Approximation of the Bivariate Renewal Function

Recently, Kambo and his co-researchers (2012) proposed a method of approximation for evaluating the one-dimensional renewal function based on the first three moments. Their method is simple and elegant, which gives exact values for well-known distributions. In this article, we propose an analogous method for the evaluation of bivariate renewal function based on the first two moments of the variables and their joint moment. The proposed method yields exact results for certain widely used bivariate distributions like bivariate exponential distribution, bivariate Weibull distributions, and bivariate Pareto distributions. An illustrative example in the form of a two-dimensional warranty problem is considered and comparisons of our method are made with the results of other models.     

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.     

Forms for the distribution of ss ilkptxcxtt statistic for bivariate sphericity

This paper examines the nonnuU distribution of the ellipticity statistic used for testing sphericity in a bivariate normal population. The distribution function is expressed as mixtures of F distribution functions, where the mixing probabilities are binomial and mgatiire binomial. A simple approximation to the distribution is suggested and examined     

Uniform asymptotics for ruin probability of a two-dimensional dependent renewal risk model

ABSTRACT

In this paper we consider the tail behavior of a two-dimensional dependent renewal risk model with two dependent classes of insurance business, in which the claim sizes are governed by a common renewal counting process, and their inter-arrival times are dependent, identically distributed. For the case that the claim size distribution belongs to the intersection of long-tailed distribution class and dominant variation class, we obtain an asymptotic formula, which holds uniformly for all time in an infinite interval. Moreover, we point out that the formula still holds uniformly for all time in an infinite interval for widely dependent random variables (r.v.s) under some conditions.     

Some bivariate uniform distributions

Bivariate uniform distributions with dependent components are readily derived by distribution function transformations of the components of non-uniform dependent continuous bivariate random variables (X,Y). Contour plots of joint density functions show the various, and varying, forms of dependence which can arise from different distributional forms for (X,Y) and aids the choice of bivariate uniform distributions as empirical models.     

A Bivariate Generalization of Gamma Distribution

In this article, a bivariate generalisation of the gamma distribution is proposed by using an unsymmetrical bivariate characteristic function; an extension to the non central case also receives attention. The probability density functions of the product and ratio of the correlated components of this distribution are also derived. The benefits of introducing this generalized bivariate gamma distribution and the distributions of the product and the ratio of its components will be demonstrated by graphical representations of their density functions. An example of this generalized bivariate gamma distribution to rainfall data for two specific districts in the North West province is also given to illustrate the greater versatility of the new distribution.     

On univariate and bivariate generalized gamma convolutions

This paper has two parts. In the first part some results for generalized gamma convolutions (GGCs) are reviewed. A GGC is a limit distribution for sums of independent gamma variables. In the second part, bivariate gamma distributions and bivariate GGCs are considered. New bivariate gamma distributions are derived from shot-noise models. The remarkable property hyperbolic complete monotonicity (HCM) for a function is considered both in the univariate case and in the bivariate case.     

Moments of the product and ratio of two correlated chi-square variables

The exact probability density function of a bivariate chi-square distribution with two correlated components is derived. Some moments of the product and ratio of two correlated chi-square random variables have been derived. The ratio of the two correlated chi-square variables is used to compare variability. One such application is referred to. Another application is pinpointed in connection with the distribution of correlation coefficient based on a bivariate t distribution.      

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 applying Möbius transformation to cardioid random variables

Möbius transformation from the unit circle onto itself is applied to a cardioid random variable. The distribution function, cosine and sine moments, conditions for modality and symmetry of the resulting distribution are studied. The paper proposes a bivariate cardioid distribution with symmetric cardioid marginals. The distribution is generated from a circular-circular structural model or a method of trivariate reduction introduced and controlled by a Möbius transformation link between common parts of the two variables. An illustration is given for wind direction data as an application of the bivariate cardioid distribution.     

On applying Möbius transformation to cardioid random variables

Möbius transformation from the unit circle onto itself is applied to a cardioid random variable. The distribution function, cosine and sine moments, conditions for modality and symmetry of the resulting distribution are studied. The paper proposes a bivariate cardioid distribution with symmetric cardioid marginals. The distribution is generated from a circular-circular structural model or a method of trivariate reduction introduced and controlled by a Möbius transformation link between common parts of the two variables. An illustration is given for wind direction data as an application of the bivariate cardioid distribution.     

