Mathematical forms of the distribution of the product of two normal variables

This paper finds the mathematical forms of the distribution of the product where x and x follow a bivariate normal distribution In this paper the distribution when PT0 is expressed as an integral, a new, fundamental result. From this general form, six different cases can be distinguished depending on what is known about the parameters and p. The special cases are Aroian $year:1959 and (6) Additionally, we prove that if and as the distribution of the product approaches the Type III distribution. When p=0# Aroian $year:1959 and Aroian and Meeker $year:1977, give tables for various values of 6., 6 . The results in this paper will be used to provide brief tables for p^O in a separate paper     

A test for bivariate symmetry based on the empirical distribution function

Hollander (1970) proposed a conditionally distribution-free test of bivariate symmetry based on the empirical distribution function. In this paper Hollander’s test statistic is examined In greater detail: in particular; its conditional asymptotic distribution is derived under the null hypothesis as well as under a sequence of local alternatives. Percentage points of the asymptotic distribution are presented; a power comparison between Hollander’s statistic and the likelihood ratio criterion in testing a variant of the sphericity hypothesis in multivariate analysis is made.     

How non-normality affects the quadratic discriminant function

The quadratic discriminant function is commonly used for the two group classification problem when the covariance matrices in the two populations are substantially unequal. This procedure is optimal when both populations are multivariate normal with known means and covariance matrices. This study examined the robustness of the QDF to non-normality. Sampling experiments were conducted to estimate expected actual error rates for the QDF when sampling from a variety of non-normal distributions. Results indicated that the QDF was robust to non-normality except when the distributions were highly skewed, in which case relatively large deviations from optimal were observed. In all cases studied the average probabilities of misclassification were relatively stable while the individual population error rates exhibited considerable variability.     

Subset selection based on likelihood from uniform and related popupulations

Let π_i (i=1,2,…, k) be charceterized by the uniform distribution on (a_i;b_i), where exactly one of a_i and b_i is unknown. With unequal sample sizes, suppose that from the k (>=2) given populations, we wish to select a random-size subset containing the one with the smllest value of θ_i= b_i - a_i. RuleR_i selects π if a likelihood-based k-dimensional confidence region for the unknown (θ₁,… θ_k) contains at least one point having θ_i as its smallest component. A second rule, R , is derived through a likelihood ratio and turns out to be that of Barr and prabhu whenthe sample sizes are equal. Numerical comparisons are made. The results apply to the larger class of densities g ( z ; θ_i) =M(z)Q(θ_i) if a(θ_i) < z <b(θ_i). Extensions to the cases when both a_i and b_i are unknown and when θ_j isof interest are indicated. 1<=j<=k     

Computation of the poisson-inverse gaussian distribution

Recursion relations suitable for rapid computation are derived for the probabilities of the compound Poisson distribution when the compounder is the inverse-Gaussian distribution. Series representation of the probabilities are given. Asymptotic results as well as approximations for probabilities, compared with the exact values, are investigated.     

On multimodal distributions

As a sequel to khinchine's definition of unimodality a multimodal distribution function is defined. A characterization for such a distribution is given. The convolution of two such distributions is studied.     

Qdf misclassification probabilities for known population parameters

Approximated QDF misclassification probabilities have been derived for bivariate normal populations with known parameter values. Tne effect of unequal covariances and population distance on the misclassification probabilities are examined     

Probability density function of quotient of order statistics from the pareto,power and weibull distributions

This paper deals with the probability density functions of quotient of order statistics. We use the Mellin transform technique, to find the distribution of the quotient Z= X/Xwhere X.,X(i < j) are the ith and jth order statistics from the Pareto, Power and Weibull distributions     

Asymptotic distribution of regression type estimators of parameters of stable laws

Asymptotic distributions of regression-type estimators for the parameters of stable distributions am obtained. The asymptotic normalized standard deviations of the estimators are computed for various values of the parameters and various choices of the number of points used in getting the regression estimates.