Generalization of the robust bivariate t2 statistic to multivariate populations

The robust bivariate Hotelling–type T² statistics proposed by Tiku and Balakrishnan (1988) is extendend to p–variate (p ≧ 3) populations.     

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.     

On the behaviour of some transforms of the sample correlation coefficient:discretebivariate populations

The present paper investigates for the first time, the robustness of some of the familiar transformations of the sample correlation coefficient when the parent population is discrete. Three specific cases examined are:The bivariate Poisson (BVP):the bivariate negative binomial (BNB):The trinomial (TN). Investigation of the (near) normality of the transformed statistics is done by the techniques considered by Subrahmaniam and Gajjar. In addition, an empirical examination of their behaviour is carried out by the density estimation technique due to Tarter and Kronmal.     

Numerical computation of rectangular bivariate and trivariate normal and t probabilities

Algorithms for the computation of bivariate and trivariate normal and t probabilities for rectangles are reviewed. The algorithms use numerical integration to approximate transformed probability distribution integrals. A generalization of Plackett's formula is derived for bivariate and trivariate t probabilities. New methods are described for the numerical computation of bivariate and trivariate t probabilities. Test results are provided, along with recommendations for the most efficient algorithms for single and double precision computations.     

A robust procedure for testing the equality of mean vectors of two bivariate populations with unequal covariance matrices

A robust test is developed for testing equality of the mean vectors of two bivariate (multivariate) populations when the variance-covariance matrices are not necessarily equal. The test is an extension of the univariate robust test given by Tiku and Singh (1981).     

On bivariate transformation of scale distributions

ABSTRACT

Elsewhere, I have promoted (univariate continuous) “transformation of scale” (ToS) distributions having densities of the form 2g(Π^?1(x)) where g is a symmetric distribution and Π is a transformation function with a special property. Here, I develop bivariate (readily multivariate) ToS distributions. Univariate ToS distributions have a transformation of random variable relationship with Azzalini-type skew-symmetric distributions; the bivariate ToS distribution here arises from marginal variable transformation of a particular form of bivariate skew-symmetric distribution. Examples are given, as are basic properties—unimodality, a covariance property, random variate generation—and connections with a bivariate inverse Gaussian distribution are pointed out.     

Weighted bivariate logarithmic series distributions

In this paper, we are interested in the weighted distributions of a bivariate three parameter logarithmic series distribution studied by Kocherlakota and Kocherlakota (1990). The weighted versions of the model are derived with weight W(x,y) = x^[r] y^[s]. Explicit expressions for the probability mass function and probability generating functions are derived in the case r = s = l. The marginal and conditional distributions are derived in the general case. The maximum likelihood estimation of the parameters, in both two parameter and three parameter cases, is studied. A procedure for computer generation of bivariate data from a discrete distribution is described. This enables us to present two examples, in order to illustrate the methods developed, for finding the maximum likelihood estimates.     

Estimation in a bivariate integer-valued autoregressive process

ABSTRACT

A bivariate integer-valued autoregressive time series model is presented. The model structure is based on binomial thinning. The unconditional and conditional first and second moments are considered. Correlation structure of marginal processes is shown to be analogous to the ARMA(2, 1) model. Some estimation methods such as the Yule–Walker and conditional least squares are considered and the asymptotic distributions of the obtained estimators are derived. Comparison between bivariate model with binomial thinning and bivariate model with negative binomial thinning is given.     

On the distribution of hotelling's t2 and multiple correlation r2when sampling from a mixture of two normals

In this paper the non-null distribution of Hotelling's T² and the null distribution of multiple correlation R² are derived when the sample is taken from a mixture of two p-component multivariate normal distributions with mean vectors μ₁ and μ₂ respectively and common covariance matrix ∑, ∑. In a special case the non-null distribution of R² is a l s o given, while the general noncentral distribution is given i n Awan (1981). These results have been used to study the robustness of T² and R² tests by Srivastava and Awan (1982), and Awan and Srivastava (1982) respectively.     

