Negative exponential disparity based family of goodness-of-fit tests for multinomial models

This paper investigates a new family of goodness-of-fit tests based on the negative exponential disparities. This family includes the popular Pearson's chi-square as a member and is a subclass of the general class of disparity tests (Basu and Sarkar, 1994) which also contains the family of power divergence statistics. Pitman efficiency and finite sample power comparisons between different members of this new family are made. Three asymptotic approximations of the exact null distributions of the negative exponential disparity famiiy of tests are discussed. Some numerical results on the small sample perfomance of this family of tests are presented for the symmetric null hypothesis. It is shown that the negative exponential disparity famiiy, Like the power divergence family, produces a new goodness-of-fit test statistic that can be a very attractive alternative to the Pearson's chi-square. Some numerical results suggest that, application of this test statistic, as an alternative to Pearson's chi-square, could be preferable to the I ^2/3 statistic of Cressie and Read (1984) under the use of chi-square critical values.     

A test for lack-of-fit of zero-inflated negative binomial models

When a count data set has excessive zero counts, nonzero counts are overdispersed, and the effect of a continuous covariate might be nonlinear, for analysis a semiparametric zero-inflated negative binomial (ZINB) regression model is proposed. The unspecified smooth functional form for the continuous covariate effect is approximated by a cubic spline. The semiparametric ZINB regression model is fitted by maximizing the likelihood function. The likelihood ratio procedure is used to evaluate the adequacy of a postulated parametric functional form for the continuous covariate effect. An extensive simulation study is conducted to assess the finite-sample performance of the proposed test. The practicality of the proposed methodology is demonstrated with data of a motorcycle survey of traffic regulations conducted in 2007 in Taiwan by the Ministry of Transportation and Communication.     

A Cautionary Note on Beta Families of Distributions and the Aliases Within

In this note, we examine the four parameter beta family of distributions in the context of the beta-normal and beta-logistic distributions. In the process, we highlight the concept of numerical and limiting alias distributions, which in turn relate to numerical instabilities in the numerical maximum likelihood fitting routines for these families of distributions. We conjecture that the numerical issues pertaining to fitting these multiparameter distributions may be more widespread than has originally been reported across several families of distributions.     

The probability contours and a goodness-of-fit test for the singly truncated bivariate normal distribution

The necessary statistic for constructing (1-α)% contour semiellipses of the distribution surface corresponding to the singly truncated bivariate normal is derived and 1t5 percentages tabulated, An approximate goodness-of-fit test which uses the derived statistic is indicated and an example given.     

Entropy-based goodness-of-fit tests for the Pareto I distribution

In this article, having observed the generalized order statistics in a sample, we construct a test for the hypothesis that the underlying distribution is the Pareto I distribution. The Shannon entropy of generalized order statistics is used to test the null hypothesis.     

Penalized likelihood-ratio test for finite mixture models with multinomial observations

Due to the irregularity of finite mixture models, the commonly used likelihood-ratio statistics often have complicated limiting distributions. We propose to add a particular type of penalty function to the log-likelihood function. The resulting penalized likelihood-ratio statistics have simple limiting distributions when applied to finite mixture models with multinomial observations. The method is especially effective in addressing the problems discussed by Chernoff and Lander (1995). The theory developed and simulations conducted show that the penalized likelihood method can give very good results, better than the well-known C(α) procedure, for example. The paper does not, however, fully explore the choice of penalty function and weight. The full potential of the new procedure is to be explored in the future.     

An empirical likelihood goodness-of-fit test for time series

Summary. Standard goodness-of-fit tests for a parametric regression model against a series of nonparametric alternatives are based on residuals arising from a fitted model. When a parametric regression model is compared with a nonparametric model, goodness-of-fit testing can be naturally approached by evaluating the likelihood of the parametric model within a nonparametric framework. We employ the empirical likelihood for an α -mixing process to formulate a test statistic that measures the goodness of fit of a parametric regression model. The technique is based on a comparison with kernel smoothing estimators. The empirical likelihood formulation of the test has two attractive features. One is its automatic consideration of the variation that is associated with the nonparametric fit due to empirical likelihood's ability to Studentize internally. The other is that the asymptotic distribution of the test statistic is free of unknown parameters, avoiding plug-in estimation. We apply the test to a discretized diffusion model which has recently been considered in financial market analysis.     

