Combined estimators for the correlation coefficient

Three combined estimators for the bivariate normal correlation parameter are considered. The data consist of k independent sample correlation coefficients and it is assumed that the underlying correlation parameters are all equal to ρ. Based upon the joint density function of the sample correlations a combined estimator of ρ is obtained as an approximation to the maximum likelihood solution. Two linearly combined estimators are also considered. One of them is based on Fisher's z-transformation of the sample correlations and the other on an unbiased estimator of ρ. The comparison of these three estimators indicates that the combined (approximate) MLE has a slightly smaller estimated mean squared error relative to the other two combined methods of estimation, but it does so at the expense of a relatively larger bias.     

On marginal likelihood inference for the intra-class correlation coefficient

A p-component set of responses have been constructed by a location-scale transformation to a p-component set of error variables, the covariance matrix of the set of error variables being of intra-class covariance structure:all variances being unity, and covariance being equal [IML0001]. A sample of size n has been described as a conditional structural model, conditional on the value of the intra-class correlation coefficient ρ. The conditional technique of structural inference provides the marginal likelihood function of ρ based on the standardized residuals. For the normal case, the marginal likelihood function of ρ is seen to be dependent on the standardized residuals through the sample intra-class correlation coefficient. By the likelihood modulation technique, the nonnull distribution of the sample intra-class correlation coefficient has also been obtained.     

The approximate null distribution of the likelihood ratio test for a mixture of two bivariate normal distributions with equal co variance

We define the mixture likelihood approach to clustering by discussing the sampling distribution of the likelihood ratio test of the null hypothesis that we have observed a sample of observations of a variable having the bivariate normal distribution versus the alternative that the variable has the bivariate normal mixture with unequal means and common within component covariance matrix. The empirical distribution of the likelihood ratio test indicates that convergence to the chi-squared distribution with 2 df is at best very slow, that the sample size should be 5000 or more for the chi-squared result to hold, and that for correlations between 0.1 and 0.9 there is little, if any, dependence of the null distribution on the correlation. Our simulation study suggests a heuristic function based on the gamma.     

Optimum stratification with two variates

For the 2×2 rectilinear stratification of a bivariate normal distribution with proportional and optimum allocation the dependence of the objective function z(x₁;y₁) on the coefficient of correlation ρ and the sampling fraction q=n/N is investigated. With proportional allocation for great values of ρ (but already for q=0) a so-called ρ-effect arises, which results in a saddle-point of z as “optimum” stratification point in the center of gravity of the distribution and two additional minima. With optimum allocation first for smaller values of q also the ρ-effect arises; for grater values of q a so-called q-effect is superposed, which results in a multitude of minima, saddle-points and maxima of z. All these points satisfy the generalized conditions of Dalenius, but for practical use only the global minimum is of interest.     

Matching a correlation coefficient by a Gaussian copula

A Gaussian copula is widely used to define correlated random variables. To obtain a prescribed Pearson correlation coefficient of ρ_x between two random variables with given marginal distributions, the correlation coefficient ρ_z between two standard normal variables in the copula must take a specific value which satisfies an integral equation that links ρ_x to ρ_z. In a few cases, this equation has an explicit solution, but in other cases it must be solved numerically. This paper attempts to address this issue. If two continuous random variables are involved, the marginal transformation is approximated by a weighted sum of Hermite polynomials; via Mehler’s formula, a polynomial of ρ_z is derived to approximate the function relationship between ρ_x and ρ_z. If a discrete variable is involved, the marginal transformation is decomposed into piecewise continuous ones, and ρ_x is expressed as a polynomial of ρ_z by Taylor expansion. For a given ρ_x, ρ_z can be efficiently determined by solving a polynomial equation.     

A generalization of the bivariate Beta-Binomial distribution

This paper presents a new bivariate discrete distribution that generalizes the bivariate Beta-Binomial distribution. It is generated by Appell hypergeometric function F₁ and can be obtained as a Binomial mixture with an Exton's Generalized Beta distribution. The model has different marginal distributions which are, together with the conditional distributions, more flexible than the Beta-Binomial distribution. It has non-linear regression curves and is useful for random variables with positive correlation. These features make the model very adequate to fit observed data as the two applications included show.     

