Testing the Equality of Two Exponential Distributions

This paper proposes an overlapping-based test statistic for testing the equality of two exponential distributions with different scale and location parameters. The test statistic is defined as the maximum likelihood estimate of the Weitzman's overlapping coefficient, which estimates the agreement of two densities. The proposed test statistic is derived in closed form. Simulated critical points are generated for the proposed test statistic for various sample sizes and significance levels via Monte Carlo Simulations. Statistical powers of the proposed test are computed via simulation studies and compared to those of the existing Log likelihood ratio test.     

Testing the similarity of two normal populations with application to the bioequivalence problem

The problem of testing the similarity of two normal populations is reconsidered, in this article, from a nonclassical point of view. We introduce a test statistic based on the maximum likelihood estimate of Weitzman's overlapping coefficient. Simulated critical points are provided for the proposed test for various sample sizes and significance levels. Statistical powers of the proposed test are computed via simulation studies and compared to those of the existing tests. Furthermore, Type-I error robustness of the proposed and the existing tests are studied via simulation studies when the underlying distributions are non-normal. Two data sets are analyzed for illustration purposes. Finally, the proposed test has been implemented to assess the bioequivalence of two drug formulations.     

A New Test of Randomness for Lehmer Generators Based on the Manhattan Distance Between Pairs of Consecutive Random Numbers

This article considers the Marsaglia effect by proposing a new test of randomness for Lehmer random number generators. Our test is based on the Manhattan distance criterion between consecutive pairs of random numbers rather than the usually adopted Euclidian distance. We derive the theoretical distribution functions for the Manhattan distance for both overlapping (two dimensional) as well as non-overlapping cases. Extensive goodness-of-fit testing as well as empirical experimentation provides ample proof of the merits of the proposed criterion.     

A Comparison of Different Methods for Representing Categorical Data

We first compare correspondence analysis, which uses chi-square distance, and an alternative approach using Hellinger distance, for representing categorical data in a contingency table. We propose a coefficient which globally measures the similarity between these two approaches. This coefficient can be decomposed into several components, one component for each principal dimension, indicating the contribution of the dimensions to the difference between the two representations. We also make comparisons with the logratio approach based on compositional data. These three methods of representation can produce quite similar results. Two illustrative examples are given.     

Limiting distribution of the number of clumps of palindromes in DNA

An important aspect in the analysis of long DNA sequences is to identify whether palindromes are over- or under-represented. The essential step in that direction is the analysis of the limiting distribution of the number of clumps of palindromes where clump is defined as the overlapping occurrence of palindromes. Using the Chen–Stein method, it is shown in this paper that the limiting distribution, under suitable conditions and a type of heterogeneous sequence, is the Poisson distribution. Moreover, error bounds and the rate of convergence are derived in terms of total variation distance between two probability distributions.     

Gini correlation as a measure of monotonicity and two of its usages

In this paper it is shown that the Gini correlation (suggested recently in this Journal) is a special case of a general family of measures of monotonicity. Two new usages of the GINI coefficient will be discussed: 1. for measuring convexity of time series, and 2. for measuring overlapping between groups. The asymptotic variances of the above two measures will be derived.     

Discriminating Between the Log-Normal and Log-Logistic Distributions

Log-normal and log-logistic distributions are often used to analyze lifetime data. For certain ranges of the parameters, the shape of the probability density functions or the hazard functions can be very similar in nature. It might be very difficult to discriminate between the two distribution functions. In this article, we consider the discrimination procedure between the two distribution functions. We use the ratio of maximized likelihood for discrimination purposes. The asymptotic properties of the proposed criterion are investigated. It is observed that the asymptotic distributions are independent of the unknown parameters. The asymptotic distributions are used to determine the minimum sample size needed to discriminate between these two distribution functions for a user specified probability of correct selection. We perform some simulation experiments to see how the asymptotic results work for small sizes. For illustrative purpose, two data sets are analyzed.     

On the distribution of rin samples from the mixtures of bivariate normal populations

The distributions of some transformations of the sample correlation coefficient r are studied here, when the parent population is a mixture of two standard bivariate normals. The behavior of these transformations is assessed through the first four standard moments. It is shown that there is a close relationship between the behavior of the transformed variables and the lack of normality as evinced by the 'kurtosis' defined in the bivariate population     

Posterior Distributions for the Gini Coefficient Using Grouped Data

When available data comprise a number of sampled households in each of a number of income classes, the likelihood function is obtained from a multinomial distribution with the income class population proportions as the unknown parameters. Two methods for going from this likelihood function to a posterior distribution on the Gini coefficient are investigated. In the first method, two alternative assumptions about the underlying income distribution are considered, namely a lognormal distribution and the Singh–Maddala (1976) income distribution. In these cases the likelihood function is reparameterized and the Gini coefficient is a nonlinear function of the income distribution parameters. The Metropolis algorithm is used to find the corresponding posterior distributions of the Gini coefficient from a sample of Bangkok households. The second method does not require an assumption about the nature of the income distribution, but uses (a) triangular prior distributions, and (b) beta prior distributions, on the location of mean income within each income class. By sampling from these distributions, and the Dirichlet posterior distribution of the income class proportions, alternative posterior distributions of the Gini coefficient are calculated.     

Discriminating between the Weibull and log-normal distributions for Type-II censored data

Log-normal and Weibull distributions are the two most popular distributions for analysing lifetime data. In this paper, we consider the problem of discriminating between the two distribution functions. It is assumed that the data are coming either from log-normal or Weibull distributions and that they are Type-II censored. We use the difference of the maximized log-likelihood functions, in discriminating between the two distribution functions. We obtain the asymptotic distribution of the discrimination statistic. It is used to determine the probability of correct selection in this discrimination process. We perform some simulation studies to observe how the asymptotic results work for different sample sizes and for different censoring proportions. It is observed that the asymptotic results work quite well even for small sizes if the censoring proportions are not very low. We further suggest a modified discrimination procedure. Two real data sets are analysed for illustrative purposes.     