A new goodness-of-fit test for certain bivariate distributions applicable to traffic accidents

T. Cacoullos and H. Papageorgiou [On some bivariate probability models applicable to traffic accidents and fatalities, Int. Stat. Rev. 48 (1980) 345–356] studied a special class of bivariate discrete distributions appropriate for modeling traffic accidents, and fatalities resulting therefrom. The corresponding random variable may be written as $Z={(N,Y)}^{\prime }$, with $Y={\sum }_{j=1}^N{X}_j$, where ${\left\{X_j\right\}}_{j=1}^N$, are independent copies of a (discrete) random variable $X$, and $N$ is independent of ${\left\{X_j\right\}}_{j=1}^N$, and follows a Poisson law. If $X$ follows a Poisson law (resp. Binomial law), the resulting distribution is termed Poisson–Poisson (resp. Poisson–Binomial). L2-type goodness-of-fit statistics are constructed for the ‘general distribution’ of this kind, where $X$ may be an arbitrary discrete nonnegative random variable. The test statistics utilize a simple characterization involving the corresponding probability generating function, and are shown to be consistent. The proposed procedures are shown to perform satisfactorily in simulated data, while their application to accident data leads to positive conclusions regarding the modeling ability of this class of bivariate distributions.     

Empirical likelihood ratio-based goodness-of-fit test for the logistic distribution

The logistic distribution has been used to model growth curves in survival analysis and biological studies. In this article, we propose a goodness-of-fit test for the logistic distribution based on the empirical likelihood ratio. The test is constructed based on the methodology introduced by Vexler and Gurevich [17 A. Vexler and G. Gurevich, Empirical likelihood ratios applied to goodness-of-fit tests based on sample entropy, Comput. Stat. Data Anal. 54 (2010), pp. 531–545. doi: 10.1016/j.csda.2009.09.025[Crossref], [Web of Science ®] , [Google Scholar]]. In order to compute the test statistic, parameters of the distribution are estimated by the method of maximum likelihood. Power comparisons of the proposed test with some known competing tests are carried out via simulations. Finally, an illustrative example is presented and analyzed.     

A Note on Comparing Several Poisson Means

There has been a renewed interest lately in comparing the means (rates) of several Poisson distributions (processes) due to its wide applicability in various fields, especially in life sciences. In this article we first review the recent developments in the literature, and then propose a parametric bootstrap method which performs as good as, if not better than, the existing methods. Results of our comprehensive simulation study have been provided to compare the relevant methods. Finally these methods have been used to four real-life datasets including a most recent one obtained from a clinical trial on the Intrinsa hormone patch developed by Proctor & Gamble.     

Semiparametric mixture models and repeated measures: the multinomial cut point model

Summary.  Suppose that we have m repeated measures on each subject, and we model the observation vectors with a finite mixture model.  We further assume that the repeated measures are conditionally independent. We present methods to estimate the shape of the component distributions along with various features of the component distributions such as the medians, means and variances. We make no distributional assumptions on the components; indeed, we allow different shapes for different components.     

A modified chi-square test for Bertholon model with censored data

In this work, we propose the construction of a chi-squared goodness-of-fit test in censored data case, for Bertholon model which can analyse various competing risks of failure or death. This test is based on a modification of the Nikulin-Rao-Robson (NRR) statistic proposed by Bagdonavicius and Nikulin (2011a Bagdonavicius, V., Nikulin, M. (2011a). Chi-squared tests for general composite hypotheses from censored samples. Comptes Rendus Mathématiques: Series I 349(3–4):219–223. [Google Scholar], 2011b Bagdonavicius, V., Nikulin, M. (2011b). Chi-squared goodness-of-fit test for right censored data. International Journal of Applied Mathematics and Statistics 24:30–50. [Google Scholar]) for censored data. We applied this test to numerical examples from simulated samples and real data.     

A Jackknifed estimators for the negative binomial regression model

Shrinkage estimator is a commonly applied solution to the general problem caused by multicollinearity. Recently, the ridge regression (RR) estimators for estimating the ridge parameter k in the negative binomial (NB) regression have been proposed. The Jackknifed estimators are obtained to remedy the multicollinearity and reduce the bias. A simulation study is provided to evaluate the performance of estimators. Both mean squared error (MSE) and the percentage relative error (PRE) are considered as the performance criteria. The simulated result indicated that some of proposed Jackknifed estimators should be preferred to the ML method and ridge estimators to reduce MSE and bias.     

