalsuch sets of permutations. Each example allows the constructionof infinite families of perpendicular arrays.