Filipe J. Marques

Near-exact distributions for the independence and sphericity likelihood ratio test statistics

In this paper we show how, based on a decomposition of the likelihood ratio test for sphericity into two independent tests and a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates, we may obtain extremely well-fitting near-exact distributions for both test statistics. Since both test statistics have the distribution of the product of independent Beta random variables, it is possible to obtain near-exact distributions for both statistics in the form of Generalized Near-Integer Gamma distributions or mixtures of these distributions. For the independence test statistic, numerical studies and comparisons with asymptotic distributions proposed by other authors show the extremely high accuracy of the near-exact distributions developed as approximations to the exact distribution. Concerning the sphericity test statistic, comparisons with formerly developed near-exact distributions show the advantages of these new near-exact distributions.

The distribution of the product of powers of independent uniform random variables - A simple but useful tool to address and better understand the structure of some distributions

What is the distribution of the product of given powers of independent uniform (0, 1) random variables? Is this distribution useful? Is this distribution commonly used in some contexts? Is this distribution somehow related to the distribution of the product of other random variables? Are there some test statistics with this distribution? This paper will give the answers to the above questions. It will be seen that the answer to the last four questions above is: yes! We will show how particular choices of the numbers of variables involved and their powers will result in interesting and useful distributions and how these distributions may help us to shed some new light on some well-known distributions and also how it may help us to address, in a much simpler way, some distributions usually considered to be rather complicated as is the case with the exact distribution of a number of statistics used in Multivariate Analysis, including some whose exact distribution up until now is not available in a concise and manageable form.

Near-exact Distributions for Certain Likelihood Ratio Test Statistics

In this paper we will show how, using an expansion of a Logbeta distribution as an infinite mixture of Gamma distributions we are able to obtain near-exact distributions for the negative logarithm of the l.r.t. (likelihood ratio test) statistics used in Multivariate Analysis to test the independence of several sets of variables, the equality of several mean vectors, sphericity and the equality of several variance-covariance matrices which will match as many of the exact moments as we wish and for which we will be able to have an a priori upper-bound for the difference between their c.d.f. and the exact c.d.f.. These near-exact distributions also display very good performance, with an agreement with the exact distribution which may virtually be taken as far as we wish and which it is not possible to obtain with the usual asymptotic distributions. Furthermore, based on the results presented it will be easy to build near-exact distributions for any l.r.t. statistics which may be built as the product of the above l.r.t. statistics.

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.

Near-exact distributions for the likelihood ratio test statistic of the multi-sample block-matrix sphericity test

Abstract The multi-sample block-matrix sphericity test and its particular cases have wide applications in different areas of research, as for example to test the error structure in several multivariate linear models. However, the practical implementation of this test has been hindered by difficulties in handling the exact distribution of the associated statistic and the non-availability in the literature of well-fitting asymptotic distributions. We use a decomposition of the null hypothesis into three null hypotheses which will induce a factorization of the likelihood ratio test (l.r.t.) statistic. We then use the induced factorization of the characteristic function (c.f.) of the logarithm of the l.r.t. statistic to obtain very well-fitting and highly manageable near-exact distributions for the l.r.t. statistic of this test and its particular cases. These near-exact distributions will allow for the easy computation of very accurate near-exact quantiles and p -values, enabling this way a more frequent practical use of these tests. A measure of proximity between distributions, based on the corresponding characteristic functions, is used to assess the performance of the near-exact distributions.

On the distribution of linear combinations of independent Gumbel random variables

The distribution of linear combinations of independent Gumbel random variables is of great interest for modeling risk and extremes in the most different areas of application. In this paper we develop near-exact approximations for the distribution of linear combination of independent Gumbel random variables based on a shifted generalized near-integer gamma distribution and on the distribution of the difference of two independent generalized integer gamma distributions. These near-exact distributions are computationally appealing and numerical studies confirm their accuracy, as assessed by a proximity measure used in related studies. We illustrate the proposed approximations on applied problems in networks engineering, computational biology, and flood risk management.

A note on the distribution of the linear combination of independent Gamma random variables

In this paper we show that, when the coefficients are not all positive, the distribution of the linear combination of independent Gamma random variables may be represented in form of an infinite mixture of differences of two independent Gamma random variables. Based on truncations of this mixture two precise approximations are developed. Numerical computational studies are conducted to access the quality of these approximations.

The Multi‐sample Block‐matrix Sphericity Test

The multi‐sample block‐matrix sphericity test and its particular cases have wide applications in several areas. However, the practical implementation of this test has been hindered by difficulties in handling the exact distribution of the associated statistic and the non‐availability in the literature of asymptotic distributions. We use a decomposition of the null hypothesis into three null hypotheses to obtain very well‐fit and highly manageable near‐exact distributions for the likelihood ratio test statistic of this test and its particular cases. These distributions will allow for the easy computation of well‐fit near‐exact quantiles and p‐values.

On the Exact, Asymptotic and Near‐exact Distributions for the Likelihood Ratio Statistics to Test Equality of Several Exponential Distributions

The distribution of the likelihood ratio test statistic to test the equality of several one or two‐parameter Exponential distributions either for censored or non‐censored samples has been studied by several authors. These statistics are of interest in many areas, namely in reliability and lifetime studies. We propose several near‐exact distributions for these statistics, which provide very accurate but yet very manageable approximations to the exact distribution, much adequate for practical purposes.

Near-Exact Distributions for Likelihood Ratio Statistics Used in the Simultaneous Test of Conditions on Mean Vectors and Patterns of Covariance Matrices

The authors address likelihood ratio statistics used to test simultaneously conditions on mean vectors and patterns on covariance matrices. Tests for conditions on mean vectors, assuming or not a given structure for the covariance matrix, are quite common, since they may be easily implemented. But, on the other hand, the practical use of simultaneous tests for conditions on the mean vectors and a given pattern for the covariance matrix is usually hindered by the nonmanageability of the expressions for their exact distribution functions. The authors show the importance of being able to adequately factorize the c.f. of the logarithm of likelihood ratio statistics in order to obtain sharp and highly manageable near-exact distributions, or even the exact distribution in a highly manageable form. The tests considered are the simultaneous tests of equality or nullity of means and circularity, compound symmetry, or sphericity of the covariance matrix. Numerical studies show the high accuracy of the near-exact distributions and their adequacy for cases with very small samples and/or large number of variables. The exact and near-exact quantiles computed show how the common chi-square asymptotic approximation is highly inadequate for situations with small samples or large number of variables.