Maximum likelihood estimation for arma models in the presence of ARMA errors

An ARMA(p, q) process observed with an ARMA(c, d) error has an ARMA (p + c, k) representation with k = max(c + q, p + d) whose parameters satisfy some nonlinear constraints. Identification of the model is discussed. We develop Newton-Raphson estimators for the ARMA(p + c, k) process which take advantage of the information contained in the nonlinear restrictions. Explicit expressions for the derivatives of the restrictions are derived.     

A goodness-of-fit test for logistic-normal models using nonparametric smoothing method

Logistic-normal models can be applied for analysis of longitudinal binary data. The aim of this article is to propose a goodness-of-fit test using nonparametric smoothing techniques for checking the adequacy of logistic-normal models. Moreover, the leave-one-out cross-validation method for selecting the suitable bandwidth is developed. The quadratic form of the proposed test statistic based on smoothing residuals provides a global measure for checking the model with categorical and continuous covariates. The formulae of expectation and variance of the proposed statistics are derived, and their asymptotic distribution is approximated by a scaled chi-squared distribution. The power performance of the proposed test for detecting the interaction term or the squared term of continuous covariates is examined by simulation studies. A longitudinal dataset is utilized to illustrate the application of the proposed test.     

Parameter estimation in semi-linear models using a maximal invariant likelihood function

In this paper, we consider the problem of estimation of semi-linear regression models. Using invariance arguments, Bhowmik and King [2007. Maximal invariant likelihood based testing of semi-linear models. Statist. Papers 48, 357–383] derived the probability density function of the maximal invariant statistic for the non-linear component of these models. Using this density function as a likelihood function allows us to estimate these models in a two-step process. First the non-linear component parameters are estimated by maximising the maximal invariant likelihood function. Then the non-linear component, with the parameter values replaced by estimates, is treated as a regressor and ordinary least squares is used to estimate the remaining parameters. We report the results of a simulation study conducted to compare the accuracy of this approach with full maximum likelihood and maximum profile-marginal likelihood estimation. We find maximising the maximal invariant likelihood function typically results in less biased and lower variance estimates than those from full maximum likelihood.     

A simulation comparison on spatial Poisson mixed models

The statistical methods for analyzing spatial count data have often been based on random fields so that a latent variable can be used to specify the spatial dependence. In this article, we introduce two frequentist approaches for estimating the parameters of model-based spatial count variables. The comparison has been carried out by a simulation study. The performance is also evaluated using a real dataset and also by the simulation study. The simulation results show that the maximum likelihood estimator appears to be with the better sampling properties.     

One-Sided Estimating and Testing Problems for Scale Models from Grouped Samples

Abstract

This article mainly analyzes estimating and testing problems for scale models from grouped samples. Suppose the support region of a density function, which does not depend on parameters, is divided into some disjoint intervals, grouped samples are the number of observations falling in each intervals respectively. The studying of grouped samples may be dated back to the beginning of the century, in which only one sample location and/or scale models is considered. (Shi, N.-Z., Gao, W., Zhang, B.-X. (2001 Shi, N.-Z., Gao, W. and Zhang, B.-X. 2001. One-sided estimating and testing problems for location models from grouped samples. Comm. Statist.—Simul. Comput, 30(4): 895–898.  [Google Scholar]). One-sided estimating and testing problems for location models from grouped samples. Comm. Statist.—Simul. Comput. 30(4)) had investigated one-sided problems for location models, this article discusses one-sided estimating and testing problems for scale models. Some algorithms for obtaining the maximum likelihood estimates of the parameters subject to order restrictions are proposed.     

Two-stage informative cluster sampling—estimation and prediction with applications for small-area models

This paper considers the effects of informative two-stage cluster sampling on estimation and prediction. The aims of this article are twofold: first to estimate the parameters of the superpopulation model for two-stage cluster sampling from a finite population, when the sampling design for both stages is informative, using maximum likelihood estimation methods based on the sample-likelihood function; secondly to predict the finite population total and to predict the cluster-specific effects and the cluster totals for clusters in the sample and for clusters not in the sample. To achieve this we derive the sample and sample-complement distributions and the moments of the first and second stage measurements. Also we derive the conditional sample and conditional sample-complement distributions and the moments of the cluster-specific effects given the cluster measurements. It should be noted that classical design-based inference that consists of weighting the sample observations by the inverse of sample selection probabilities cannot be applied for the prediction of the cluster-specific effects for clusters not in the sample. Also we give an alternative justification of the Royall [1976. The linear least squares prediction approach to two-stage sampling. Journal of the American Statistical Association 71, 657–664] predictor of the finite population total under two-stage cluster population. Furthermore, small-area models are studied under informative sampling.