Two-moment approximations to some null distributions in order-restricted inference: Unequal sample sizes

Bartholomew's statistics for testing homogeneity of normal means with ordered alternatives have null distributions which are mixtures of chi-squared or beta distributions according as the variances are known or not. If the sample sizes are not equal, the mixing coefficients can be difficult to compute. For a simple order and a simple tree ordering, approximations to the significance levels of these tests have been developed which are based on patterns in the weight sets. However, for a moderate or large number of means, these approximations can be tedious to implement. Employing the same approach that was used in the development of these approximations, two-moment chisquared and beta approximations are derived for these significance levels. Approximations are also developed for the testing situation in which the order restriction is the null hypothesis. Numerical studies show that in each of the cases the two-moment approximation is quite satisfactory for most practical purposes.     

Testing a covariance matrix: exact null distribution of its likelihood criterion

For X～ N _p (μ, Σ) testing H _o:Σ = Σ ₀, with Σ ₀ known, relies at present on an approximation of the null-distribution of the likelihood ratio statistic.

We present here the exact null distribution and also its computation, hence providing a precise tool that can be used in small sample cases.     

A monte carlo investigation of a statistic for a bivariate missing data problem

Testing the equal means hypothesis of a bivariate normal distribution with homoscedastic varlates when the data are incomplete is considered. If the correlational parameter, ρ, is known, the well-known theory of the general linear model is easily employed to construct the likelihood ratio test for the two sided alternative. A statistic, T, for the case of ρ unknown is proposed by direct analogy to the likelihood ratio statistic when ρ is known. The null and nonnull distribution of T is investigated by Monte Carlo techniques. It is concluded that T may be compared to the conventional t distribution for testing the null hypothesis and that this procedure results in a substantial increase in power-efficiency over the procedure based on the paired t test which ignores the incomplete data. A Monte Carlo comparison to two statistics proposed by Lin and Stivers (1974) suggests that the test based on T is more conservative than either of their statistics.     

A mixture approximation to the distribution of a weighted sum of chi-squared variables

The problem of Computing the Values of the distribution functions of linear combinations of chi-squared random variables and their percentile points was previously studied by Grad and Solomon (1955), Imhof (1961). Jensen and Solomon (1972). Solomon and Stephens (1977) and others. All the methods available in the literature either demand lengthy computations or are insufficiently accurate. In the present paper a new method is proposed which reduces the amount of needed computations and yet yields sufficiently accurate results.     

Transmuted generalized exponential distribution: A generalization of the exponential distribution with applications to survival data

In this article, we investigate the potential usefulness of the three-parameter transmuted generalized exponential distribution for analyzing lifetime data. We compare it with various generalizations of the two-parameter exponential distribution using maximum likelihood estimation. Some mathematical properties of the new extended model including expressions for the quantile and moments are investigated. We propose a location-scale regression model, based on the log-transmuted generalized exponential distribution. Two applications with real data are given to illustrate the proposed family of lifetime distributions.     

Maximum Likelihood Estimation of a Poisson Parameter in a Model of Equioverlapping Samples:A Simulation Study

In a model of equioverlapping samples maximum likelihood estimation of a Poisson parameter is examined and compared with two linear unbiased estimations by mean squared error. Since a likelihood estimator is not explicitly available in general, a simulation study has been performed and the results are illustrated     

An exact decomposition for a class of exponential dispersion models: Application to sequential testing

We define the Wishart distribution on the cone of positive definite matrices and an exponential distribution on the Lorentz cone as exponential dispersion models. We show that these two distributions possess a property of exact decomposition, and we use this property to solve the following problem: given q samples (y_il,… y_iNj), i = l,…,q, from a N(μ_i,Σ_i,) distribution, test H:Σ₁ = Σ₂ = … = σ_q. Using the exact decomposition property, the classical test statistic for H, involving q parameters p_i = (N_i, - l)/2, i = 1,…,q, is replaced by a sequence of q - l test statistics for the sequence of tests H_i,:σ₁ =σ₂ = … =σ_i given that H_i-1 is true, i = 2,…,q. Each one of these test statistics involves two parameters only, p._i-1 = p₁ + … + p_i-1 and p_i. We also use the exact decomposition property to test equality of the “direction parameters” for q sample points from the exponential distribution on the Lorentz cone. We give a table of critical values for the distribution on the three-dimensional Lorentz cone. Tables of critical values in higher dimensions can easily be computed following the same method as in dimension three.     

