Approximate Confidence Limits for a Proportion of the Pólya Distribution

The problem of constructing approximate confidence limits for a proportion parameter of the Pólya distribution is discussed. Three different methods for determining approximate one-sided and two-sided confidence limits for that parameter of the Pólya distribution have been proposed and compared. Particular cases of those confidence bounds are confidence intervals for the parameter of the binomial and the hypergeometric distributions.     

Geiger Counter-Type Pólya–Eggenberger Distributions

Motivated by the paper of Dandekar (1955), a one-urn model with Polya–Eggenberger sampling scheme is developed, which yields a large number of discrete distributions, including the Dandekar's modified binomial distribution, as particular cases. The model is further modified to some new generalized distributions of order k. Some probable applications of these models were discussed in Dandekar (1955) and Feller (1968) in fields of fertility study and radioactivity. It also has applications in premium determination in insurance sector.     

On Markov-Pólya Distribution and the Katz Family of Distributions

This article is concerned with the Markov-Pólya distribution and its links with the Katz family of distributions. The Katz family is defined through a first-order recursion of remarkable form; it (only) covers the Poisson, negative binomial and binomial distributions. The Markov-Pólya distribution arises in the study of certain urn or population models that incorporate (anti)contagion effects. The present work is motivated by questions and applications in actuarial sciences. First, the Markov-Pólya distribution is presented as a claim frequency model. This distribution is then shown to satisfy a Katz-like recursion. As a consequence, a simple recursion is derived for computing a compound sum distribution that generalizes the Panjer algorithm in risk theory. The Katz family is also obtained as a limit of the Markov-Pólya distribution. Finally, an observed frequency of car accidents is fitted by a Markov-Pólya distribution.     

Estimating the Pólya process

Traditionally, a Pólya process is approached from a probability point of view. No prior inference work has been done on them. In this study, we approach the continuous-time Pólya process from an estimation point of view. We construct efficient estimators for the replacement matrix of certain classes of Pólya processes.     

Characterizations of asymptotic distributions of continuous-time Pólya processes

Abstract

We propose an elementary but effective approach to studying a general class of Poissonized tenable and balanced urns on two colors. We characterize the asymptotic behavior of the process via a partial differential equation that governs the process, coupled with the method of moments applied in a bootstrapped manner. We show that the limiting distribution of the process underlying the Bagchi-Pal urn is gamma. We also look into the tenable and balanced processes associated with randomized replacement matrix. Similar results carry over to the process, with minor modifications in the methods of proof, done mutatis mutandis.     

Pólya–Aeppli of Order k Risk Model

In this study, we define the Pólya–Aeppli process of order k as a compound Poisson process with truncated geometric compounding distribution with success probability 1 ? ρ > 0 and investigate some of its basic properties. Using simulation, we provide a comparison between the sample paths of the Pólya–Aeppli process of order k and the Poisson process. Also, we consider a risk model in which the claim counting process {N(t)} is a Pólya-Aeppli process of order k, and call it a Pólya—Aeppli of order k risk model. For the Pólya–Aeppli of order k risk model, we derive the ruin probability and the distribution of the deficit at the time of ruin. We discuss in detail the particular case of exponentially distributed claims and provide simulation results for more general cases.     

Inference for Type II bivariate Pólya–Aeppli distribution

In this article, we discuss the estimation of model parameters of the Type II bivariate Pólya–Aeppli distribution using the method of moments and the maximum likelihood method. We also compare some interval estimation methods. We then carry out a Monte Carlo simulation study to evaluate the performance of the proposed point and interval estimation methods. Finally, we present an example to illustrate all the inferential methods developed here.     

Likelihood inference for Type I bivariate Pólya–Aeppli distribution

This article discusses likelihood inference for the Type I bivariate Pólya–Aeppli distribution. The Type I bivariate Pólya–Aeppli distribution was derived by Minkova and Balakrishnan by using compounding with geometric random variables and the trivariate reduction method. They also discussed the moment estimation of the parameters of the Type I bivariate Pólya–Aeppli distribution. Here, we carry out a simulation study to compare the performance of the developed Maximum Likelihood Estimation (MLE) method with the moment estimation. The obtained results show that, through the MLEs require more computational time compared to the moment estimates (MoM), the MLEs perform better, in most of the settings, than the MoM. Finally, we apply the Type I bivariate Pólya–Aeppli model to a real dataset containing the frequencies of railway accidents in two subsequent six-year periods for the purpose of illustration. We also carry out some hypothesis tests using the Wald test statistic. From these results, we conclude that the two variables belong to the same univariate Pólya–Aeppli distribution, but are correlated.     

