Efficient computation of location depth contours by methods of computational geometry

The concept of location depth was introduced as a way to extend the univariate notion of ranking to a bivariate configuration of data points. It has been used successfully for robust estimation, hypothesis testing, and graphical display. The depth contours form a collection of nested polygons, and the center of the deepest contour is called the Tukey median. The only available implemented algorithms for the depth contours and the Tukey median are slow, which limits their usefulness. In this paper we describe an optimal algorithm which computes all bivariate depth contours in O(n ²) time and space, using topological sweep of the dual arrangement of lines. Once these contours are known, the location depth of any point can be computed in O(log² n) time with no additional preprocessing or in O(log n) time after O(n ²) preprocessing. We provide fast implementations of these algorithms to allow their use in everyday statistical practice.     

In Memoriam: Bernard George Greenberg 1919-1985

The bivariate normal density with unit variance and correlation ρ is well known. We show that by integrating out ρ, the result is a function of the maximum norm. The Bayesian interpretation of this result is that if we put a uniform prior over ρ, then the marginal bivariate density depends only on the maximal magnitude of the variables. The square-shaped isodensity contour of this resulting marginal bivariate density can also be regarded as the equally weighted mixture of bivariate normal distributions over all possible correlation coefficients. This density links to the Khintchine mixture method of generating random variables. We use this method to construct the higher dimensional generalizations of this distribution. We further show that for each dimension, there is a unique multivariate density that is a differentiable function of the maximum norm and is marginally normal, and the bivariate density from the integral over ρ is its special case in two dimensions.     

Dependence function for continuous bivariate densities

The notion of cross-product ratio for discrete two-way contingency table is extended to the case of continuous bivariate densities. This results in the “local dependence function” that measues the margin-free dependence between bivariate random variables. Properties and examples of the dependence function are discussed. The bivariate normal density plays a special role since it has constant dependence. Continuous bivariate densities can be constructed by specifying the dependence function along with two marginals in analogy to the construction of two-way contingency tables given marginals and patterns of interaction. The dependence function provides a partial ordering on bivariate dependence.     

An error-bounded polynomial approximation for bivariate normal probabilities

Although the bivariate normal distribution is frequently employed in the development of screening models, the formulae for computing bivariate normal probabilities are quite complicated. A simple and accurate error-bounded, noniterative approximation for bivariate normal probabilities based on a simple univariate normal quadratic or cubic approximation is developed for use in screening applications. The approximation, which is most accurate for large absolute correlation coefficients, is especially suitable for screening applications (e.g., in quality control), where large absolute correlations between performance and screening variables are desired. A special approximation for conditional bivariate normal probabilities is also provided which in quality control screening applications improves the accuracy of estimating the average outgoing product quality. Some anomalies in computing conditional bivariate normal probabilities using BNRDF and NORDF in IMSL are also discussed.     

An approximation to percentiles of a variable of the bivariate normal distribution when the other variable is truncated,with applications

A Cornish-Fisher expansion is used to approximate the per-centiles of a variable of the bivariate normal distribution when the other variable is truncated. The expression is in terms of the bivariate cumulants of a singly truncated bivariate normal distribution. The percentiles are useful in the problem of personnel selection where we use a screening variable to screen applicants for employment and a correlated performance variable to screen employees for rehiring. This paper provides a bivariate cumulants table for determining the cutoff score of the performance variable. The following two problems are also con¬sidered: (1) determine the proportion of applicants who would have been successful had no screening been applied, and (2) determine the proportion of individuals being rejected byscreening who would have been successful had they been hired, The variable that is used to measure job performance and the variable that measures the outcome of an aptitude test are assumed to be jointly normally distributed with correlation ρ     

Characterizations of Bivariate and Multivariate Life Distributions Based on Reciprocal Subtangent

The subtangent is the projection of the tangent upon the axis of abscissa. The usefulness of the reciprocal subtangent as a measure of the survival and density curves has earlier been reported in the literature for univariate distributions. This measure was generalized for bivariate and multivariate setups and related characterization problems were examined. The conditionally specified bivariate exponential distribution has been uniquely determined from the local constancy of the bivariate reciprocal subtangents. The case of global constancy and other related results have been studied.

Conditionally specified bivariate Lomax distribution and normal distribution were also studied. Further, the conditionally specified multivariate exponential distribution was uniquely determined from the local constancy of the multivariate reciprocal subtangents.     

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.     

Distribution of a linear function of correlated ordered variables

In this paper, we have considered the problem of finding the distribution of a linear combination of the minimum and the maximum for a general bivariate distribution. The general results are used to obtain the required distribution in the case of bivariate normal, bivariate exponential of Arnold and Strauss, absolutely continuous bivariate exponential distribution of Block and Basu, bivariate exponential distribution of Raftery, Freund's bivariate exponential distribution and Gumbel's bivariate exponential distribution. The distributions of the minimum and maximum are obtained as special cases.     