Approximations to the powers of the likelihood ratio tests: the loop ordering and slippage alternatives

Approximations to the power functions of the likelihood ratio tests of homogeneity of normal means against the simple loop ordering at slippage alternatives are considered. If a researcher knows which mean is smallest and which is largest, but does not know how the other means are ordered, then a simple loop ordering is appropriate. The accuracy of the several moment approximations are studied for the case of known variances and it is found that for powers in the range typically of interest, the two-moment approximation seems quite adequate. Approximations based on mixtures of noncentral F variables are developed for the case of unknown variances. The critical values of the test statistics are also tabulated for selected levels of significance.     

A multi sample test for the equality of coefficients of variation in normal populations

Two tests are derived for the hypothesis that the coefficients of variation of k normal populations are equal. The k samples may be of unequal size. The first test is the likelihood ratio test with the usual X²-approximation. A simulation study shows that the small sample behaviour under the null hypothesis is unsatisfactory. An alternative test, based on the sample coefficients of variation, appears to have somewhat better properties.     

A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution

Summary.  A useful discrete distribution (the Conway–Maxwell–Poisson distribution) is revived and its statistical and probabilistic properties are introduced and explored. This distribution is a two-parameter extension of the Poisson distribution that generalizes some well-known discrete distributions (Poisson, Bernoulli and geometric). It also leads to the generalization of distributions derived from these discrete distributions (i.e. the binomial and negative binomial distributions). We describe three methods for estimating the parameters of the Conway–Maxwell–Poisson distribution. The first is a fast simple weighted least squares method, which leads to estimates that are sufficiently accurate for practical purposes. The second method, using maximum likelihood, can be used to refine the initial estimates. This method requires iterations and is more computationally intensive. The third estimation method is Bayesian. Using the conjugate prior, the posterior density of the parameters of the Conway–Maxwell–Poisson distribution is easily computed. It is a flexible distribution that can account for overdispersion or underdispersion that is commonly encountered in count data. We also explore two sets of real world data demonstrating the flexibility and elegance of the Conway–Maxwell–Poisson distribution in fitting count data which do not seem to follow the Poisson distribution.     

Sample size selection in clinical trials when population means are subject to a partial order: one-sided ordered alternatives

The statistical methodology under order restriction is very mathematical and complex. Thus, we provide a brief methodological background of order-restricted likelihood ratio tests for the normal theoretical case for the basic understanding of its applications, and relegate more technical details to the appendices. For data analysis, algorithms for computing the order-restricted estimates and computation of p-values are described. A two-step procedure is presented for obtaining the sample size in clinical trials when the minimum power, say 0.80 or 0.90 is specified, and the normal means satisfy an order restriction. Using this approach will result in reduction of 14-24% in the sample size required when one-sided ordered alternatives are used, as illustrated by several examples.     

Tests for the skewness parameter of two-piece double exponential distribution in the presence of nuisance parameters

In this paper, tests for the skewness parameter of the two-piece double exponential distribution are derived when the location parameter is unknown. Classical tests like Neyman structure test and likelihood ratio test (LRT), that are generally used to test hypotheses in the presence of nuisance parameters, are not feasible for this distribution since the exact distributions of the test statistics become very complicated. As an alternative, we identify a set of statistics that are ancillary for the location parameter. When the scale parameter is known, Neyman–Pearson's lemma is used, and when the scale parameter is unknown, the LRT is applied to the joint density function of ancillary statistics, in order to obtain a test for the skewness parameter of the distribution. Test for symmetry of the distribution can be deduced as a special case. It is found that power of the proposed tests for symmetry is only marginally less than the power of corresponding classical optimum tests when the location parameter is known, especially for moderate and large sample sizes.     

