A proportional-reduction-in-impurity measure of association for categorical variables

This paper presents a proportional-reduction-in-impurity (PRI) measure for categorical association, that employs application-dependent loss functions which make the measure widely applicable. The well-known proportional -reduction-in-error (PRE) measure is shown to be a special case of the new PRI measure. Moreover, the asymptotic variance of the maximum likelihood estimator (MLE) of the measure is derived to facilitate its use for statistical inference. An extension of the PRI measure to compositional association is made to show that it can have a variety of applications. Selected loss functions are treated to illustrate the derivation of the measure.     

On weighted measure of inaccuracy for doubly truncated random variables

Recently, authors have studied the weighted version of Kerridgeinaccuracy measure for left/right truncated distributions. In the present communication we introduce the notion of weighted interval inaccuracy measure and study it in the context of two-sided truncated random variables. In reliability theory and survival analysis, this measure may help to study the various characteristics of a system/component when it fails between two time points. We study various properties of this measure, including the effect of monotone transformations, and obtain its upper and lower bounds. It is shown that the proposed measure can uniquely determine the distribution function and characterizations of some important life distributions have been provided. This new measure is a generalization of recent weighted residual (past) inaccuracy measure.     

A comparison between two simple measures of skewness

In 1995, Arnold and Groeneveld introduced the measure of skewness gammaM in terms of F(mode)-the cumulative probability of a random variable less than or equal to the mode of the distribution. They assumed that the mode of a distribution exists and is unique. Independently, in 1996, the present author arrived at the measure of skewness T, which is given in terms of F(mean). This measure possesses desirable properties and is equally simple. The measure gammaM satisfies - 1 gammaM 1 , with 1 (- 1) indicating extreme right (left) skewness. However, the measure T can take on any value on the real line; hence, an equivalent measure gammaT is considered and is compared with gammaM. We consider a variety of families of distributions and include in our study other measures of skewness of interest. Skewness values are easily obtained using MINITAB programs.     

A NEW MEASURE OF ROTATABILITY OF RESPONSE SURFACE DESIGNS

This paper proposes a new measure of rotatability of a response surface design. It is based on a comparison between the design moments of a given design and the design moments of a rotatable design closest to the given design in the least squares sense. This measure is easier to calculate than Khur?s measure. An alternative easier non-geometrical way of deriving Khuri's measure is also presented. Both measures are calculated to illustrate some designs considered by Khuri.     

Study of the quality of several asymptotic and near-exact approximations based on moments for the distribution of the Wilks Lambda statistic

In this paper a measure of proximity of distributions, when moments are known, is proposed. Based on cases where the exact distribution is known, evidence is given that the proposed measure is accurate to evaluate the proximity of quantiles (exact vs. approximated). The measure may be applied to compare asymptotic and near-exact approximations to distributions, in situations where although being known the exact moments, the exact distribution is not known or the expression for its probability density function is not known or too complicated to handle. In this paper the measure is applied to compare newly proposed asymptotic and near-exact approximations to the distribution of the Wilks Lambda statistic when both groups of variables have an odd number of variables. This measure is also applied to the study of several cases of telescopic near-exact approximations to the exact distribution of the Wilks Lambda statistic based on mixtures of generalized near-integer gamma distributions.     

Quantitative convergence assessment for Markov chain Monte Carlo via cusums

Yu (1995) provides a novel convergence diagnostic for Markov chain Monte Carlo (MCMC) which provides a qualitative measure of mixing for Markov chains via a cusum path plot for univariate parameters of interest. The method is based upon the output of a single replication of an MCMC sampler and is therefore widely applicable and simple to use. One criticism of the method is that it is subjective in its interpretation, since it is based upon a graphical comparison of two cusum path plots. In this paper, we develop a quantitative measure of smoothness which we can associate with any given cusum path, and show how we can use this measure to obtain a quantitative measure of mixing. In particular, we derive the large sample distribution of this smoothness measure, so that objective inference is possible. In addition, we show how this quantitative measure may also be used to provide an estimate of the burn-in length for any given sampler. We discuss the utility of this quantitative approach, and highlight a problem which may occur if the chain is able to remain in any one state for some period of time. We provide a more general implementation of the method to overcome the problem in such cases.     

A nonparametric measure of interclass correlation

A nonparametric measure of interclass correlation is considered and its unbiased estimator and a test based on the estimator are studied. Hie measure is an analogue of the Kendall's measure of dependence. It is shown that the variance of the estimator is small and the information loss of the test based on the estimator is not serious relative to a standard parametric test in the sense of the Pitman asymptotic relative efficiency. Furthermore, the approximate variance of the estimator is given in the normal model.     

Power-divergence-type measure of departure from symmetry for square contingency tables that have nominal categories

For square contingency tables that have nominal categories, Tomizawa considered two kinds of measure to represent the degree of departure from symmetry. This paper proposes a generalization of those measures. The proposed measure is expressed by using the average of the power divergence of Cressie and Read, or the average of the diversity index of Patil and Taillie. Special cases of the proposed measure include Tomizawa's measures. The proposed measure would be useful for comparing the degree of departure from symmetry in several tables.     

Local dependence functions for extreme value distributions

Kotz & Nadarajah (2002) introduced a measure of local dependence which is a localized version of the Pearson's correlation coefficient. In this paper we provide detailed analyses (both algebraic and numerical) of the form of the measure for the class of bivariate extreme value distributions. We consider, in particular, five families of bivariate extreme value distributions. We also discuss two applications of the new measure. In the first application we introduce an overall measure of correlation and produce evidence to suggest that it is superior than the usual Pearson's correlation coefficient. The second application introduces two new concepts for ordering of bivariate dependence.     

