Fiducial inference in the classical errors-in-variables model

For the slope parameter of the classical errors-in-variables model, existing interval estimations with finite length will have confidence level equal to zero because of the Gleser–Hwang effect. Especially when the reliability ratio is low and the sample size is small, the Gleser–Hwang effect is so serious that it leads to the very liberal coverages and the unacceptable lengths of the existing confidence intervals. In this paper, we obtain two new fiducial intervals for the slope. One is based on a fiducial generalized pivotal quantity and we prove that this interval has the correct asymptotic coverage. The other fiducial interval is based on the method of the generalized fiducial distribution. We also construct these two fiducial intervals for the other parameters of interest of the classical errors-in-variables model and introduce these intervals to a hybrid model. Then, we compare these two fiducial intervals with the existing intervals in terms of empirical coverage and average length. Simulation results show that the two proposed fiducial intervals have better frequency performance. Finally, we provide a real data example to illustrate our approaches.     

Approximate confidence and tolerance limits for the discrete Pareto distribution for characterizing extremes in count data

Statistical tolerance intervals for discrete distributions are widely employed for assessing the magnitude of discrete characteristics of interest in applications like quality control, environmental monitoring, and the validation of medical devices. For such data problems, characterizing extreme counts or outliers is also of considerable interest. These applications typically use traditional discrete distributions, like the Poisson, binomial, and negative binomial. The discrete Pareto distribution is an alternative yet flexible model for count data that are heavily right‐skewed. Our contribution is the development of statistical tolerance limits for the discrete Pareto distribution as a strategy for characterizing the extremeness of observed counts in the tail. We discuss the coverage probabilities of our procedure in the broader context of known coverage issues for statistical intervals for discrete distributions. We address this issue by applying a bootstrap calibration to the confidence level of the asymptotic confidence interval for the discrete Pareto distribution's parameter. We illustrate our procedure on a dataset involving cyst formation in mice kidneys.     

Beyond point forecasting: Evaluation of alternative prediction intervals for tourist arrivals

This paper evaluates the performances of prediction intervals generated from alternative time series models, in the context of tourism forecasting. The forecasting methods considered include the autoregressive (AR) model, the AR model using the bias-corrected bootstrap, seasonal ARIMA models, innovations state space models for exponential smoothing, and Harvey’s structural time series models. We use thirteen monthly time series for the number of tourist arrivals to Hong Kong and Australia. The mean coverage rates and widths of the alternative prediction intervals are evaluated in an empirical setting. It is found that all models produce satisfactory prediction intervals, except for the autoregressive model. In particular, those based on the bias-corrected bootstrap perform best in general, providing tight intervals with accurate coverage rates, especially when the forecast horizon is long.     

The Rule of Three, its Variants and Extensions

The Rule of Three (R3) states that 3/ n  is an approximate 95% upper limit for the binomial parameter, when there are no events in  n  trials. This rule is based on the one-sided Clopper–Pearson exact limit, but it is shown that none of the other popular frequentist methods lead to it. It can be seen as a special case of a Bayesian R3, but it is shown that among common choices for a non-informative prior, only the Bayes–Laplace and Zellner priors conform with it. R3 has also incorrectly been extended to 3 being a "reasonable" upper limit for the number of events in a future experiment of the same (large) size, when, instead, it applies to the binomial mean. In Bayesian estimation, such a limit should follow from the posterior predictive distribution. This method seems to give more natural results than—though when based on the Bayes–Laplace prior technically converges with—the method of prediction limits, which indicates between 87.5% and 93.75% confidence for this extended R3. These results shed light on R3 in general, suggest an extended Rule of Four for a number of events, provide a unique comparison of Bayesian and frequentist limits, and support the choice of the Bayes–Laplace prior among non-informative contenders.     

Poisson–geometric INAR(1) process for modeling count time series with overdispersion

In this paper, we propose a new first‐order non‐negative integer‐valued autoregressive [INAR(1)] process with Poisson–geometric marginals based on binomial thinning for modeling integer‐valued time series with overdispersion. Also, the new process has, as a particular case, the Poisson INAR(1) and geometric INAR(1) processes. The main properties of the model are derived, such as probability generating function, moments, conditional distribution, higher‐order moments, and jumps. Estimators for the parameters of process are proposed, and their asymptotic properties are established. Some numerical results of the estimators are presented with a discussion of the obtained results. Applications to two real data sets are given to show the potentiality of the new process.     

Maximum likelihood estimation of a binomial proportion using one‐sample misclassified binary data

In this article, we construct two likelihood‐based confidence intervals (CIs) for a binomial proportion parameter using a double‐sampling scheme with misclassified binary data. We utilize an easy‐to‐implement closed‐form algorithm to obtain maximum likelihood estimators of the model parameters by maximizing the full‐likelihood function. The two CIs are a naïve Wald interval and a modified Wald interval. Using simulations, we assess and compare the coverage probabilities and average widths of our two CIs. Finally, we conclude that the modified Wald interval, unlike the naïve Wald interval, produces close‐to‐nominal CIs under various simulations and, thus, is preferred in practice. Utilizing the expressions derived, we also illustrate our two CIs for a binomial proportion parameter using real‐data example.     

