Likelihood-based tests in zero-inflated power series models

We address the issue of performing testing inference in the class of zero-inflated power series models. These models provide a straightforward way of modelling count data and have been widely used in practical situations. The likelihood ratio, Wald and score statistics provide the basis for testing the parameter of inflation of zeros in this class of models. In this paper, in addition to the well-known test statistics, we also consider the recently proposed gradient statistic. We conduct Monte Carlo simulation experiments to evaluate the finite-sample performance of these tests for testing the parameter of inflation of zeros. The numerical results show that the new gradient test we propose is more reliable in finite samples than the usual likelihood ratio, Wald and score tests. An empirical application to real data is considered for illustrative purposes.     

Minimum variance unbiased estimation for modified power series distribution

In this paper we study the problem of finding the minimum variance unbiased (MVU) estimators of the functions of the para-meters of the modified power series distributions (MPSD). A theorem giving the necessary and sufficient conditions for the existence of the MVU estimators has been proved. Also, the estimators for a number of estimable functions of a parameter are obtained. Two other theorems dealing with the MVU estimation of the left truncated MPSD with unknown truncation point are also given. The particular case of the Lagrangian Poisson, the Lagrangian binomial and the Borel-Tanner distributions are considered and tables are also provided for the MVU estimators for some functions of the parameters. The variances of the estimators are also given for some cases.     

On a data-dependent choice of the tuning parameter appearing in certain goodness-of-fit tests

We propose a data-dependent method for choosing the tuning parameter appearing in many recently developed goodness-of-fit test statistics. The new method, based on the bootstrap, is applicable to a class of distributions for which the null distribution of the test statistic is independent of unknown parameters. No data-dependent choice for this parameter exists in the literature; typically, a fixed value for the parameter is chosen which can perform well for some alternatives, but poorly for others. The performance of the new method is investigated by means of a Monte Carlo study, employing three tests for exponentiality. It is found that the Monte Carlo power of these tests, using the data-dependent choice, compares favourably to the maximum achievable power for the tests calculated over a grid of values of the tuning parameter.     

Computation of the percentage points and the power for the two-sided Kolmogorov-Smirnov one sample test

Two recursive schemes are presented for the calculation of the probabilityP(g(x)≤S _n (x)≤h(x) for allx∈®), whereS _n is the empirical distribution function of a sample from a continuous distribution andh, g are continuous and isotone functions. The results are specialized for the calculation of the distribution and the corresponding percentage points of the test statistic of the two-sided Kolmogorov-Smirnov one sample test. The schemes allow the calculation of the power of the test too. Finally an extensive tabulation of percentage points for the Kolmogorov-Smirnov test is given.     

Correlation between the numbers of two types of children in a family with the mpsd for the family size

The family size (sibship size) N is regarded as an integer-valued random variable having a Modified Power Series distribution (MPSD) of Gupta (1974). The family produces two types of children, with probabilities p and q (p+q =1) . It is proved that the correlation between the numbers B and C of these children is positive or negative according as the function log f(θ) is convex or concave with respect to the function g(θ), (see Section 2). This condition is a simple and a natural extension of the one given by Rao et al (1973). Several examples are discussed to illustrate the result.     

Some factors affecting statistical power of approximate tests in the linear mixed model for longitudinal data

Different longitudinal study designs require different statistical analysis methods and different methods of sample size determination. Statistical power analysis is a flexible approach to sample size determination for longitudinal studies. However, different power analyses are required for different statistical tests which arises from the difference between different statistical methods. In this paper, the simulation-based power calculations of F-tests with Containment, Kenward-Roger or Satterthwaite approximation of degrees of freedom are examined for sample size determination in the context of a special case of linear mixed models (LMMs), which is frequently used in the analysis of longitudinal data. Essentially, the roles of some factors, such as variance–covariance structure of random effects [unstructured UN or factor analytic FA0], autocorrelation structure among errors over time [independent IND, first-order autoregressive AR1 or first-order moving average MA1], parameter estimation methods [maximum likelihood ML and restricted maximum likelihood REML] and iterative algorithms [ridge-stabilized Newton-Raphson and Quasi-Newton] on statistical power of approximate F-tests in the LMM are examined together, which has not been considered previously. The greatest factor affecting statistical power is found to be the variance–covariance structure of random effects in the LMM. It appears that the simulation-based analysis in this study gives an interesting insight into statistical power of approximate F-tests for fixed effects in LMMs for longitudinal data.     

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.