Measuring association between nominal categorical variables: an alternative to the Goodman–Kruskal lambda

As a measure of association between two nominal categorical variables, the lambda coefficient or Goodman–Kruskal's lambda has become a most popular measure. Its popularity is primarily due to its simple and meaningful definition and interpretation in terms of the proportional reduction in error when predicting a random observation's category for one variable given (versus not knowing) its category for the other variable. It is an asymmetric measure, although a symmetric version is available. The lambda coefficient does, however, have a widely recognized limitation: it can equal zero even when there is no independence between the variables and when all other measures take on positive values. In order to mitigate this problem, an alternative lambda coefficient is introduced in this paper as a slight modification of the Goodman–Kruskal lambda. The properties of the new measure are discussed and a symmetric form is introduced. A statistical inference procedure is developed and a numerical example is provided.     

Edgeworth expansions for nonparametric distribution estimation with applications

In this paper, we will investigate the nonparametric estimation of the distribution function F of an absolutely continuous random variable. Two methods are analyzed: the first one based on the empirical distribution function, expressed in terms of i.i.d. lattice random variables and, secondly, the kernel method, which involves nonlattice random vectors dependent on the sample size n; this latter procedure produces a smooth distribution estimator that will be explicitly corrected to reduce the effect of bias or variance. For both methods, the non-Studentized and Studentized statistics are considered as well as their bootstrap counterparts and asymptotic expansions are constructed to approximate their distribution functions via the Edgeworth expansion techniques. On this basis, we will obtain confidence intervals for F(x) and state the coverage error order achieved in each case.     

On a system of nonhomogeneous components sharing a common frailty

The components of a reliability system subjected to a common random environment usually have dependent lifetimes. This paper studies the stochastic properties of such a system with lifetimes of the components following multivariate frailty models and multivariate mixed proportional reversed hazard rate (PRHR) models, respectively. Through doing stochastic comparison, we devote to throwing a new light on how the random environment affects the number of working components of a reliability system and on assessing the performance of a k-out-of-n system.     

On characterizations based on record values and order statistics

Consider an infinite sequence of independent random variables having common continuous c.d.f. F. For 1 ⩽ i ⩽ n, let X_i:n denote the ith order statistic of the first n random variables, and let {X(n), n ⩾ 1} be the sequence of upper record values. We examine the similarities and differences between the dependence structures of the X_i:n's and the X(n)'s, with an emphasis on the latter. We present an interesting situation involving a characterization of F using the moment sequence of records. We obtain characterizations based on the properties of certain regression functions associated with order statistics, record values, and the original observations. We discuss the resemblance between some known and some new characterizations based on order statistics, record values and those based on the properties of truncated F.     

On rates of convergence to infinitely divisible laws for dependent random variables

Large O and small o approximations of the expected value of a class of functions (modified K-functional and Lipschitz class) of the normalized partial sums of dependent random variables by the expectation of the corresponding functions of infinitely divisible random variables have been established. As a special case, we have obtained rates of convergence to the Stable Limit Laws and to the Weak Laws of Large Numbers. The technique used is the conditional version of the operator method of Trotter and the Taylor expansion.     

Structure-preserving mappings of uniform order statistics

There are several well-known mappings which transform the first r common order statistics in a sample of size n from a standard uniform distribution to a full vector of dimension r of order statistics in a sample of size r from a uniform distribution. Continuing the results reported in a previous paper by the authors, it is shown that transformations of these types do not lead to order statistics from an i.i.d. sample of random variables, in general, when being applied to order statistics from non-uniform distributions. By accepting the loss of one dimension, a structure-preserving transformation exists for power function distributions.     

Characterizations of exponentiality within the HNBUE class and related tests

A harmonic new better than used in expectation (HNBUE) variable is a random variable which is dominated by an exponential distribution in the convex stochastic order. We use a recently obtained condition on stochastic equality under convex domination to derive characterizations of the exponential distribution and bounds for HNBUE variables based on the mean values of the order statistics of the variable. We apply the results to generate discrepancy measures to test if a random variable is exponential against the alternative that is HNBUE, but not exponential.     

Characterizations based on regression assumptions of order statistics

Let X₍₁₎≤X₍₂₎≤···≤X_(n) be the order statistics from independent and identically distributed random variables {X_i, 1≤i≤n} with a common absolutely continuous distribution function. In this work, first a new characterization of distributions based on order statistics is presented. Next, we review some conditional expectation properties of order statistics, which can be used to establish some equivalent forms for conditional expectations for sum of random variables based on order statistics. Using these equivalent forms, some known results can be extended immediately.     

Importance of Components for a System

Which component is most important for a system's survival? We answer this question by ranking the information relationship between a system and its components. The mutual information (M) measures dependence between the operational states of the system and a component for a mission time as well as between their life lengths. This measure ranks each component in terms of its expected utility for predicting the system's survival. We explore some relationships between the ordering of importance of components by M and by Zellner's Maximal Data Information (MDIP) criterion. For many systems the bivariate distribution of the component and system lifetimes does not have a density with respect to the two-dimensional Lebesgue measure. For these systems, M is not defined, so we use a modification of a mutual information index to cover such situations. Our results for ordering dependence are general in terms of binary structures, sum of random variables, and order statistics.     

