Negative moments of a modified power series distribution and bias of the maximum likelihood estimator

The class of Modified Power Series distributions (MPSD) containing Lagrangian Poisson (LPD) (Consul and Jain, 1973) and Lagrangian binomial distributions (LBD) (Jain and Consul, 1971) was studied by Gupta (1974). We investigate the problem of finding the negative momentsE[X^-r ], of displaced and decapitated Modified Power Series Distributions. We derive the relationship between rand (r-1) negative moments. The negative moments of the decapitated and displaced LPD are obtained. These results are, then, used to find the exact amount of bias in the ML estimators of the parameters in the LPD and the LBD. We have also given the variances of the ML estimator and the minimum variance unbiased estimator of the parameter in the LPD.     

Inflated modified power series distributions with applications

In certain applications involving discrete data, it is sometimes found that X = 0 is observed with a frequency significantly higher than predicted by the assumed model. Zero inflated Poisson, binomial and negative binomial models have been employed in some clinical trials and in some regression analysis problems.

In this paper, we study the zero inflated modified power series distributions (IMPSD) which include among others the generalized Poisson and the generalized negative binomial distributions and hence the Poisson, binomial and negative binomial distributions. The structural properties along with the distribution of the sum of independent IMPSD variables are studied. The maximum likelihood estimation of the parameters of the model is examined and the variance-covariance matrix of the estimators is obtained. Finally, examples are presented for the generalized Poisson distribution to illustrate the results.     

On power and sample size calculations for Wald tests in generalized linear models

A Wald test-based approach for power and sample size calculations has been presented recently for logistic and Poisson regression models using the asymptotic normal distribution of the maximum likelihood estimator, which is applicable to tests of a single parameter. Unlike the previous procedures involving the use of score and likelihood ratio statistics, there is no simple and direct extension of this approach for tests of more than a single parameter. In this article, we present a method for computing sample size and statistical power employing the discrepancy between the noncentral and central chi-square approximations to the distribution of the Wald statistic with unrestricted and restricted parameter estimates, respectively. The distinguishing features of the proposed approach are the accommodation of tests about multiple parameters, the flexibility of covariate configurations and the generality of overall response levels within the framework of generalized linear models. The general procedure is illustrated with some special situations that have motivated this research. Monte Carlo simulation studies are conducted to assess and compare its accuracy with existing approaches under several model specifications and covariate distributions.     

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.     

Sensitivity analysis of reliability functions of the exponential power series lifetime distribution

ABSTRACT

Hazard rate functions are often used in modeling of lifetime data. The Exponential Power Series (EPS) family has a monotone hazard rate function. In this article, the influence of input factors such as time and parameters on the variability of hazard rate function is assessed by local and global sensitivity analysis. Two different indices based on local and global sensitivity indices are presented. The simulation results for two datasets show that the hazard rate functions of the EPS family are sensitive to input parameters. The results also show that the hazard rate function of the EPS family is more sensitive to the exponential distribution than power series distributions.     

Weighted bivariate logarithmic series distributions

In this paper, we are interested in the weighted distributions of a bivariate three parameter logarithmic series distribution studied by Kocherlakota and Kocherlakota (1990). The weighted versions of the model are derived with weight W(x,y) = x^[r] y^[s]. Explicit expressions for the probability mass function and probability generating functions are derived in the case r = s = l. The marginal and conditional distributions are derived in the general case. The maximum likelihood estimation of the parameters, in both two parameter and three parameter cases, is studied. A procedure for computer generation of bivariate data from a discrete distribution is described. This enables us to present two examples, in order to illustrate the methods developed, for finding the maximum likelihood estimates.     

A non-normal class of distribution function for dose-binary response curve*

A non-normal class of distribution (Edgeworth Series distribution) function in three and four parameters has been considered for dose-binary response relationship. This class accounts for the non-normality (expressed in terms of skewness and kurtosis) present in the relationship in addition to the usual location and scale parameters (generally considered by two parameter models). We present the maximum likelihood method of estimation of the parameters and test of probit (normal distribution) hypothesis. Edgeworth Series distribution when fitted to the data of Milicer & Szczotka (1966) showed an excellent closeness to the observed values, significant improvement over probit and logit fit (Aranda-Ordaz, 1981), and better fit compared to Prentice (1976) model.     