Bounds for the error in an approximation to the distribution of continuous random vectors with bounded supports

An approximation for the distributions of m-dimensional random variables with absolutely continuous cumulative distribution functions and bounded supports is presented. In addition, two easily computed bounds for the error in using the approximation rather than the actual distribution are provided for any measurable region.     

A continuous general multivariate distribution and its properties

Starting from two known continuous univariate distributions, a bivariate distribution is constructed depending on a parameter which measures the degree of stochastic dependence between the two random variables. From the foregoing construction we then pass to a multivariate-type distribution, constructed using only univariate distributions and an association matrix. Some properties of the multivariate and bivariate case are studied.     

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.     

Estimating the Cost of a Warranty

The cost of certain types of warranties is closely related to functions that arise in renewal theory. The problem of estimating the warranty cost for a random sample of size n can be reduced to estimating these functions. In an earlier paper, I gave several methods of estimating the expected number of renewals, called the renewal function. This answered an important accounting question of how to arrive at a good approximation of the expected warranty cost. In this article, estimation of the renewal function is reviewed and several extensions are given. In particular, a resampling estimator of the renewal function is introduced. Further, I argue that managers may wish to examine other summary measures of the warranty cost, in particular the variability. To estimate this variability, I introduce estimators, both parametric and nonparametric, of the variance associated with the number of renewals. Several numerical examples are provided.     

A polar coordinate transformation for estimating bivariate survival functions with randomly censored and truncated data

This paper proposes a new estimator for bivariate distribution functions under random truncation and random censoring. The new method is based on a polar coordinate transformation, which enables us to transform a bivariate survival function to a univariate survival function. A consistent estimator for the transformed univariate function is proposed. Then the univariate estimator is transformed back to a bivariate estimator. The estimator converges weakly to a zero-mean Gaussian process with an easily estimated covariance function. Consistent truncation probability estimate is also provided. Numerical studies show that the distribution estimator and truncation probability estimator perform remarkably well.     

Bivariate distributions of some ratios of independent noncentral chi-square random variables

The bivariate distributions of three pairs of ratios of in¬dependent noncentral chi-square random variables are considered. These ratios arise in the problem of computing the joint power function of simultaneous F-tests in balanced ANOVA and ANCOVA. The distributions obtained are generalizations to the noncentral case of existing results in the literature. Of particular note is the bivariate noncentral F distribution, which generalizes a special case of Krishnaiah*s (1964,1965) bivariate central F distribution. Explicit formulae for the cdf's of these distribu¬tions are given, along with computational procedures     

Vine copula approximation: a generic method for coping with conditional dependence

Pair-copula constructions (or vine copulas) are structured, in the layout of vines, with bivariate copulas and conditional bivariate copulas. The main contribution of the current work is an approach to the long-standing problem: how to cope with the dependence structure between the two conditioned variables indicated by an edge, acknowledging that the dependence structure changes with the values of the conditioning variables. The changeable dependence problem, though recognized as crucial in the field of multivariate modelling, remains widely unexplored due to its inherent complication and hence is the motivation of the current work. Rather than resorting to traditional parametric or nonparametric methods, we proceed from an innovative viewpoint: approximating a conditional copula, to any required degree of approximation, by utilizing a family of basis functions. We fully incorporate the impact of the conditioning variables on the functional form of a conditional copula by employing local learning methods. The attractions and dilemmas of the pair-copula approximating technique are revealed via simulated data, and its practical importance is evidenced via a real data set.     

Copules archimédiennes et families de lois bidimensionnelles dont les marges sont données

Every bivariate distribution function with continuous marginals can be represented in terms of a unique copula, that is, in terms of a distribution function on the unit square with uniform marginals. This paper is concerned with a special class of copulas called Archimedean, which includes the uniform representation of many standard bivariate distributions. Conditions are given under which these copulas are stochastically ordered and pointwise limits of sequences of Archimedean copulas are examined. We also provide two new one-parameter families of bivariate distributions which include as limiting cases the Frechet bounds and the independence distribution.     

Testing for bivariate gumbel against bivariate new better than used in expectation

It is shown that a bivariate survival function is both New Better than Used in Expectation (NBUE) and New Worse than Used in Expectation (NWUE) if and only if it is a bivariate Gumbel distribution. Statistical procedures are then presented to test whether that, within the class of bi-variate NBUE survival functions, a survival function is a Gumbel's bivariate exponential.