Statistics for a New Test of Discordance in Circular Data

This article considers the derivation of approximate distributions for two types of statistics that can be used in developing new tests of discordance in circular samples from the von Mises distribution. An alternative test of discordance is proposed based on the circular distance between sample points. The advantage of the test is that it allows users to detect possible outliers in both univariate and bivariate circular data. For illustration, the test is applied to two real circular data sets.     

A Robust and Almost Fully Efficient M-Estimator

Simultaneous robust estimates of location and scale parameters are derived from a class of M-estimating equations. A coefficient p ( p > 0), which plays a role similar to that of a tuning constant in the theory of M-estimation, determines the estimating equations. These estimating equations may be obtained as the gradient of a strictly convex criterion function. This article shows that the estimators are uniquely defined, asymptotically bi-variate normal and have positive breakdown for some choices of p . When p = 0.12 and p = 0.3, the estimators are almost fully efficient for normal and exponential distributions: efficiencies with respect to the maximum likelihood estimators are 1.00 and 0.99, respectively. It is shown that the location estimator for known scale has the maximum breakdown point 0.5 independent of p , when the target model is symmetric. Also it is shown that the scale estimator has a positive breakdown point which depends on the choice of p . A simulation study finds that the proposed location estimator has smaller variance than the Hodges–Lehmann estimator, Huber's minimax and bisquare M-estimators.     

Distance measures in a one-parameter class of density functions

In a class of density functions with parameter λ > 0 , which includes Weibull and Gamma distributions, three distance measures are examined. Several properties of these measures and relationships between them are proved. Furthermore the exponential distribution is characterized in a one- parameter class of Weibull distributions.     

Posterior Predictive Comparisons for the Two-sample Problem

The two-sample problem of inferring whether two random samples have equal underlying distributions is formulated within the Bayesian framework as a comparison of two posterior predictive inferences rather than as a problem of model selection. The suggested approach is argued to be particularly advantageous in problems where the objective is to evaluate evidence in support of equality, along with being robust to the priors used and being capable of handling improper priors. Our approach is contrasted with the Bayes factor in a normal setting and finally, an additional example is considered where the observed samples are realizations of Markov chains.     

Energy statistics: A class of statistics based on distances

Energy distance is a statistical distance between the distributions of random vectors, which characterizes equality of distributions. The name energy derives from Newton's gravitational potential energy, and there is an elegant relation to the notion of potential energy between statistical observations. Energy statistics are functions of distances between statistical observations in metric spaces. Thus even if the observations are complex objects, like functions, one can use their real valued nonnegative distances for inference. Theory and application of energy statistics are discussed and illustrated. Finally, we explore the notion of potential and kinetic energy of goodness-of-fit.     

Quantitative comparisons between finitary posterior distributions and Bayesian posterior distributions

The main object of Bayesian statistical inference is the determination of posterior distributions. Sometimes these laws are given for quantities devoid of empirical value. This serious drawback vanishes when one confines oneself to considering a finite horizon framework. However, assuming infinite exchangeability gives rise to fairly tractable a posteriori quantities, which is very attractive in applications. Hence, with a view to a reconciliation between these two aspects of the Bayesian way of reasoning, in this paper we provide quantitative comparisons between posterior distributions of finitary parameters and posterior distributions of allied parameters appearing in usual statistical models.     

A Hellinger distance approach to MCMC diagnostics

Bayesian analysis often requires the researcher to employ Markov Chain Monte Carlo (MCMC) techniques to draw samples from a posterior distribution which in turn is used to make inferences. Currently, several approaches to determine convergence of the chain as well as sensitivities of the resulting inferences have been developed. This work develops a Hellinger distance approach to MCMC diagnostics. An approximation to the Hellinger distance between two distributions f and g based on sampling is introduced. This approximation is studied via simulation to determine the accuracy. A criterion for using this Hellinger distance for determining chain convergence is proposed as well as a criterion for sensitivity studies. These criteria are illustrated using a dataset concerning the Anguilla australis, an eel native to New Zealand.     

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 Sarmanov family with beta and gamma marginal distributions: an application to the Bayes premium in a collective risk model

In this paper we firstly develop a Sarmanov–Lee bivariate family of distributions with the beta and gamma as marginal distributions. We obtain the linear correlation coefficient showing that, although it is not a strong family of correlation, it can be greater than the value of this coefficient in the Farlie–Gumbel–Morgenstern family. We also determine other measures for this family: the coefficient of median concordance and the relative entropy, which are analyzed by comparison with the case of independence. Secondly, we consider the problem of premium calculation in a Poisson–Lindley and exponential collective risk model, where the Sarmanov–Lee family is used as a structure function. We determine the collective and Bayes premiums whose values are analyzed when independence and dependence between the risk profiles are considered, obtaining that notable variations in premiums values are obtained even when low levels of correlation are considered.     

On Terrel's characterization of uniform distribution

Terrel (1983) (The Annals of Probability, Vol. 11, No. 3, 823–826) showed that the coefficient of correlation between the smaller and larger of a sample of size two is at most one-half, and this upper bound is attained only for continuous uniform distributions. His proof is of computational nature and is based on the properties of Legendre polynomials. We give an easier proof of Terrel's characterization and we show how our method can be used for obtaining sharper bounds within the class of discrete distributions onN points and also a characterization of the equidistant uniform distribution.