A bayesian approach to detect informative observations in an experiment

In this article Lindley's (1956) measure of average information is used to measure the loss of information due to the unavailability of a set of observations in an experiment. This measure of loss of information may be used to detect a set of most informative observations in a given design.     

A measure of disclosure risk for microdata

Summary. Protection against disclosure is important for statistical agencies releasing microdata files from sample surveys. Simple measures of disclosure risk can provide useful evidence to support decisions about release. We propose a new measure of disclosure risk: the probability that a unique match between a microdata record and a population unit is correct. We argue that this measure has at least two advantages. First, we suggest that it may be a more realistic measure of risk than two measures that are currently used with census data. Second, we show that consistent inference (in a specified sense) may be made about this measure from sample data without strong modelling assumptions. This is a surprising finding, in its contrast with the properties of the two 'similar' established measures. As a result, this measure has potentially useful applications to sample surveys. In addition to obtaining a simple consistent predictor of the measure, we propose a simple variance estimator and show that it is consistent. We also consider the extension of inference to allow for certain complex sampling schemes. We present a numerical study based on 1991 census data for about 450 000 enumerated individuals in one area of Great Britain. We show that the theoretical results on the properties of the point predictor of the measure of risk and its variance estimator hold to a good approximation for these data.     

The stability of several measures of association in small contingency tables

Measures of association are often used to describe the relationship between row and column variables in two—dimensional contingency tables. It is not uncommon in biomedical research to categorize continuous variables to obtain a two—dimensional table. In these situations it is desirable that the measure of association not be too sensitive to changes in the number of categories or to the choice of cut points. To accomplish this objective we attempt to find a measure of association that closely approximates the corresponding measure of association for the underlying distribution.Measures that are close to the underlying measure for various table sizes andcutpoints are called stable measures.     

Some results on conditional expectations of lower record values

In this paper, we propose a measure for obtaining the expectation of time between two lower k-records under the condition that the greater one is given. Several properties of the proposed measure are derived. Some characterization results and stochastic comparisons based on the new measure are also provided.     

Mixed-spectra analysis for stationary random fields

We consider a, discrete time, weakly stationary bidimensional process, for which the spectral measure is the sum of an absolutely continuous measure, a discrete measure of finite order and a finite number of absolutely continuous measures on several lines. In this paper we are interested in estimating the spectral density of the absolutely continuous measure and of the density on the lines. For this aim, by using the double kernel method, we construct consistent estimators of these densities and we study their asymptotic behaviors in term of the mean squared error with rate.     

A measure of slope-rotatability for second order response surface experimental designs

In the design of experiments for estimating the slope of a response surface, slope-rotatability is a desirable property. In this paper, a measure is introduced that enables us to assess the degree of slope-rotatability for a given response surface design. The measure takes the value zero if and only if the design is slope-rotatable, and becomes larger as the design deviates from a slope-rotatable design. Examples of applying this measure to some response surface designs are also given.     

Utility functions that lead to the likelihood ratio as a relative model performance measure

In this paper, we adopt the point of view of a decision-maker who evaluates a probabilistic model based on the test-sample averaged utility of the expected-utility optimal strategy that the model suggests in a horse race setting. After briefly reviewing the basic properties of the resulting performance measure for probabilistic models, we show that such a measure ranks two models according to their likelihood ratio if and only if the performance measure is based on a generalized logarithmic utility function. Thus, we provide a new decision theoretic motivation of the likelihood ratio as a model performance measure.     

A suggestion for measuring heteroskedasticity

The lack of a generally-accepted measure of the degree of severity of heteroskedasticity is shown to have caused some Monte Carlo studies to draw misleading conclusions. An attractive measure of heteroskedasticity is suggested.     

Renyi's entropy for residual lifetime distribution

In the present paper, we introduce and study Renyi's information measure (entropy) for residual lifetime distributions. It is shown that the proposed measure uniquely determines the distribution. We present characterizations for some lifetime models. Further, we define two new classes of life distributions based on this measure. Various properties of these classes are also given.     

The influence in comparing covariance matrices

The influence measure for the likelihood ratio test for comparing two covariance matrices is derived using the influence curve approach under the normality assumption. The influence measure for testing the equality of covariance matrices against the arbitrariness of them is partitioned into three influence measures: one for testing the equality of covariance matrices against the proportionality of them, another for testing the proportionality against the equality of correlations between them and the other for testing the equality of correlations against the arbitrariness. This partition implies that an observation can be influential in performing some tests among the four tests but not in performing the remaining tests. Thus the partition is more informative than considering the influence measure for the test of equality alone. Each influence measure is useful for detecting outliers in performing the corresponding likelihood ratio test.     

A measure of aliasing and applications to two-level fractional factorials

This paper examines some properties of a measure of aliasing proposed by Hedayat, Raktoe, and Federer (1974). It is shown that in the case of balanced orthogonal designs with no repeated treatments, minimizing the alias measure is equivalent to minimizing tr Yar(ψ). Lower bounds are found for fixed eigenvalues of the design matrix. These results are applied to two-level fractional factorials to show that in certain cases classical fractional-factorial designs yield minimal solutions for the alias measure.