A new test for the presence of a normal mixture distribution based on the posterior Bayes factor

We present a new test for the presence of a normal mixture distribution, based on the posterior Bayes factor of Aitkin (1991). The new test has slightly lower power than the likelihood ratio test. It does not require the computation of the MLEs of the parameters or a search for multiple maxima, but requires computations based on classification likelihood assignments of observations to mixture components.     

The power functions of the likelihood ratio tests for a simply ordered trend in normal means

Likelihood ratio tests for the homogeneity of k normal means with the alternative restricted by an increasing trend are considered as well as the likelihood ratio tests of the null hypothesis that the means satisfy the trend. While the work is primarily a survey of results concerning the power functions of these tests, the extensions of some results to the case of not necessarily equal sample sizes are presented. For the case of known or unknown population variances, exact expressions are given for the power functions for k=3,4, and approximations are discussed for larger k. The topics of consistency, bias and monotonicity of the power functions are included. Also, Bartholomew's conjectures concerning minimal and maximal powers are investigated, with results of a new numerical study given.     

A modified Kolmogorov-Smirnov test for a rectangular distribution with unknown parameters: Computation of the distribution of the test statistic

A recursive scheme for the calculation of the distribution of the test statistic of a modified Kolmogorov-Smirnov-test for a rectangular distribution with unknown parameters is given.     

A Discrete Distribution Arising as a Solution of a Linear Difference Equation: Extension of the Non Central Negative Binomial Distribution

This article considers a discrete distribution that arises as the dominant solution of a linear difference equation. Basic properties and various chance mechanisms that lead to this distribution are given. In particular, its formulation as a weighted distribution and a mixed Poisson process are proposed. Parameter estimation by (a) using a combination of observed frequencies and moments and (b) maximum likelihood are examined. An example of goodness of fit is considered.     

A Bayesian predictive approach to determining the number of components in a mixture distribution

This paper describes a Bayesian approach to mixture modelling and a method based on predictive distribution to determine the number of components in the mixtures. The implementation is done through the use of the Gibbs sampler. The method is described through the mixtures of normal and gamma distributions. Analysis is presented in one simulated and one real data example. The Bayesian results are then compared with the likelihood approach for the two examples.     

The power functions of the likelihood ratio tests for a simple tree ordering in normal means: unequal weights

Likelihood ratio tests are considered for two testing situations; testing for the homogeneity of k normal means against the alternative restricted by a simple tree ordering trend and testing the null hypothesis that the means satisfy the trend against all alternatives. Exact expressions are given for the power functions for k = 3 and 4 and unequal sample sizes, both for the case of known and unknown population variances, and approximations are discussed for larger k. Also, Bartholomew’s conjectures concerning minimal and maximal powers are investigated for the case of equal and unequal sample sizes. The power formulas are used to compute powers for a numerical example.     

A comparison of procedures to correct for base-line differences in the analysis of continuous longitudinal data: a case-study

Summary.  The main advantage of longitudinal studies is that they can distinguish changes over time within individuals (longitudinal effects) from differences between subjects at the start of the study (base-line characteristics; cross-sectional effects). Often, especially in observational studies, subjects are very heterogeneous at base-line, and one may want to correct for this, when doing inferences for the longitudinal trends. Three procedures for base-line correction are compared in the context of linear mixed models for continuous longitudinal data. All procedures are illustrated extensively by using data from an experiment which aimed at studying the relationship between the post-operative evolution of the functional status of elderly hip fracture patients and their preoperative neurocognitive status.