Convergence rates of recombining trees for pricing options on stocks under GBM and regime-switching models with known cash dividends

In the literature there appear various kinds of binomial trees for pricing options on stocks under geometric Brownian motions (GBMs) with known cash dividends. The aim of this paper is to compare the performance of the existing binomial trees in aspect of the convergence rates, which are usually used to measure precisely how fast the approximate values converge to the exact one, and to give a theoretical proof of the convergence rates for the interpolation binomial trees which are based on a model that excludes the arbitrage possibilities. Also the paper extends the studies to the regime-switching models with known cash dividend payment.     

On bivariate generalized binomial and negative binomial distributions

The present paper deals with two types of generalized general binomial (binomial or negative binomial) distributions: (i) a univariate general binomial generalized by a bivariate distribution and (ii) a bivariate general binomial generalized by two independent univariate distributions. The probabilities, moments, conditional distributions and regression functions for these distributions are obtained in terms of bipartitional polynomials. Moreover recurrence relations for the probabilities and moments, independent of the bipartitional polynomials, are given. Finally these general results are applied to the (i) Binomial-Bivariate Poisson and (ii) Bivariate Binomial-Poissons distributions.     

Variables sampling plans for the Poisson and the binomial

Variables sampling plans based upon continuous distributions are well known. The usual assumption is that a measurable characteristic associated with a product has a normal distribution, a case which has been treated extensively in the literature. Other continuous distributions, particularly the exponential, have also been used as models. In this paper we discuss variables sampling plans for situations in which the measurable characteristic has either a Poisson or a binomial distribution.     

浅析二项分布、泊松分布和正态分布之间的关系

二项分布、泊松分布和正态分布一直是学习和研究概率统计的基础。在一定条件下,这三个分布之间存在着密切关系。文章通过求极限分布,研究了二项分布与泊松分布、二项分布与正态分布之间的关系,并利用特征函数和分布函数相互唯一确定这一性质,分析了泊松分布和正态分布之间的关系。     

ON THE ANALYSIS OF VARIANCE FOR BETA – BINOMIA

The beta–binomial distribution is reported in literature as a useful generalization of the binomial in case of heterogeneous binomial sampling. An extra model parameter is introduced to accommodate for extra–binomial variation. Some additions to results already available will be given by presenting approximate F–tests for factorial designs, where the response variable is of 0–1 type and sampling is heterogeneous binomial. These tests can be used when sample sizes are large and equal and some degrees of freedom are left from replicates or negligible interactions to estimate the extra model parameter.     

On some properties of the linear function Poisson binomial and negative binomial distributions

Summary A general model in fluctuations of sums of random variables leading, under certain assumptions, to each of the generalized and linear function Poisson, binomial and negative binomial distributions is presented. Moreover the generating functions and the factorial moments of the linear function Poisson, binomial and negative binomial distributions are obtained in close forms and certain distributional properties are discussed.     

An approximation method to derive confidence intervals for quantiles with some applications

Standard jackknife confidence intervals for a quantile Q _y (β) are usually preferred to confidence intervals based on analytical variance estimators due to their operational simplicity. However, the standard jackknife confidence intervals can give undesirable coverage probabilities for small samples sizes and large or small values of β. In this paper confidence intervals for a population quantile based on several existing estimators of a quantile are derived. These intervals are based on an approximation for the cumulative distribution function of a studentized quantile estimator. Confidence intervals are empirically evaluated by using real data and some applications are illustrated. Results derived from simulation studies show that proposed confidence intervals are narrower than confidence intervals based on the standard jackknife technique, which assumes normal approximation. Proposed confidence intervals also achieve coverage probabilities above to their nominal level. This study indicates that the proposed method can be an alternative to the asymptotic confidence intervals, which can be unknown in practice, and the standard jackknife confidence intervals, which can have poor coverage probabilities and give wider intervals.     

A local maximum likelihood estimator for Poisson regression

A local maximum likelihood estimator based on Poisson regression is presented as well as its bias, variance and asymptotic distribution. This semiparametric estimator is intended to be an alternative to the Poisson, negative binomial and zero-inflated Poisson regression models that does not depend on regularity conditions and model specification accuracy. Some simulation results are presented. The use of the local maximum likelihood procedure is illustrated on one example from the literature. This procedure is found to perform well. This research was partially supported by Calouste Gulbenkian Foundation and PRODEP III.     

