Constructing copula functions with weighted geometric means

The weighted arithmetic mean of two copulas is a copula. In some cases, geometric and harmonic means also provide copulas. There are copulas specially appropriate to be combined by using weighted geometric means. With this method of construction we combine Farlie–Gumbel–Morgentern and Ali–Mikhail–Haq copulas to obtain families of copulas which can be expressed in terms of double power series. The Gumbel–Barnett copula is also considered and a new copula is proposed, which arises as the first order approximation of the weighted geometric mean of two copulas. Invariance of two multivariate distributions (Cuadras–Augé and Johnson–Kotz) by weighted geometric and arithmetic means is also studied.     

The Multi-Sample Block-Scalar Sphericity Test: Exact and Near-Exact Distributions for Its Likelihood Ratio Test Statistic

In this article the authors show how by adequately decomposing the null hypothesis of the multi-sample block-scalar sphericity test it is possible to obtain the likelihood ratio test statistic as well as a different look over its exact distribution. This enables the construction of well-performing near-exact approximations for the distribution of the test statistic, whose exact distribution is quite elaborate and non-manageable. The near-exact distributions obtained are manageable and perform much better than the available asymptotic distributions, even for small sample sizes, and they show a good asymptotic behavior for increasing sample sizes as well as for increasing number of variables and/or populations involved.     

Study of the quality of several asymptotic and near-exact approximations based on moments for the distribution of the Wilks Lambda statistic

In this paper a measure of proximity of distributions, when moments are known, is proposed. Based on cases where the exact distribution is known, evidence is given that the proposed measure is accurate to evaluate the proximity of quantiles (exact vs. approximated). The measure may be applied to compare asymptotic and near-exact approximations to distributions, in situations where although being known the exact moments, the exact distribution is not known or the expression for its probability density function is not known or too complicated to handle. In this paper the measure is applied to compare newly proposed asymptotic and near-exact approximations to the distribution of the Wilks Lambda statistic when both groups of variables have an odd number of variables. This measure is also applied to the study of several cases of telescopic near-exact approximations to the exact distribution of the Wilks Lambda statistic based on mixtures of generalized near-integer gamma distributions.     

Tail Behavior of the Failure Rate Functions of Mixtures

The tail behavior of the failure rate of mixtures of lifetime distributions is studied. A typical result is that if the failure rate of the strongest component of the mixture decreases to a limit, then the failure rate of the mixture decreases to the same limit. For a class of distributions containing the gamma distributions this result can be improved in the sense that the behavior of the failure rate of the mixture asymptotically mirrors that of the strongest component in whether it decreases or increases to a limit. This revised version was published online in July 2006 with corrections to the Cover Date.     

A hybrid EM/Gauss-Newton algorithm for maximum likelihood in mixture distributions

A faster alternative to the EM algorithm in finite mixture distributions is described, which alternates EM iterations with Gauss-Newton iterations using the observed information matrix. At the expense of modest additional analytical effort in obtaining the observed information, the hybrid algorithm reduces the computing time required and provides asymptotic standard errors at convergence. The algorithm is illustrated on the two-component normal mixture.