Asymptotic Results for a Generalized Pólya Urn with “Multi-Updating” and Applications to Clinical Trials

In this article, a new Pólya urn model is introduced and studied; in particular, a strong law of large numbers and two central limit theorems are proved. This urn generalizes a model studied in Berti et al. (2004 Berti , P. , Pratelli , L. , Rigo , P. ( 2004 ). Limit theorems for a class of identically distributed random variables . Ann. Probab. 32 ( 3A ): 2029 – 2052 . [Google Scholar]), May et al. (2005 May , C. , Paganoni , A. M. , Secchi , P. ( 2005 ). On a two-color generalized Pólya urn . Metron 63 ( 1 ): 115 – 134 . [Google Scholar]), and in Crimaldi (2007 Crimaldi , I. ( 2007 ). Almost sure conditional convergence for a generalized Pólya urn. Submitted. Available at: http://amsacta.cib.unibo.it/archive/00002468  [Google Scholar]), and it has natural applications in clinical trials. Indeed, the model includes both delayed and missing (or null) responses. Moreover, a connection with the conditional identity in distribution of Berti et al. (2004 Berti , P. , Pratelli , L. , Rigo , P. ( 2004 ). Limit theorems for a class of identically distributed random variables . Ann. Probab. 32 ( 3A ): 2029 – 2052 . [Google Scholar]) is given.     

Efficient time/space algorithm to compute rectangular probabilities of multinomial, multivariate hypergeometric and multivariate Pólya distributions

The computation of rectangular probabilities of multivariate discrete integer distributions such as the multinomial, multivariate hypergeometric or multivariate Pólya distributions is of great interest both for statistical applications and for probabilistic modeling purpose. All these distributions are members of a broader family of multivariate discrete integer distributions for which computationaly efficient approximate methods have been proposed for the evaluation of such probabilities, but with no control over their accuracy. Recently, exact algorithms have been proposed for computing such probabilities, but they are either dedicated to a specific distribution or to very specific rectangular probabilities. We propose a new algorithm that allows to perform the computation of arbitrary rectangular probabilities in the most general case. Its accuracy matches or even outperforms the accuracy exact algorithms when the rounding errors are taken into account. In the worst case, its computational cost is the same as the most efficient exact method published so far, and is much lower in many situations of interest. It does not need any additional storage than the one for the parameters of the distribution, which allows to deal with large dimension/large counting parameter applications at no extra memory cost and with an acceptable computation time, which is a major difference with respect to the methods published so far.     

Confidence intervals and statistical power of the ‘Validation’ ratio for surrogate or intermediate endpoints

There is currently much interest in the use of surrogate endpoints in clinical trials and intermediate endpoints in epidemiology. Freedman et al. [Statist. Med. 11 (1992) 167] proposed the use of a validation ratio for judging the evidence of the validity of a surrogate endpoint. The method involves calculation of a confidence interval for the ratio. In this paper, I compare through computer simulations the performance of Fieller's method with the delta method for this calculation. In typical situations, the numerator and denominator of the ratio are highly correlated. I find that the Fieller method is superior to the delta method in coverage properties and in statistical power of the validation test. In addition, the formula for predicting statistical power seems to be much more accurate for the Fieller method than for the delta method. The simulations show that the role of validation analysis is likely to be limited in evaluating the reliability of using surrogate endpoints in clinical trials; however, it is likely to be a useful tool in epidemiology for identifying intermediate endpoints.     

A Comparison of Two Random Number Generators for the Birnbaum–Saunders Distribution

Abstract

The Birnbaum–Saunders distribution was developed to describe fatigue failure lifetimes, however, the distribution has been shown to be applicable for a variety of situations that frequently occur in the engineering sciences. In general, the distribution can be used for situations that involve stochastic wear–out failure. The distribution does not have an exponential family structure, and it is often necessary to use simulation methods to study the properties of statistical inference procedures for this distribution. Two random number generators for the Birnbaum–Saunders distribution have appeared in the literature. The purpose of this article is to present and compare these two random number generators to determine which is more efficient. It is shown that one of these generators is a special case of the other and is simpler and more efficient to use.     