SEMI-PARAMETRIC ESTIMATION OF HURDLE REGRESSION MODELS WITH AN APPLICATION TO MEDICAID UTILIZATION

This paper develops a semi-parametric estimation method for hurdle (two-part) count regression models. The approach in each stage is based on Laguerre series expansion for the unknown density of the unobserved heterogeneity. The semi-parametric hurdle model nests Poisson and negative binomial hurdle models, which have been used in recent applied literature. The empirical part of the paper evaluates the impact of managed care programmes for Medicaid eligibles on utilization of health-care services using a key utilization variable, the number of doctor and health centre visits. Health status measures and age seem to be more important in determining health-care utilization than other socio-economic and enrollment variables. The semi-parametric approach is particularly useful for the analysis of overdispersed individual level data characterized by a large proportion of non-users, and highly skewed distribution of counts for users. © 1997 John Wiley & Sons, Ltd.     

Opmerkingen en berekeningen betreffende "Poisson-proces met interrupties" van J. C. A. Zaat

Summary
In an earlier paper by J.C. A. ZAAT an interrupted Poisson process was studied. Here we clarify some of the results there obtained. The process is obtained from a stationary Poisson process on the real axis by covering the axis with a sequence of adjoining intervals which have alternatively length a and b, the first left-hand endpoint of an a-interval to the right of 0 being chosen in a stochastic point with If has a rectangular distribution on [0, a + b] the interrupted Poisson process is stationary, but the distribution of the length of an interval between two successive points in the interrupted process is now different for different intervals. For each distribution ofy over [0, a + b] the distribution of the length of the interval between the nth and the (n + I)st point to the right ofO in the interrupted Poisson process tends to the distribution function G(y) as n tends to infinity. Somehow ZAAT based his calculations exclusively on this stationary distribution.     

Zero-inflated count regression models with applications to some examples

In this paper, we employed SAS PROC NLMIXED (Nonlinear mixed model procedure) to analyze three example data having inflated zeros. Examples used are data having covariates and no covariates. The covariates utilized in this article have binary outcomes to simplify our analysis. Of course the analysis can readily be extended to situations with several covariates having multiple levels. Models fitted include the Poisson (P), the negative binomial (NB), the generalized Poisson (GP), and their zero-inflated variants, namely the ZIP, the ZINB and the ZIGP models respectively. Parameter estimates as well as the appropriate goodness-of-fit statistic (the deviance D) in this case are computed and in some cases, the Pearson’s X ² statistic, that is based on the variance of the relevant model distribution is also computed. Also obtained are the expected frequencies for the models and GOF tests are conducted based on the rule established by Lawal (Appl Stat 29:292–298, 1980). Our results extend previous results on the analysis of the chosen data in this example. Further, results obtained are very consistent with previous analyses on the data sets chosen for this article. We also present an hierarchical figure relating all the models employed in this paper. While we do not pretend that the results obtained are entirely new, however, the analyses give opportunities to researchers in the field the much needed means of implementing these models in SAS without having to resort to S-PLUS, R or Stata.     

How to poison Poisson (when approximating binomial tails)*

Summary In these days some sportsmen are given dangerous drugs in order to stimulate them into still higher efforts. The present paper wants to add some poison to the classical approximation of a binomial by a Poisson distribution. The relatively harmless drug of a little extra calculation shall be seen to result in a much better accuracy. This paper reports both on theoretical considerations, following from series expansions of Poisson-type parameters, and on extensive numerical investigations of accuracy by means of an Electrologica X8 computer. The main conclusion is that the probability of at most k successes in n Bernoulli trials with success probability p <.5 can be very closely approximated by the probability of at most k events in a Poisson distribution with expectation not np but (12–2p)n-7kζ= (12–8p)n-k+k/n^np. In the case p > 5 one should make p < .5 by a reversal, i.e. the interchanging of successes and failures, before the application of a Poisson-type approximation (with one exception mentioned in section 4). For the probability of at least j successes one should approximate the complementary probability of at most j—1 successes. After an introductory section 1 and a discussion of measures of accuracy in section 2, the existing Poisson-type approximations are treated in section 3. The fourth section introduces a new parameter and discusses the advantages of reversal in other situations than p > 5. In section 5 a series expansion for the exact Poisson parameter is derived. Two new parameters, among which is ζ, are introduced in section 6, where the series expansions for parameters given in table 4 lead to conclusions about the various approximations, summarized in table 6. The final section 7 gives some numerical results, which confirm to a large extent the conclusions from the asymptotic expansions.     

Some results on discrete a–monotonicity

Some characterizations and preservation properties of a–monotonicity are investigated. In particular it is shown that a mixture of binomial or Poisson distributions preserves the a–monotonicity of the mixing distribution. Also, a characterization of a class of infinitely divisible distributions, within the discrete α–unimodal class, is given. This characterization is the discrete version of that of ALAMATSAZ (1985) and KOTZ and STEUTEL (1988) in the continuous case.     

Seemingly Unrelated Negative Binomial Regression

This paper discusses the specification and estimation of seemingly unrelated multivariate count data models. A new model with negative binomial marginals is proposed. In contrast to a previous model based on the multivariate Poisson distribution, the new model allows for over-dispersion, a phenomenon that is frequently encountered in economic count data. Semi-parametric estimation is possible if some of the assumption of the fully specified model are violated.