Inferences Based on a Skipped Correlation Coefficient

The most popular method for trying to detect an association between two random variables is to test H ₀ ?:?ρ=0, the hypothesis that Pearson's correlation is equal to zero. It is well known, however, that Pearson's correlation is not robust, roughly meaning that small changes in any distribution, including any bivariate normal distribution as a special case, can alter its value. Moreover, the usual estimate of ρ, r, is sensitive to only a few outliers which can mask a true association. A simple alternative to testing H ₀ ?:?ρ =0 is to switch to a measure of association that guards against outliers among the marginal distributions such as Kendall's tau, Spearman's rho, a Winsorized correlation, or a so-called percentage bend correlation. But it is known that these methods fail to take into account the overall structure of the data. Many measures of association that do take into account the overall structure of the data have been proposed, but it seems that nothing is known about how they might be used to detect dependence. One such measure of association is selected, which is designed so that under bivariate normality, its estimator gives a reasonably accurate estimate of ρ. Then methods for testing the hypothesis of a zero correlation are studied.     

On the bivariate negative binomial regression model

In this paper, a new bivariate negative binomial regression (BNBR) model allowing any type of correlation is defined and studied. The marginal means of the bivariate model are functions of the explanatory variables. The parameters of the bivariate regression model are estimated by using the maximum likelihood method. Some test statistics including goodness-of-fit are discussed. Two numerical data sets are used to illustrate the techniques. The BNBR model tends to perform better than the bivariate Poisson regression model, but compares well with the bivariate Poisson log-normal regression model.     

A monte carlo investigation of a statistic for a bivariate missing data problem

Testing the equal means hypothesis of a bivariate normal distribution with homoscedastic varlates when the data are incomplete is considered. If the correlational parameter, ρ, is known, the well-known theory of the general linear model is easily employed to construct the likelihood ratio test for the two sided alternative. A statistic, T, for the case of ρ unknown is proposed by direct analogy to the likelihood ratio statistic when ρ is known. The null and nonnull distribution of T is investigated by Monte Carlo techniques. It is concluded that T may be compared to the conventional t distribution for testing the null hypothesis and that this procedure results in a substantial increase in power-efficiency over the procedure based on the paired t test which ignores the incomplete data. A Monte Carlo comparison to two statistics proposed by Lin and Stivers (1974) suggests that the test based on T is more conservative than either of their statistics.     

Robustness of the Multiple Correlation Coefficient When Sampling From a Mixture of Two Multivariate Normal Populations

The density of the multiple correlation coefficient is derived by direct integration when the sample covariance matrix has a linear non-central distribution. Using the density, we deduce the null and non-null distribution of the multiple correlation coefficient when sampling from a mixture of two multivariate normal populations with the same covariance matrix. We also compute actual significance levels of the test of the hypothesis H_o : ρ_1·2…p = 0 versus H_a:ρ_1·2…p > 0, given the mixture model.     

Bivariate Kumaraswamy distribution: properties and a new method to generate bivariate classes

In this paper, we introduce a bivariate Kumaraswamy (BVK) distribution whose marginals are Kumaraswamy distributions. The cumulative distribution function of this bivariate model has absolutely continuous and singular parts. Representations for the cumulative and density functions are presented and properties such as marginal and conditional distributions, product moments and conditional moments are obtained. We show that the BVK model can be obtained from the Marshall and Olkin survival copula and obtain a tail dependence measure. The estimation of the parameters by maximum likelihood is discussed and the Fisher information matrix is determined. We propose an EM algorithm to estimate the parameters. Some simulations are presented to verify the performance of the direct maximum-likelihood estimation and the proposed EM algorithm. We also present a method to generate bivariate distributions from our proposed BVK distribution. Furthermore, we introduce a BVK distribution which has only an absolutely continuous part and discuss some of its properties. Finally, a real data set is analysed for illustrative purposes.     

Mixture separation for mixed-mode data

One possible approach to cluster analysis is the mixture maximum likelihood method, in which the data to be clustered are assumed to come from a finite mixture of populations. The method has been well developed, and much used, for the case of multivariate normal populations. Practical applications, however, often involve mixtures of categorical and continuous variables. Everitt (1988) and Everitt and Merette (1990) recently extended the normal model to deal with such data by incorporating the use of thresholds for the categorical variables. The computations involved in this model are so extensive, however, that it is only feasible for data containing very few categorical variables. In the present paper we consider an alternative model, known as the homogeneous Conditional Gaussian model in graphical modelling and as the location model in discriminant analysis. We extend this model to the finite mixture situation, obtain maximum likelihood estimates for the population parameters, and show that computation is feasible for an arbitrary number of variables. Some data sets are clustered by this method, and a small simulation study demonstrates characteristics of its performance.     

