SMOOTH TESTS FOR THE BIVARIATE POISSON DISTRIBUTION

A theorem of Rayner & Best (1989) is generalised to permit the construction of smooth tests of goodness of fit without requiring a set of orthonormal functions on the hypothesised distribution. This result is used to construct smooth tests for the bivariate Poisson distribution. The test due to Crockett (1979) is similar to a smooth test that assesses the variance structure under the bivariate Poisson model; the test due to Loukas & Kemp (1986) is related to a smooth test that seeks to detect a particular linear relationship between the variances and covariance under the bivariate Poisson model. Using focused smooth tests may be more informative than using previously suggested tests. The distribution of the Loukas & Kemp (1986) statistic is not well approximated by the x²distribution for larger correlations, and a revised statistic is suggested.     

Sample size calculation for testing non-inferiority and equivalence under Poisson distribution

When counting the number of chemical parts in air pollution studies or when comparing the occurrence of congenital malformations between a uranium mining town and a control population, we often assume Poisson distribution for the number of these rare events. Some discussions on sample size calculation under Poisson model appear elsewhere, but all these focus on the case of testing equality rather than testing equivalence. We discuss sample size and power calculation on the basis of exact distribution under Poisson models for testing non-inferiority and equivalence with respect to the mean incidence rate ratio. On the basis of large sample theory, we further develop an approximate sample size calculation formula using the normal approximation of a proposed test statistic for testing non-inferiority and an approximate power calculation formula for testing equivalence. We find that using these approximation formulae tends to produce an underestimate of the minimum required sample size calculated from using the exact test procedure. On the other hand, we find that the power corresponding to the approximate sample sizes can be actually accurate (with respect to Type I error and power) when we apply the asymptotic test procedure based on the normal distribution. We tabulate in a variety of situations the minimum mean incidence needed in the standard (or the control) population, that can easily be employed to calculate the minimum required sample size from each comparison group for testing non-inferiority and equivalence between two Poisson populations.     

Robust Poisson likelihood estimation for frailty Cox models: A simulation study

Frailty models can be fit as mixed-effects Poisson models after transforming time-to-event data to the Poisson model framework. We assess, through simulations, the robustness of Poisson likelihood estimation for Cox proportional hazards models with log-normal frailties under misspecified frailty distribution. The log-gamma and Laplace distributions were used as true distributions for frailties on a natural log scale. Factors such as the magnitude of heterogeneity, censoring rate, number and sizes of groups were explored. In the simulations, the Poisson modeling approach that assumes log-normally distributed frailties provided accurate estimates of within- and between-group fixed effects even under a misspecified frailty distribution. Non-robust estimation of variance components was observed in the situations of substantial heterogeneity, large event rates, or high data dimensions.     

A score test for overdispersion in Poisson regression based on the generalized Poisson-2 model

Overdispersion is a common phenomenon in Poisson modeling. The generalized Poisson (GP) regression model accommodates both overdispersion and underdispersion in count data modeling, and is an increasingly popular platform for modeling overdispersed count data. The Poisson model is one of the special cases in the collection of models which may be specified by GP regression. Thus, we may derive a test of overdispersion which compares the equi-dispersion Poisson model within the context of the more general GP regression model. The score test has an advantage over the likelihood ratio test (LRT) and over the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis (the Poisson model). Herein, we propose a score test for overdispersion based on the GP model (specifically the GP-2 model) and compare the power of the test with the LRT and Wald tests. A simulation study indicates the proposed score test based on asymptotic standard normal distribution is more appropriate in practical applications.     

Testing overdispersion in the zero-inflated Poisson model

The zero-inflated negative binomial (ZINB) model is used to account for commonly occurring overdispersion detected in data that are initially analyzed under the zero-inflated Poisson (ZIP) model. Tests for overdispersion (Wald test, likelihood ratio test [LRT], and score test) based on ZINB model for use in ZIP regression models have been developed. Due to similarity to the ZINB model, we consider the zero-inflated generalized Poisson (ZIGP) model as an alternate model for overdispersed zero-inflated count data. The score test has an advantage over the LRT and the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis. This paper proposes score tests for overdispersion based on the ZIGP model and illustrates that the derived score statistics are exactly the same as the score statistics under the ZINB model. A simulation study indicates the proposed score statistics are preferred to other tests for higher empirical power. In practice, based on the approximate mean–variance relationship in the data, the ZINB or ZIGP model can be considered, and a formal score test based on asymptotic standard normal distribution can be employed for assessing overdispersion in the ZIP model. We provide an example to illustrate the procedures for data analysis.     