Comparison of experiments for a class of positively dependent random variables

Using Blackwell's definition for comparison of experiments, it is shown that some sets of positively dependent random variables are less informative than similar sets of independent random variables. It is also shown that the information content of symmetric multivariate normal random vectors with a common known variance increases as the common correlation coefficient decreases. Some results which compare members of two-parameter exponential families are also included.     

Two remarks on order statistics

Let X₍₁₎,…,X_(n) be the order statistics of n iid distributed random variables. We prove that (X_(i)) have a certain Markov property for general distributions and secondly that the order statistics have monotone conditional regression dependence. Both properties are well known in the case of continuous distributions.     

Inequalities for the partial sums of elliptical order statistics related to genetic selection

Let X₁,…,X_n be exchangeable normal variables with a common correlation p, and let X₍₁₎ > … > X_(n) denote their order statistics. The random variable σⁿ_i=_n—_k+1x_i, called the selection differential by geneticists, is of particular interest in genetic selection and related areas. In this paper we give results concerning a conjecture of Tong (1982) on the distribution of this random variable as a function of ρ. The same technique used can be applied to yield more general results for linear combinations of order statistics from elliptical distributions.     

Analogues of Khintchine, Marcinkiewicz–Zygmund and Rosenthal Inequalities for Symmetric Statistics

In this paper we prove the analogues of Khintchine, Marcinkiewicz–Zygmund and Rosenthal moment inequalities for symmetric statistics of second order in not identically distributed random variables.     

A new way of quantifying the symmetry of a random variable: Estimation and hypothesis testing

New measures of skewness for real-valued random variables are proposed. The measures are based on a functional representation of real-valued random variables. Specifically, the expected value of the transformed random variable can be used to characterize the distribution of the original variable. Firstly, estimators of the proposed skewness measures are analyzed. Secondly, asymptotic tests for symmetry are developed. The tests are consistent for both discrete and continuous distributions. Bootstrap versions improving the empirical results for moderated and small samples are provided. Some simulations illustrate the performance of the tests in comparison to other methods. The results show that our procedures are competitive and have some practical advantages.     

Sharp bounds for L-statistics from dependent samples of random length

Several distribution-free bounds on expected values of L-statistics based on the sample of possibly dependent and nonidentically distributed random variables are given in the case when the sample size is a random variable, possibly dependent on the observations, with values in the set {1,2,…}. Some bounds extend the results of Papadatos (2001a) to the case of random sample size. The others provide new evaluations even if the sample size is nonrandom. Some applications of the presented bounds are also indicated.     

On blest's measure of rank correlation

Blest (2000) proposed a new nonparametric measure of correlation between two random variables. His coefficient, which is dissymmetric in its arguments, emphasizes discrepancies observed among the first ranks in the orderings induced by the variables. The authors derive the limiting distribution of Blest's index and suggest symmetric variants whose merits as statistics for testing independence are explored using asymptotic relative efficiency calculations and Monte Carlo simulations.     

A modified Cp statistic in a system-of-equations model

A new statistic, SΓ(p), is developed for variable selection in a system-of-equations model. The standardized total mean square error in the SΓ(p)statistic is weighted by the covariance matrix of dependent variables instead of the error covariance matrix of the true model as in the original definition. The new statistic can be also used for model selection in the non-nested models. The estimate of SΓ(p), SC(p), is derived and shown to become SC_ε(p) in the similar form of C_p in a single-equation model when the covariance matrix of sampled dependent variables is replaced by the error covariance matrix under the full model.     

Determination of a new mixed variable lot-size multiple dependent state sampling plan based on the process capability index

This article proposes a new mixed variable lot-size multiple dependent state sampling plan in which the attribute sampling plan can be used in the first stage and the variables multiple dependent state sampling plan based on the process capability index will be used in the second stage for the inspection of measurable quality characteristics. The proposed mixed plan is developed for both symmetric and asymmetric fraction non conforming. The optimal plan parameters can be determined by considering the satisfaction levels of the producer and the consumer simultaneously at an acceptable quality level and a limiting quality level, respectively. The performance of the proposed plan over the mixed single sampling plan based on C_pk and the mixed variable lot size plan based on C_pk with respect to the average sample number is also investigated. Tables are constructed for easy selection of plan parameters for both symmetric and asymmetric fraction non conforming and real world examples are also given for the illustration and practical implementation of the proposed mixed variable lot-size plan.     

Bayesian inference on P(X > Y) in bivariate Rayleigh model

In the literature, assuming independence of random variables X and Y, statistical estimation of the stress–strength parameter R = P(X > Y) is intensively investigated. However, in some real applications, the strength variable X could be highly dependent on the stress variable Y. In this paper, unlike the common practice in the literature, we discuss on estimation of the parameter R where more realistically X and Y are dependent random variables distributed as bivariate Rayleigh model. We derive the Bayes estimates and highest posterior density credible intervals of the parameters using suitable priors on the parameters. Because there are not closed forms for the Bayes estimates, we will use an approximation based on Laplace method and a Markov Chain Monte Carlo technique to obtain the Bayes estimate of R and unknown parameters. Finally, simulation studies are conducted in order to evaluate the performances of the proposed estimators and analysis of two data sets are provided.     

On k-match problems

Several waiting time random variables for a duplication within a memory window of size k in a sequence of {1,2,…,m}-valued random variables are investigated. The exact distributions of the waiting time random variables are derived by the method of conditional probability generating functions. In particular, the exact distribution of the waiting time for the first k-match is obtained when the underlying sequence is generated by higher order Markov dependent trials. Examples for numerical calculations are also given.