Likelihood estimation for longitudinal zero-inflated power series regression models

In this paper, a zero-inflated power series regression model for longitudinal count data with excess zeros is presented. We demonstrate how to calculate the likelihood for such data when it is assumed that the increment in the cumulative total follows a discrete distribution with a location parameter that depends on a linear function of explanatory variables. Simulation studies indicate that this method can provide improvements in obtaining standard errors of the estimates. We also calculate the dispersion index for this model. The influence of a small perturbation of the dispersion index of the zero-inflated model on likelihood displacement is also studied. The zero-inflated negative binomial regression model is illustrated on data regarding joint damage in psoriatic arthritis.     

Bayesian analysis of zero-inflated regression models

In modeling defect counts collected from an established manufacturing processes, there are usually a relatively large number of zeros (non-defects). The commonly used models such as Poisson or Geometric distributions can underestimate the zero-defect probability and hence make it difficult to identify significant covariate effects to improve production quality. This article introduces a flexible class of zero inflated models which includes other familiar models such as the Zero Inflated Poisson (ZIP) models, as special cases. A Bayesian estimation method is developed as an alternative to traditionally used maximum likelihood based methods to analyze such data. Simulation studies show that the proposed method has better finite sample performance than the classical method with tighter interval estimates and better coverage probabilities. A real-life data set is analyzed to illustrate the practicability of the proposed method easily implemented using WinBUGS.     

A simplified estimation procedure based on the EM algorithm for the power series cure rate model

The family of power series cure rate models provides a flexible modeling framework for survival data of populations with a cure fraction. In this work, we present a simplified estimation procedure for the maximum likelihood (ML) approach. ML estimates are obtained via the expectation-maximization (EM) algorithm where the expectation step involves computation of the expected number of concurrent causes for each individual. It has the big advantage that the maximization step can be decomposed into separate maximizations of two lower-dimensional functions of the regression and survival distribution parameters, respectively. Two simulation studies are performed: the first to investigate the accuracy of the estimation procedure for different numbers of covariates and the second to compare our proposal with the direct maximization of the observed log-likelihood function. Finally, we illustrate the technique for parameter estimation on a dataset of survival times for patients with malignant melanoma.     

Serial correlation and distributions on the shere

Estimation and tests for serial correlation in recation and regression models with normal error have been derive from various points of view; for example: Anderson (1948), Durbi for Watson (1950, 1951, 1971), Theil (1965), Durbin (1970), Haq (1970), Kadiyala (1970), Abrahamse & Louter (1971), Levenbac (1972), Berenblut & Webb (1973), Phillips & Harvey (1974), a Sims (1975). In this paper we derive likelihood functions and most powerful tests for serial correclation in Locationa and regression models with arbitrary but specificed error; the methods extend to include the determination of the likelihood for the parameter of the error distribution.

In Section 2, we survey the modthods that have been used in deriving the various tests and estimates in the literature. In Section 2, we introduce the stataistical model that directly describes the error distribution and we obtain the likelihood function for error correlation and determine locally and specifically kost powerful tests for correlation. In Section 3 we consider the case with normal error derive a normal distribution on the sphere by radial projection. The likelihood function and test are then specialized to the case of normal error in Section 4. The computational procedures for the tests and related power functions are examined in Section 5. Power comparisons for the textile data of Theil and Nagar (1961), the consumption data of Kelin (1950), and the plums and the wheat data of Hildreth & Lu (1960) are presented in Section 6, while the likelihood functions for correlation in these data are given in Section 7.     

Effect of length-biased sampling on the modeling error

In statistical data analysis, the choice of an appropriate model is a very important factor. An inappropriate model leads to a different kind of error in the analysis. This error has been called by C. R. Rao as type III error or modeling error as opposed to type I and type II errors in statistical inference.In This paper we Study the relative errors in Incurred by Erroneously Assuming the Distribution of the Family Size N as P(n) While in fact it is the Length-biased (Weighted) Version of P(n).An Analytical Expression for the Relative Error,When the Distribution of N Belongs to the Class of Modified Power Series Distributions, is Derived. More Specifically, the Effect of length-biasing on the Relative Error is Investigated, When N Follows a Generalized Poisson Distribution. These Results are Compared With the Case When N Follows a Poisson Distribution.     

