The distribution of the first j digits of beta related random variables

This paper gives the discrete distribution of the first j significant digits of two random variables: (1) a beta variable with integer parameter n and the other parameter m > 0, and (2) the reciprocal of (1). As a special case for n=1, we obtain the distribution of the first j significant digits of the pwoers of uniformly distributed random variables. These generalize the results of Kennard and Reith (1981) and Friedberg (1984), who considered only uniformly distributed random variables.     

The effect of dependence on distribution of the functions of random variables

In this article, we study the effect of dependence on the distributional properties of functions of two random variables. Expressions for the cumulative distribution functions of the linear combinations, products, and ratios of two dependent random variables in terms of their associated copula are derived. We discuss the effect of dependence on quantities such as the variances of linear combinations of functions, the value-at-risk measure, and the stress–strength parameter. Several examples, a simulation study, and a real data analysis are provided to illustrate the result.     

The distribution of the product of two independent generalized trapezoidal random variables

ABSTRACT

In this article, we derive the probability density function (pdf) of the product of two independent generalized trapezoidal random variables having different supports, in closed form, by considering all possible cases. We also show that the results for the product of two triangular and uniform random variables follow as special cases of our main result. As an illustration, we obtain pdf of product for a suitably constrained set of parameters and plot some graphs using MATLAB, which express variation in pdf with change in different parameters of the generalized trapezoidal distribution.     

On the convergence rate for arrays of row-wise NOD random variables

Abstract

In this article, the complete convergence results of weighted sums for arrays of rowwise negatively orthant dependent (NOD) random variables are investigated. Some sufficient conditions for complete convergence for arrays of rowwise NOD random variables are presented without assumption of identical distribution.     

Two new confidence intervals for the coefficient of variation in a normal distribution

In this article we introduce an approximately unbiased estimator for the population coefficient of variation, τ, in a normal distribution. The accuracy of this estimator is examined by several criteria. Using this estimator and its variance, two approximate confidence intervals for τ are introduced. The performance of the new confidence intervals is compared to those obtained by current methods.     

Positive dependence of random variables with a common marginal distribution

In this paper we review some notions of positive dependence of random variables with a common univariate marginal distribution and describe the related moment and probability inequalities. We first present a comparison between i.i.d. random variables and exchangeable random variables via an application of de Finetti's theorem, then describe some useful probability inequalities via partial orderings of the strength of their positive dependence. Finally, we state a result for random variables which are not necessarily exchangeable. Special applications to the multivariate normal distribution will be discussed, and the results involve only the correlation matrix of the distribution.     

Stein’s method and the distribution of the product of zero mean correlated normal random variables

Abstract

Over the last 80?years there has been much interest in the problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables. Motivated by this historical interest, we use a recent technique from the Stein’s method literature to obtain a simple new proof, which also serves as an exposition of a general method that may be useful in related problems.     

Distribution of products of independently distributed pathway random variables

A lot of work has been done on products and ratios of random variables by Provost and his co-workers, see, for example, Provost [S.B. Provost, The exact distribution of the ratio of a linear combination of chi-square variables over the root of a product of chi-square variables, Canad. J. Statist. 14 (1986), pp. 61–67; S.B. Provost, The distribution function of a statistic for testing the equality of scale parameters in two gamma populations, Metrika 36 (1989), pp. 337–345]. Here, we extend this idea by introducing a pathway model. The exact density functions of the products of pathway random variables are obtained using the Mellin transform technique. Their computable series forms are derived. The particular cases of the derived results are shown to be associated with the thermonuclear functions and reaction rate probability integral in the theory of nuclear reaction rate, Krätzel integral in applied analyses and inverse Gaussian density in stochastic processes. Graphical representations of the density functions of the product of random variables for the different values of the pathway parameters are shown. The new probability model is fitted to revenue data.     

