Carlos A. Coelho

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 Wrapped Gamma Distribution and Wrapped Sums and Linear Combinations of Independent Gamma and Laplace Distributions

In this paper we first obtain an expression for the probability density function of the wrapped or circular Gamma distribution and then we show how it may be seen, both for integer and non-integer shape parameter, as a mixture of truncated Gamma distributions. Some other properties of the wrapped Gamma distribution are studied and it is shown how this distribution and mixtures of these distributions may be much useful tools in modelling directional data in biology and meteorology. Based on the results obtained, namely the ones concerning mixtures, and on some properties of the distributions of the sum of independent Gamma random variables, the wrapped versions of the distributions of such sums, for both integer and non-integer shape parameters are derived. Also the wrapped sum of independent generalized Laplace distributions is introduced as a particular case of a mixture of wrapped Gamma distributions. Among the particular cases of the distributions introduced there are symmetrical, slightly skewed and highly skewed wrapped distributions as well as the recently introduced wrapped Exponential and Laplace distributions.

A mixture of Generalized Integer Gamma distributions as the exact distribution of the product of an odd number of independent Beta random variables: applications ⁄

Abstract In this paper we show first how the distribution of the logarithm of a random variable with a Beta distribution may be expressed either as a mixture of Gamma distributions or as a mixture of Generalized Integer Gamma (GIG) distributions and then how the exact distribution of the product of an odd number of independent Beta random variables whose first parameter evolves by 1/2 and whose second parameter is the half of an odd integer may be expressed as a mixture of GIG distributions. Some particularities of these mixtures are analysed. The results are then used to obtain the exact distribution of the logarithm of the Wilks Λ statistic to test the independence of two sets of variables, both with an odd number of variables, and the exact distribution of the logarithm of the generalized Wilks Λ statistic to test the independence of several sets of variables, in the case where two or three of them have an odd number of variables. A discussion of relative advantages and disadvantages of the use of the exact versus near-exact distributions is carried out.

Multilingual Document Clustering, Topic Extraction and Data Transformations

This paper describes a statistics-based approach for clustering documents and for extracting cluster topics. Relevant Expressions (REs) are extracted from corpora and used as clustering base features. These features are transformed and then by using an approach based on Principal Components Analysis, a small set of document classification features is obtained. The best number of clusters is found by Model-Based Clustering Analysis. Data transformations to approximate to normal distribution are done and results are discussed. The most important REs are extracted from each cluster and taken as cluster topics.

Learning Verbal Transitivity Using LogLinear Models

In this paper we show how loglinear models can be used to cluster verbs based on their subcategorization preferences. We describe how the information about the phrases or clauses a verb goes with can be computationally learned from an automatically tagged corpus with 9,333,555 words. We will use loglinear modeling to describe the relation between the acquired counts for the part-of-speech tags co-occurring with the verbs on predetermined positions.Based on these results an unsupervised clustering algorithm will be proposed.

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.

A Generalized Integer Gamma Distribution as an Asymptotic Replacement for a Logbeta Distribution & Applications

SYNOPTIC ABSTRACT The use of the concept of near-exact distribution enables us to obtain distributions which are very close to the exact distribution but which are far more manageable. Indeed such near-exact distributions are usually far more accurate than any of the usual asymptotic distributions, being thus much useful in the computation of quantiles and as a tool in testing procedures. The use of the concept of near-exact distributions may be the solution to several approximation problems to distributions. In this paper it is shown how we may asymptotically replace the distribution of a random variable whose exponential has a Beta distribution by a particular Generalized Integer (GI) Gamma distribution. This result is then applied in obtaining the near-exact distribution of the product of an odd number of particular independent Beta random variables. A random variable with a distribution of this type is the generalized Wilks Lambda statistic, when two or more sets with an odd number of variables are considered. From the results obtained a near-exact distribution for the generalized Wilks Lambda is obtained under the form of a particular GI Gamma distribution.

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.