Density based exploration of bivariate data

The difficulties of assessing details of the shape of a bivariate distribution, and of contrasting subgroups, from a raw scatterplot are discussed. The use of contours of a density estimate in highlighting features of distributional shape is illustrated on data on the development of aircraft technology. The estimated density height at each observation imposes an ordering on the data which can be used to select contours which contain specified proportions of the sample. This leads to a display which is reminiscent of a boxplot and which allows simple but effective comparison of different groups. Some simple properties of this technique are explored.Interesting features of a distribution such as arms and multimodality are found along the directions where the largest probability mass is located. These directions can be quantified through the modes of a density estimate based on the direction of each observation.     

Approximating probability integrals of multivariate normal using association models

The bivariate plane is symmetrically partitioned into fine rectangular regions, and a symmetric uniform association model is used to represent the resulting discretized bivariate normal probabilities. A new algorithm is developed by utilizing a quadrature and the above association model to approximate the diagonal probabilities. The off-diagonal probabilities are then approximated using the model. This method is an alternative to Wang's (1987) approach, computationally advantageous and relatively easy to extend to higher dimensions. Bivariate and trivariate normal probabilities approximated by our method are observed to agree very closely with the corresponding known results.     

Regression mean residual life of a parallel system of dependent components

In this paper, we consider a system consisting of two dependent components and we are interested in the average remaining life of the component that fails last when (i) the first failure occurs at time t and (ii) the first failure occurs after time t. For both the cases, expressions are derived in the case of general bivariate normal distribution and a class of bivariate exponential distribution including bivariate exponential distribution of Arnold and Strauss, absolutely continuous bivariate exponential distribution of Block and Basu, bivariate exponential distribution of Raftery, Freund's bivariate exponential distribution and Gumbel's bivariate exponential distribution.     

On the distribution of quotient of random variables conditioned to the positive quadrant

Abstract

Motivated by Caginalp and Caginalp [Physica A—Statistical Mechanics and Its Applications, 499, 2018, 457–471], we derive the exact distribution of X/Y conditioned on X?>?0, Y?>?0 for more than ten classes of distributions, including the bivariate t, bivariate Cauchy, bivariate Lomax, Arnold and Strauss’ bivariate exponential, Balakrishna and Shiji’s bivariate exponential, Mohsin et al.’s bivariate exponential, Morgenstern type bivariate exponential, bivariate gamma exponential and bivariate alpha skew normal distributions. The results can be useful in finance and other areas.     

A latent variable regression model for asymmetric bivariate ordered categorical data

In many areas of medical research, especially in studies that involve paired organs, a bivariate ordered categorical response should be analyzed. Using a bivariate continuous distribution as the latent variable is an interesting strategy for analyzing these data sets. In this context, the bivariate standard normal distribution, which leads to the bivariate cumulative probit regression model, is the most common choice. In this paper, we introduce another latent variable regression model for modeling bivariate ordered categorical responses. This model may be an appropriate alternative for the bivariate cumulative probit regression model, when postulating a symmetric form for marginal or joint distribution of response data does not appear to be a valid assumption. We also develop the necessary numerical procedure to obtain the maximum likelihood estimates of the model parameters. To illustrate the proposed model, we analyze data from an epidemiologic study to identify some of the most important risk indicators of periodontal disease among students 15-19 years in Tehran, Iran.     

Selection procedures based on normal and nonnormal variables with single and multiple characters

The performance of selection procedures using a single screening variable are assessed in the presence of nonnormality, in particular mixtures of bivariate normal distributions and the bivariate Edgeworth series distribution.

Screening with multiple characters in the normal situation is studied using principal components.     

The Bivariate Normal Copula

We collect well-known and less-known facts about the bivariate normal distribution and translate them into copula language. In addition, we provide various (equivalent) expressions for the bivariate normal copula, we compute its Gini's gamma, and we derive improved bounds and approximations on its diagonal.     

Weak convergence of bivariate sequence to bivariate normal

It is shown that under certain conditions the distributions of a bivariate sequence of random vectors converge weakly to that of a bivariate normal distribution.     

An investigation of Kendall's τ modified for censored data with applications

To accommodate testing for independence in bivariate data subject to censoring, several modifications of Kendall's τ are discussed. An extensive computer simulation is done to investigate power properties of these modifications under alternatives of the bivariate normal or bivariate exponential types. The statistics are then applied to available heart pacemaker patient survival data.     

An Integral of the Bivariate Normal and an Application

A formula to evaluate the integral of the bivariate normal density over finite area regions of the plane is developed. It is then used to compare regression estimates when bivariate normality is appropriate.