Distribution of the quotient of noncentral normal random variables

The Mellin convolution is used to derive in analytical form an exact 3-parameterprobabilitydensity function of the quotient of two noncentral normal random variables. In contrast with the 5-parameter probability density function previously derivedby Fieller (1932) and Hinkley (1969), this 3-parameter probability density function is feasible for computer evaluation of the mean and cumulative distribution function, which are needed, for example, when dealing with estimation and distribution problems in regression analysis and sampling theory. When the normal variables are independent, the probability density function reduces to a 2-parameter function, for which a computer program is operational. An illustrative example is given for one set of parameters when the normal variables are independent, in which themean and functional form of the probability density function are presented, together with a brief tabulation of the probability density function.     

A synthetic control chart for the coefficient of variation

The coefficient of variation (CV) control chart has recently been introduced in the literature. Here, the salient features of this chart and the conforming run length chart are integrated to produce a synthetic coefficient of variation (SynCV) chart. The run length profile of the SynCV chart is numerically compared with the originally proposed chart and the upward EWMA-γ² chart. The SynCV chart outperforms the original CV chart, while the EWMA-γ² outperforms the SynCV chart for small increases in the CV. However, for large increases in the CV, the SynCV chart outperforms the EWMA-γ² chart.     

Convergence for weighted sums of widely orthant dependent random variables

ABSTRACT

In this article, a complete convergence result and a complete moment convergence result are obtained for the weighted sums of widely orthant dependent random variables under mild conditions. As corollaries, the corresponding results are also obtained under the extended negatively orthant dependent setup. In particular, the complete convergence result generalizes and improves the related known works in the literature.     

A comment on mckay's approximation for the coefficient of variation

Some clarification of statistics based on McKay's x² approximation for the distribution of the sample coefficient of variation is presented. The conclusions of Warren (1982) are shown to result from the confusion of two definitions for the sample coefficient of variation.     

The strong laws of large numbers for weighted sums of extended negatively dependent random variables

In this paper, the strong laws of large numbers for maximum value of weighted sums of extended negatively dependent random variables are obtained, which improve and extend the corresponding ones for independent random variables and some dependent random variables.     

Inferences on the coefficient of variation of an inverse gaussian distribution

The likelihood ratio test for a characteristic parameter of the inverse Gaussian distribution is derived. The parameter of interest characterizes the coefficient of variation, the skewness and the kurtosis of the distribution. The distribution of the test statistic is presented in a simplified form. Useful quanfiles of the distribution are given. Methods for constructing confidence bounds for the parameter, including Bayes highest posterior density intervals, are considered.     

Complete convergence and the strong laws of large numbers for pairwise NQD random variables

The authors study the strong convergence for sequences of pairwise negatively quadrant dependent (NQD) random variables under some wide conditions, and present some new theorems on the complete convergence and the strong laws of large numbers. The obtained results extend and improve some theorems in existing literature.     

A remark on estimating the mean of a normal distribution with known coefficient of variation

Let X₁, X₂, …, X_n be iid N(μ, aμ²) (a>0) random variables with an unknown mean μ>0 and known coefficient of variation (CV) √a. The estimation of μ is revisited and it is shown that a modified version of an unbiased estimator of μ [cf. Khan RA. A note on estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1968;63:1039–1041] is more efficient. A certain linear minimum mean square estimator of Gleser and Healy [Estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1976;71:977–981] is also modified and improved. These improved estimators are being compared with the maximum likelihood estimator under squared-error loss function. Based on asymptotic consideration, a large sample confidence interval is also mentioned.     

Confidence interval estimation for the population coefficient of variation using ranked set sampling: a simulation study

In this paper, an evaluation of the performance of several confidence interval estimators of the population coefficient of variation (τ) using ranked set sampling compared to simple random sampling is performed. Two performance measures are used to assess the confidence intervals for τ, namely: width and coverage probabilities. Simulated data were generated from normal, log-normal, skew normal, Gamma, and Weibull distributions with specified population parameters so that the same values of τ are obtained for each distribution, with sample sizes n=15, 20, 25, 50, 100. A real data example representing birth weight of 189 newborns is used for illustration and performance comparison.