Simulated annealing for the bounds of Kendall's τ and Spearman's ρ

It has long been known that, for many joint distributions, Kendall's τ and Spearman's ρ have different values, as they measure different aspects of the dependence structure. Although the classical inequalities between Kendall's τ and Spearman's ρ for pairs of random variables are given, the joint distributions which can attain the bounds between Kendall's τ and Spearman's ρ are difficult to find. We use the simulated annealing method to find the bounds for ρ in terms of τ and its corresponding joint distribution which can attain those bounds. Furthermore, using this same method, we find the improved bounds between τ and ρ, which is different from that given by Durbin and Stuart.     

A Method for Multivariate Probability Distributions Construction via Parameter Dependence

The nature of stochastic dependence in the classic bivariate normal density framework is analyzed. In the case of this distribution we stress the way the conditional density of one of the random variables depends on realizations of the other. Typically, in the bivariate normal case this dependence takes the form of a parameter (here the “expected value”) of one probability density depending continuously (here linearly) on realizations of the other random variable. Our point is that such a pattern does not need to be restricted to that classical case of bivariate normal. We show that this paradigm can be generalized and viewed in ways that allows us to extend it far beyond the bivariate normal distributions class.     

Modified Sarhan–Balakrishnan singular bivariate distribution

Recently Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527] introduced a new bivariate distribution using generalized exponential and exponential distributions. They discussed several interesting properties of this new distribution. Unfortunately, they did not discuss any estimation procedure of the unknown parameters. In this paper using the similar idea as of Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527], we have proposed a singular bivariate distribution, which has an extra shape parameter. It is observed that the marginal distributions of the proposed bivariate distribution are more flexible than the corresponding marginal distributions of the Marshall–Olkin bivariate exponential distribution, Sarhan–Balakrishnan's bivariate distribution or the bivariate generalized exponential distribution. Different properties of this new distribution have been discussed. We provide the maximum likelihood estimators of the unknown parameters using EM algorithm. We reported some simulation results and performed two data analysis for illustrative purposes. Finally we propose some generalizations of this bivariate model.     

On the distribution of the correlation coefficient when sampling from a mixture of two bivariate normal densities: Robustness and the effect of outliers

The distribution of the sample correlation coefficient is derived when the population is a mixture of two bivariate normal distributions with zero mean but different covariances and mixing proportions 1 - λ and λ respectively; λ will be called the proportion of contamination. The test of ρ = 0 based on Student's t, Fisher's z, arcsine, or Ruben's transformation is shown numerically to be nonrobust when λ, the proportion of contamination, lies between 0.05 and 0.50 and the contaminated population has 9 times the variance of the standard (bivariate normal) population. These tests are also sensitive to the presence of outliers.     

Cure rate model with bivariate interval censored data

A mixture model is proposed to analyze a bivariate interval censored data with cure rates. There exist two types of association related with bivariate failure times and bivariate cure rates, respectively. A correlation coefficient is adopted for the association of bivariate cure rates and a copula function is applied for bivariate survival times. The conditional expectation of unknown quantities attributable to interval censored data and cure rates are calculated in the E-step in ES (Expectation-Solving algorithm) and the marginal estimates and the association measures are estimated in the S-step through a two-stage procedure. A simulation study is performed to evaluate the suggested method and a real data from HIV patients is analyzed as a real data example.     

Asymptotic Normality in Mixtures of Power Series Distributions

Abstract.  The problem of estimating the individual probabilities of a discrete distribution is considered. The true distribution of the independent observations is a mixture of a family of power series distributions. First, we ensure identifiability of the mixing distribution assuming mild conditions. Next, the mixing distribution is estimated by non-parametric maximum likelihood and an estimator for individual probabilities is obtained from the corresponding marginal mixture density. We establish asymptotic normality for the estimator of individual probabilities by showing that, under certain conditions, the difference between this estimator and the empirical proportions is asymptotically negligible. Our framework includes Poisson, negative binomial and logarithmic series as well as binomial mixture models. Simulations highlight the benefit in achieving normality when using the proposed marginal mixture density approach instead of the empirical one, especially for small sample sizes and/or when interest is in the tail areas. A real data example is given to illustrate the use of the methodology.     

Manova with covariates and an application in profile analysis

Replacing one of the two marginal distributions in a bivariate normal by a family of symmetrical distributions, we obtain a new family of symmetric bivariate distributions. We use the Tiku - Suresh (1990) method to estimate the parameters of this new bivariate family. We define a Hotelling - type statistic to test the mean vector and evaluate the asymptotic power of this statistic relative to the Hotelling T² statistic. We show that the former is considerably more powerful.