Federico J. O'Reilly

Asymptotic normality of MRPP statistics from invariance principles of u-statistics

Multi-response permutation procedures (MRPP) were recently introduced to test differences between a priori classified groups of objects ( Mielke, Berry Johnson, 1976; Mielke, 1979 ). The null distributions of the MRPP statistics were initially conjectured to be asymptotically normal for some specified conditions within the setting of a sequence of finite populations due to Madow ( 1948 ). Asymptotic normality of a class of MRPP statistics (under the null hypothesis) is shown in two cases: (i) the setting which considers the populations to be the samples resulting from sequential independent identically distributed (i.i.d.) sampling (sampling from infinite populations) and (ii) the setting of a sequence of increasingly large finite populations (sampling from finite populations). The results are direct applications of the weak convergence of a U-statistic process in the i.i.d. case to a Brownian motion (Bhattacharyya and Sen, 1977) and of the weak convergence of a U-statistic process in the finite populatio...

TESTS OF FIT FOR DISCRETE DISTRIBUTIONS BASED ON THE PROBABILITY GENERATING FUNCTION

The use of the probability generating function in testing the fit of discrete distributions was proposed by Kocherlakota & Kocherlakota (1986), and further studied by Marques and Perez-Abreu (1989). In Rueda et al. (1991), a quadratic statistic to test the fit of a discrete distribution was proposed using the probability generating function and its empirical counterpart. This was illustrated for the Poisson case with known parameter. Here, we deal with some extensions: the Poisson case with unknown parameter and the negative Binomial distribution with known or unknown parameter p. We find the asymptotic distribution of the test statistic in each case, and show with the aid of some Monte Carlo studies the closeness of these asymptotic distributions. A connection is established between this quadratic test and the Cramer von Mises test of fit described in Spinelli (1994) and Spinelli and Stephens (1997), thus providing additional insight into these procedures. Also, a correction is made on the expression of ...

Goodness of fit for the inverse Gaussian distribution

For testing the fit of the inverse Gaussian distribution with unknown parameters, the empirical distribution-function statistic A2 is studied. Two procedures are followed in constructing the test statistic; they yield the same asymptotic distribution. In the first procedure the parameters in the distribution function are directly estimated, and in the second the distribution function is estimated by its Rao-Blackwell distribution estimator. A table is given for the asymptotic critical points of A2 These are shown to depend only on the ratio of the unknown parameters. An analysis is provided of the effect of estimating the ratio to enter the table for A2. This analysis enables the proposal of the complete operating procedure, which is sustained by a Monte Carlo study.

On the Conditional Distribution of Goodness-of-Fit Tests

ABSTRACT This manuscript advocates the use of the conditional distribution of the goodness-of-fit test, given the value of the minimal sufficient statistic for the parameters, in the problem of testing the fit of a distribution known only in its form. In such a setting, since the parameters themselves are not of interest, they are considered nuisance and so conditioning seems to be appropriate. Some comments are made regarding this procedure and emphasis is placed on the fact that with this approach, there is no need for sets of tables but rather for just an algorithm, based on a special simulation which produces the “exact” conditional p-value. So it may be claimed to be an exact level α, finite-n procedure, in the continuous case. It may be used in the discrete case also but the level would be approximate. As an example, the inverse Gaussian is discussed, comparing the results of the proposed procedure with recent work, by means of some simulations, showing that for the alternatives studied, there is an increase of power.

Transforming censored samples for testing fit

One approach to testing the goodness of fit of a completely specified continuous distribution when only a subset of the ordered sample is available is to transform that subset into a complete uniform ordered sample of smaller size. After this, any of the classical tests for uniformity may be used. Two systematic procedures for transforming a subset of the ordered sample are studied, and they are illustrated when the subset results from single censoring of the data on the left or right or at both extremes. The procedures are derived from Rosenblatts transformation and are shown to coincide for Type II singly censored data at one extreme with a procedure proposed earlier by Michael and Schucany (1979). A Monte Carlo power study was conducted to analyze the relative merits of the procedures followed by the Anderson-Darling AZ test for uniformity for some types of censoring.

Exact Conditional Tests and Approximate Bootstrap Tests for the von Mises Distribution

Exact and approximate tests of fit are compared for testing that a given sample comes from the von Mises distribution. For the exact test, Gibbs sampling is used to generate samples from the conditional distribution of sample data, given the values of the sufficient statistics. The samples, called co-sufficient samples, are used to estimate the distribution of Watson’s statistic, and hence to find the exact p-value for the given sample. The test is compared to the approximate test using the parametric bootstrap. Two examples are analyzed, and the p-values of the two tests are compared. When more examples are examined, an unexpectedly high correlation is discovered between the two sets of p-values, suggesting a strong mathematical connection.

A set of independent sequential residuals for the multivariate regression model

Abstract In this paper a set of residuals for the multivariate linear regression model is introduced. These residuals are shown to be independent with known distributions which do not depend on the parameters of the model. Transformations of the mentioned residuals may be used to construct exact α goodness-of-fit tests for the multivariate regression model.

A note on the fit for the levy distribution

The problem of testing the fit of the Levy distribution with unknown scale parameter is addressed. The corresponding empirical process is analysed, and the Cramer-von Mises W2 and the Anderson-Darlings A2 statistics are used. Well known results regarding the relationship between the Levy distribution and the gamma and inverse Gaussian are exploited. Some remarks are made regarding the use of the Rao-Blackwell estimator of the distribution function in the empirical process.

On the mvue for the distribution in the multivariate normal regression

Abstract In the multivariate normal regression setting, the estimability of a distribution is studied generalizing earlier results for the univariate case. The MVUE of an estimable distribution is obtained.

Inference Using Latent Variables for Mixtures of Distributions for Censored Data with Partial Identification

Abstract In this article two methods are proposed to make inferences about the parameters of a finite mixture of distributions in the context of partially identifiable censored data. The first method focuses on a mixture of location and scale models and relies on an asymptotic approximation to a suitably constructed augmented likelihood; the second method provides a full Bayesian analysis of the mixture based on a Gibbs sampler. Both methods make explicit use of latent variables and provide computationally efficient procedures compared to other methods which deal directly with the likelihood of the mixture. This may be crucial if the number of components in the mixture is not small. Our proposals are illustrated on a classical example on failure times for communication devices first studied by Mendenhall and Hader (Mendenhall, W., Hader, R. J. (1958). Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika 45:504–520.). In addition, we study the coverage of the confidence intervals obtained from each of the methods by means of a small simulation exercise.