Bayesian inference for Poisson and multinomial log-linear models

Categorical data frequently arise in applications in the Social Sciences. In such applications, the class of log-linear models, based on either a Poisson or (product) multinomial response distribution, is a flexible model class for inference and prediction. In this paper we consider the Bayesian analysis of both Poisson and multinomial log-linear models. It is often convenient to model multinomial or product multinomial data as observations of independent Poisson variables. For multinomial data, Lindley (1964) [20] showed that this approach leads to valid Bayesian posterior inferences when the prior density for the Poisson cell means factorises in a particular way. We develop this result to provide a general framework for the analysis of multinomial or product multinomial data using a Poisson log-linear model. Valid finite population inferences are also available, which can be particularly important in modelling social data. We then focus particular attention on multivariate normal prior distributions for the log-linear model parameters. Here, an improper prior distribution for certain Poisson model parameters is required for valid multinomial analysis, and we derive conditions under which the resulting posterior distribution is proper. We also consider the construction of prior distributions across models, and for model parameters, when uncertainty exists about the appropriate form of the model. We present classes of Poisson and multinomial models, invariant under certain natural groups of permutations of the cells. We demonstrate that, if prior belief concerning the model parameters is also invariant, as is the case in a ‘reference’ analysis, then the choice of prior distribution is considerably restricted. The analysis of multivariate categorical data in the form of a contingency table is considered in detail. We illustrate the methods with two examples.     

An exact Kolmogorov–Smirnov test for the Poisson distribution with unknown mean

We develop an exact Kolmogorov–Smirnov goodness-of-fit test for the Poisson distribution with an unknown mean. This test is conditional, with the test statistic being the maximum absolute difference between the empirical distribution function and its conditional expectation given the sample total. Exact critical values are obtained using a new algorithm. We explore properties of the test, and we illustrate it with three examples. The new test seems to be the first exact Poisson goodness-of-fit test for which critical values are available without simulation or exhaustive enumeration.     

Analysis of discrete data by Conway–Maxwell Poisson distribution

In this paper, we further study the Conway–Maxwell Poisson distribution having one more parameter than the Poisson distribution and compare it with the Poisson distribution with respect to some stochastic orderings used in reliability theory. Likelihood ratio test and the score test are developed to test the importance of this additional parameter. Simulation studies are carried out to examine the performance of the two tests. Two examples are presented, one showing overdispersion and the other showing underdispersion, to illustrate the procedure. It is shown that the COM-Poisson model fits better than the generalized Poisson distribution.     

The Poisson Birnbaum–Saunders model with long-term survivors

In this paper, we propose a cure rate survival model by assuming that the number of competing causes of the event of interest follows the Poisson distribution and the time to event has the Birnbaum–Saunders (BS) distribution. We define the Poisson BS distribution and provide two useful representations for its density function which facilitate to obtain some mathematical properties. Two closed-form expressions for the moments of the new distribution are given. We estimate the parameters of the model with cure rate using maximum likelihood. For different parameter settings, sample sizes and censoring percentages, several simulations are performed. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform a global influence study. We analyse a real data set from the medical area.     

Analysis of survival data by an exponential-generalized Poisson distribution

In this paper, we consider the distribution of life length of a series system with random number of components, say Z. Considering the distribution of Z as generalized Poisson, an exponential-generalized Poisson (EGP) distribution is developed. The generalized Poisson distribution is a generalization of the Poisson distribution having one extra parameter. The structural properties of the resulting distribution are presented and the maximum likelihood estimation of the parameters is investigated. Extensive simulation studies are carried out to study the performance of the estimates. The score test is developed to test the importance of the extra parameter. For illustration, two real data sets are examined and it is shown that the EGP model, presented here, fits better than the exponential–Poisson distribution.     

Proportional hazards models with discrete frailty

We extend proportional hazards frailty models for lifetime data to allow a negative binomial, Poisson, Geometric or other discrete distribution of the frailty variable. This might represent, for example, the unknown number of flaws in an item under test. Zero frailty corresponds to a limited failure model containing a proportion of units that never fail (long-term survivors). Ways of modifying the model to avoid this are discussed. The models are illustrated on a previously published set of data on failures of printed circuit boards and on new data on breaking strengths of samples of cord.     

Parameter Change Test for Poisson Autoregressive Models

In this paper, we consider the problem of testing for a parameter change in Poisson autoregressive models. We suggest two types of cumulative sum (CUSUM) tests, namely, those based on estimates and residuals. We first demonstrate that the conditional maximum likelihood estimator (CMLE) is strongly consistent and asymptotically normal and then construct the CMLE‐based CUSUM test. It is shown that under regularity conditions, its limiting null distribution is a function of independent Brownian bridges. Next, we construct the residual‐based CUSUM test and derive its limiting null distribution. Simulation results are provided for illustration. A real‐data analysis is performed on data for polio incidence and campylobacteriosis infections.     