Zero-modified power series distribution and its Hurdle distribution version

This paper presents a new family of distributions for count data, the so called zero-modified power series (ZMPS), which is an extension of the power series (PS) distribution family, whose support starts at zero. This extension consists in modifying the probability of observing zero of each PS distribution, enabling the new zero-modified distribution to appropriately accommodate data which have any amount of zero observations (for instance, zero-inflated or zero-deflated data). The Hurdle distribution version of the ZMPS distribution is presented. PS distributions included in the proposed ZMPS family are the Poisson, Generalized Poisson, Geometric, Binomial, Negative Binomial and Generalized Negative Binomial distributions. The paper also describes the properties and particularities of the new distribution family for count data. The distribution parameters are estimated via maximum likelihood method and the use of the new family is illustrated in three real data sets. We emphasize that the new distribution family can accommodate sets of count data without any previous knowledge on the characteristic of zero-inflation or zero-deflation present in the data.     

Tests for comparing time series of unequal lengths

This paper deals with hypothesis testing for independent time series with unequal length. It proposes a spectral test based on the distance between the periodogram ordinates and a parametric test based on the distance between the parameter estimates of fitted autoregressive moving average models. Both tests are compared with a likelihood ratio test based on the pooled spectra. In all cases, the null hypothesis is that the two series under consideration are generated by the same stochastic process. The performance of the three tests is investigated by a Monte Carlo simulation study.     

Structural inference on the parameter of the rayleigh distribution from doubly censored samples

We consider the problem of statistical inference on the scale parameter of the Rayleigh Distribution from a type II doubly censored sample, using a Structural Inference approach. We derive the Structural Distribution for the scale parameter. The properties of this distribution are used to obtain different inferential statements about the parameter.     

Simplified Likelihood Based Goodness-of-fit Tests for the Weibull Distribution

The aim of this paper is to present new likelihood based goodness-of-fit tests for the two-parameter Weibull distribution. These tests consist in nesting the Weibull distribution in three-parameter generalized Weibull families and testing the value of the third parameter by using the Wald, score, and likelihood ratio procedures. We simplify the usual likelihood based tests by getting rid of the nuisance parameters, using three estimation methods. The proposed tests are not asymptotic. A comprehensive comparison study is presented. Among a large range of possible GOF tests, the best ones are identified. The results depend strongly on the shape of the underlying hazard rate.     

The power series distribution wite unknown truncation parameter: Two sample problem

The uniformly most powerful unbiased tests are formulated for the two sample problem of the power series distribution with unknown truncation parameter.     

Inverse Weibull power series distributions: properties and applications

Inverse Weibull (IW) distribution is one of the widely used probability distributions for nonnegative data modelling, specifically, for describing degradation phenomena of mechanical components. In this paper, by compounding IW and power series distributions we introduce a new lifetime distribution. The compounding procedure follows the same set-up carried out by Adamidis and Loukas [A lifetime distribution with decreasing failure rate. Stat Probab Lett. 1998;39:35–42]. We provide mathematical properties of this new distribution such as moments, estimation by maximum likelihood with censored data, inference for a large sample and the EM algorithm to determine the maximum likelihood estimates of the parameters. Furthermore, we characterize the proposed distributions using a simple relationship between two truncated moments and maximum entropy principle under suitable constraints. Finally, to show the flexibility of this type of distributions, we demonstrate applications of two real data sets.     

Maximum likelihood estimation of the logarithmic series distribution

This paper discusses the maximum likelihood estimation of the parameter of the logarithmic series distribution. The univariate case is treated in Part I, the multivariate case in Part II. A simple numerical estimation procedure is suggested using a fixed point approach. Convergence to the maximum likelihood estimator is shown. In Part III convergence rate is proven to be linear which is also demonstrated through example. In addition, comparisons with Newton’s method and the secant method in the univariate case, and with Newton’s method and the projected gradient method in the multivariate case are provided.     

Exponentiated power Lindley power series class of distributions: Theory and applications

ABSTRACT

We introduce a new four-parameter generalization of the exponentiated power Lindley (EPL) distribution, called the exponentiated power Lindley power series (EPLPS) distribution. The new distribution arises on a latent complementary risks scenario, in which the lifetime associated with a particular risk is not observable; rather, we observe only the minimum lifetime value among all risks. The distribution exhibits a variety of bathtub-shaped hazard rate functions. It contains as particular cases several lifetime distributions. Various properties of the distribution are investigated including closed-form expressions for the density function, cumulative distribution function, survival function, hazard rate function, the rth raw moment, and also the moments of order statistics. Expressions for the Rényi and Shannon entropies are also given. Moreover, we discuss maximum likelihood estimation and provide formulas for the elements of the Fisher information matrix. Finally, two data applications are given showing flexibility and potentiality of the EPLPS distribution.