What Difference Does the Dependence Between Durations Make? Insights for Population Studies of Aging

The interpretation of age-specific changes in hazards, relative risks, genetic parameters and other indicators of aging calculated from data on related individuals should take into account the regularities of bivariate selection. Due to such selection the hazard rate calculated for twins who have survived to a certain age may be lower than for singletons, even if marginal chances of survival for all individuals are the same. In a mixed population of relatives the proportion of pairs with closer family links increases with age, even if all marginal individual chances of survival are the same. The proportion of chronic conditions for MZ twins observed in a cross-sectional study may be different from that of DZ twins. The age-dependence of relative risks calculated in genetic-epidemiological studies of twins does not necessarily reflect changes in genetic influence on individual susceptibility to disease and death during the aging process. The age-related changes in heritability of susceptibility estimated in twin studies may have nothing to do with changes in the genetic determination of diseases with age. These issues are illustrated by empirical graphs together with the results of modeling and statistical analysis.     

Confidence regions for the stationary point of a quadratic response surface based on the asymptotic distribution of its MLE

Response surface methodology aims at finding the combination of factor levels which optimizes a response variable. A second order polynomial model is typically employed to make inference on the stationary point of the true response function. A suitable reparametrization of the polynomial model, where the coordinates of the stationary point appear as the parameter of interest, is used to derive unconstrained confidence regions for the stationary point. These regions are based on the asymptotic normal approximation to the sampling distribution of the maximum likelihood estimator of the stationary point. A simulation study is performed to evaluate the coverage probabilities of the proposed confidence regions. Some comparisons with the standard confidence regions due to Box and Hunter are also showed.     

Optimal designs for discriminating between some extensions of the Michaelis–Menten model

In this paper some results on the computation of optimal designs for discriminating between nonlinear models are provided. In particular, some typical deviations of the Michaelis–Menten model are considered. A common deviation of this pharmacokinetic model consists on adding a linear term. If two linear models differ in one parameter the T-optimal design for discriminating between them is c-optimal for estimating the added linear term. This is not the case for nonlinear models.     

Estimation of Pr(X < Y) Using Record Values in the One and Two Parameter Exponential Distributions

We consider the problem of estimating the stress-strength reliability when the available data is in the form of record values. The one parameter and two parameters exponential distribution are considered. In the case of two parameters exponential distributions we considered the case where the location parameter is common and the case where the scale parameter is common. The maximum likelihood estimators and the associated confidence intervals are derived.     

Approximate confidence intervals for the weighted kappa coefficient of a binary diagnostic test subject to a case–control design

Case–control design to assess the accuracy of a binary diagnostic test (BDT) is very frequent in clinical practice. This design consists of applying the diagnostic test to all of the individuals in a sample of those who have the disease and in another sample of those who do not have the disease. The sensitivity of the diagnostic test is estimated from the case sample and the specificity is estimated from the control sample. Another parameter which is used to assess the performance of a BDT is the weighted kappa coefficient. The weighted kappa coefficient depends on the sensitivity and specificity of the diagnostic test, on the disease prevalence and on the weighting index. In this article, confidence intervals are studied for the weighted kappa coefficient subject to a case–control design and a method is proposed to calculate the sample sizes to estimate this parameter. The results obtained were applied to a real example.     

The simplicity of likelihood based inferences for P(X < Y) and for the ratio of means in the exponential model

The profile likelihood of the reliability parameter θ = P(X < Y) or of the ratio of means, when X and Y are independent exponential random variables, has a simple analytical expression and is a powerful tool for making inferences. Inferences about θ can be given in terms of likelihood-confidence intervals with a simple algebraic structure even for small and unequal samples. The case of right censored data can also be handled in a simple way. This is in marked contrast with the complicated expressions that depend on cumbersome numerical calculations of multidimensional integrals required to obtain asymptotic confidence intervals that have been traditionally presented in scientific literature.     

On the use of between–within models to adjust for confounding due to unmeasured cluster-level covariates

Between–within models are generalized linear mixed models (GLMMs) for clustered data that incorporate a random intercept together with fixed effects for within-cluster and between-cluster covariates; the between-cluster covariates represent the cluster means of the within-cluster covariates. One popular use of these models is to adjust for confounding of the effect of within-cluster covariates due to unmeasured between-cluster covariates. Previous research has shown via simulations that using this approach can yield inconsistent estimators. We present theory and simulations as evidence that a primary cause of the inconsistency is heteroscedasticity of the linearized version of the GLMM used for estimation.     

Estimation of P(Y < X) for the Three-Parameter Generalized Exponential Distribution

This article considers the estimation of R = P(Y < X) when X and Y are distributed as two independent three-parameter generalized exponential (GE) random variables with different shape parameters but having the same location and scale parameters. A modified maximum likelihood method and a Bayesian technique are used to estimate R on the basis of independent complete samples. The Bayes estimator cannot be obtained in explicit form, and therefore it has been determined using an importance sampling procedure. An analysis of a real life data set is presented for illustrative purposes.