Some rememarks on locally most powerful unbiased tests for the detection of mixtures of poisson or binomial distributions

In t h i s note mixture models are used to represent overdispersion relative to Poisson or binomial distributions. We flnd a sufflclent condition on the mixing distribution underich the detection of mixture departures from the Poisson or binomial adrnits a locally most powerful unbiased test. The conditions specify plynoria: relations between the variance and mean of Le glxing distribution.     

A class of count models and a new consistent test for the Poisson distribution

We define a class of count distributions which includes the Poisson as well as many alternative count models. Then the empirical probability generating function is utilized to construct a test for the Poisson distribution, which is consistent against this class of alternatives. The limit distribution of the test statistic is derived in case of a general underlying distribution, and efficiency considerations are addressed. A simulation study indicates that the new test is comparable in performance to more complicated omnibus tests.     

Bridging the Gap: A Generalized Stochastic Process for Count Data

The Bernoulli and Poisson processes are two popular discrete count processes; however, both rely on strict assumptions. We instead propose a generalized homogenous count process (which we name the Conway–Maxwell–Poisson or COM-Poisson process) that not only includes the Bernoulli and Poisson processes as special cases, but also serves as a flexible mechanism to describe count processes that approximate data with over- or under-dispersion. We introduce the process and an associated generalized waiting time distribution with several real-data applications to illustrate its flexibility for a variety of data structures. We consider model estimation under different scenarios of data availability, and assess performance through simulated and real datasets. This new generalized process will enable analysts to better model count processes where data dispersion exists in a more accommodating and flexible manner.     

Testing for a finite mixture model with two components

Summary.  We consider a finite mixture model with k components and a kernel distribution from a general one-parameter family. The problem of testing the hypothesis k =2 versus k 3 is studied. There has been no general statistical testing procedure for this problem. We propose a modified likelihood ratio statistic where under the null and the alternative hypotheses the estimates of the parameters are obtained from a modified likelihood function. It is shown that estimators of the support points are consistent. The asymptotic null distribution of the modified likelihood ratio test proposed is derived and found to be relatively simple and easily applied. Simulation studies for the asymptotic modified likelihood ratio test based on finite mixture models with normal, binomial and Poisson kernels suggest that the test proposed performs well. Simulation studies are also conducted for a bootstrap method with normal kernels. An example involving foetal movement data from a medical study illustrates the testing procedure.     

Analyzing clustered count data with a cluster-specific random effect zero-inflated Conway–Maxwell–Poisson distribution

Count data analysis techniques have been developed in biological and medical research areas. In particular, zero-inflated versions of parametric count distributions have been used to model excessive zeros that are often present in these assays. The most common count distributions for analyzing such data are Poisson and negative binomial. However, a Poisson distribution can only handle equidispersed data and a negative binomial distribution can only cope with overdispersion. However, a Conway–Maxwell–Poisson (CMP) distribution [4] can handle a wide range of dispersion. We show, with an illustrative data set on next-generation sequencing of maize hybrids, that both underdispersion and overdispersion can be present in genomic data. Furthermore, the maize data set consists of clustered observations and, therefore, we develop inference procedures for a zero-inflated CMP regression that incorporates a cluster-specific random effect term. Unlike the Gaussian models, the underlying likelihood is computationally challenging. We use a numerical approximation via a Gaussian quadrature to circumvent this issue. A test for checking zero-inflation has also been developed in our setting. Finite sample properties of our estimators and test have been investigated by extensive simulations. Finally, the statistical methodology has been applied to analyze the maize data mentioned before.     

The empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior under Stein's loss function

For the hierarchical Poisson and gamma model, we calculate the Bayes posterior estimator of the parameter of the Poisson distribution under Stein's loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein's Loss (PESL). We also obtain the Bayes posterior estimator of the parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior by two methods. In numerical simulations, we have illustrated: The two inequalities of the Bayes posterior estimators and the PESLs; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we exploit the attendance data on 314 high school juniors from two urban high schools to illustrate our theoretical studies.     

Hidden Markov model for discrete circular–linear wind data time series

ABSTRACT

In this work, we deal with a bivariate time series of wind speed and direction. Our observed data have peculiar features, such as informative missing values, non-reliable measures under a specific condition and interval-censored data, that we take into account in the model specification. We analyse the time series with a non-parametric Bayesian hidden Markov model, introducing a new emission distribution, suitable to model our data, based on the invariant wrapped Poisson, the Poisson and the hurdle density. The model is estimated on simulated datasets and on the real data example that motivated this work.     

New inference procedures for generalized Poisson distributions

A common feature for compound Poisson and Katz distributions is that both families may be viewed as generalizations of the Poisson law. In this paper, we present a unified approach in testing the fit to any distribution belonging to either of these families. The test involves the probability generating function, and it is shown to be consistent under general alternatives. The asymptotic null distribution of the test statistic is obtained, and an effective bootstrap procedure is employed in order to investigate the performance of the proposed test with real and simulated data. Comparisons with classical methods based on the empirical distribution function are also included.