Asymptotic expaxsioxs for the joint distribution of cirrelated hotellings t2 statlstics under normality

Let T² _i=z′_iS^?1z_i, i==,…k be correlated Hotelling's T² statistics under normality. where z=(z′_i,…,z′_k)′ and nS are independently distributed as N_kp((O,ρ?∑) and Wishart distribution W_p(∑, n), respectively. The purpose of this paper is to study the distribution function F(x₁,…,x_k) of (T² _i,…,T² _k) when n is large. First we derive an asymptotic expansion of the characteristic function of (T² _i,…,T² _k) up to the order n^?2. Next we give asymptotic expansions for (T² _i,…,T² _k) in two cases (i)ρ=I_k and (ii) k=2 by inverting the expanded characteristic function up to the orders n^?2 and n^?1, respectively. Our results can be applied to the distribution function of max (T² _i,…,T² _k) as a special case.     

An optimal sign test for one-sample bivariate location model using an alternative bivariate ranked-set sample

The aim of this paper is to find an optimal alternative bivariate ranked-set sample for one-sample location model bivariate sign test. Our numerical and theoretical results indicated that the optimal designs for the bivariate sign test are the alternative designs with quantifying order statistics with labels {((r+1)/2, (r+1)/2)}, when the set size r is odd and {(r/2+1, r/2), (r/2, r/2+1)} when the set size r is even. The asymptotic distribution and Pitman efficiencies of these designs are derived. A simulation study is conducted to investigate the power of the proposed optimal designs. Illustration using real data with the Bootstrap algorithm for P-value estimation is used.     

The Prediction Sum of Squares as a General Measure for Regression Diagnostics

Statistics that usually accompany the regression model do not provide insight into the quality of the data or the potential influence of the individual observations on the estimates. In this study, the Q² statistic is used as a criterion for detecting influential observations or outliers. The statistic is derived from the jackknifed residuals, the squared sum of which is generally known as the prediction sum of squares or PRESS. This article compares R ² with Q² and suggests that the latter be used as part of the data-quality check. It is shown, for two separate data sets obtained from regional cost of living and U.S. food industry studies, that in the presence of outliers the Q² statistic can be negative, because it is sensitive to the choice of regressors and the inclusion of influential observations. Once the outliers are dropped from the sample, the discrepancy between Q² and R ² values is negligible.     

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     

Preservation of dependence concepts under bivariate weighted distributions

ABSTRACT

In this paper, we provide conditions under which some bivariate dependence structures are preserved under bivariate weighted distributions. Bivariate weighted distributions whose dependence structure is the same as the original distribution are characterized. Finally, we discuss some examples to show the usefulness of our results.     

Asymptotics of bivariate penalised splines

We study the class of bivariate penalised splines that use tensor product splines and a smoothness penalty. Similar to Claeskens, G., Krivobokova, T., and Opsomer, J.D. [(2009), ‘Asymptotic Properties of Penalised Spline Estimators’, Biometrika, 96(3), 529–544] for the univariate penalised splines, we show that, depending on the number of knots and penalty, the global asymptotic convergence rate of bivariate penalised splines is either similar to that of tensor product regression splines or to that of thin plate splines. In each scenario, the bivariate penalised splines are found rate optimal in the sense of Stone, C.J. [(12, 1982), ‘Optimal Global Rates of Convergence for Nonparametric Regression’, The Annals of Statistics, 10(4), 1040–1053] for a corresponding class of functions with appropriate smoothness. For the scenario where a small number of knots is used, we obtain expressions for the local asymptotic bias and variance and derive the point-wise and uniform asymptotic normality. The theoretical results are applicable to tensor product regression splines.     

A bivariate F distribution with marginals on arbitrary numerator and denominator degrees of freedom, and related bivariate beta and t distributions

The classical bivariate F distribution arises from ratios of chi-squared random variables with common denominators. A consequent disadvantage is that its univariate F marginal distributions have one degree of freedom parameter in common. In this paper, we add a further independent chi-squared random variable to the denominator of one of the ratios and explore the extended bivariate F distribution, with marginals on arbitrary degrees of freedom, that results. Transformations linking F, beta and skew t distributions are then applied componentwise to produce bivariate beta and skew t distributions which also afford marginal (beta and skew t) distributions with arbitrary parameter values. We explore a variety of properties of these distributions and give an example of a potential application of the bivariate beta distribution in Bayesian analysis.     

Note on estimation of reliability under bivariate pareto stress-strength model

In this paper, we discuss the problem of estimating reliability (R) of a component based on maximum likelihood estimators (MLEs). The reliability of a component is given byR=P[Y<X]. Here X is a random strength of a component subjected to a random stress(Y) and (X,Y) follow a bivariate pareto(BVP) distribution. We obtain an asymptotic normal(AN) distribution of MLE of the reliability(R).