Zero-truncated Poisson Distribution Research Articles

On Shewhart Control Charts for Zero-Truncated Negative Binomial Distributions

<p><span>The negative binomial distribution (NBD) is extensively used for the<br /><span>description of data too heterogeneous to be fitted by Poisson<br /><span>distribution. Observed samples, however may be truncated, in the<br /><span>sense that the number of individuals falling into zero class cannot be<br /><span>determined, or the observational apparatus becomes active when at<br /><span>least one event occurs. Chakraborty and Kakoty (1987) and<br /><span>Chakraborty and Singh (1990) have constructed CUSUM and<br /><span>Shewhart charts for zero-truncated Poisson distribution respectively.<br /><span>Recently, Chakraborty and Khurshid (2011 a, b) have constructed<br /><span>CUSUM charts for zero-truncated binomial distribution and doubly<br /><span>truncated binomial distribution respectively. Apparently, very little<br /><span>work has specifically addressed control charts for the NBD (see, for<br /><span>example, Kaminsky et al., 1992; Ma and Zhang, 1995; Hoffman, 2003;<br /><span>Schwertman. 2005).<br /></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p><p><span><span><span><span><span><span><span><span><span><span><span><span><span><span><span>The purpose of this paper is to construct Shewhart control charts<br /><span>for zero-truncated negative binomial distribution (ZTNBD). Formulae<br /><span>for the Average run length (ARL) of the charts are derived and studied<br /><span>for different values of the parameters of the distribution. OC curves<br /><span>are also drawn.</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><br /><br class="Apple-interchange-newline" /></span></p>

Bootstrapping with R to Determine Variances of Mixture Model Estimates in Predicting Confidence Intervals for Population Sizes

It is not easy to find the variance estimates of mixture model via the theoretical derivation directly. Instead, bootstrapping denoting a resampling technique from an original sample dataset with replacement allocation is used to calculate variances of mixture estimates of zero-truncated Poisson distributions in a prediction of population size and its confidence interval. The application is the estimation of the number of drug (opium) users in Thailand 2007 under surveillance data of counts of treatment episodes in a case. The results indicated that there were 3,262 observed opium cases who received treatments, the estimate of the unobserved number of opium users without receiving any treatment was 3,931, leading to total population size estimate of 7,193 opium users. The 95% confidence intervals as a by-product of bootstrapping were 6,674-7,712 under bootstrap normal base, 6,782-7,761 under bootstrap 95% percentiles, and 6,626-7,605 under bootstrap t intervals. Bootstrapping algorithm with R program is available here.

An All-Stage Truncated Multiple Change Point Model for Software Reliability Assessment

We propose a software reliability growth modeling framework with multiple change point occurrence environment. Especially in our modeling framework, the probability distribution of the initial fault content follows a zero-truncated Poisson distribution. Therefore, our modeling approach in this paper can derive the proper mean time between software failures (MTBF), which is one of the important reliability assessment measures and is not able to derive in the usual nonhomogeneous Poisson process modeling, which is one of the well-known software reliability modeling approach. This paper also show numerical examples of application of our proposed multiple change point model to software reliability assessment by using actual fault counting data.

On Zero-Truncation of Poisson and Poisson-Lindley Distributions and Their Applications

In this paper, a general expression for ther th factorial moment of zero-truncated Poisson-Lindley distribution (ZTPLD) has been obtained and hence the first four moments about origin has been given. A very simple and alternative method for finding moments of ZTPLD has also been suggested. The expression for the moment generating function of ZTPLD has been obtained. Both ZTPD (Zero- truncated Poisson distribution) and ZTPLD have been fitted using maximum likelihood estimate to a number of data- sets from demography, biological sciences and social sciences and it has been observed that in most cases ZTPLD gives much closer fits than ZTPD while in some cases ZTPD gives much closer fits than ZTPLD.

On compounded geometric distributions and their applications

ABSTRACTHere, we introduce two-parameter compounded geometric distributions with monotone failure rates. These distributions are derived by compounding geometric distribution and zero-truncated Poisson distribution. Some statistical and reliability properties of the distributions are investigated. Parameters of the proposed distributions are estimated by the maximum likelihood method as well as through the minimum distance method of estimation. Performance of the estimates by both the methods of estimation is compared based on Monte Carlo simulations. An illustration with Air Crash casualties demonstrates that the distributions can be considered as a suitable model under several real situations.

Public Ideology, Minority Threat, and Felony Collateral Sanctions

Federal and state felony collateral sanctions directly impact the opportunities and resources available to ex-felons and in turn their ability to successfully reenter society after conviction and possible incarceration. Because felony collateral sanctions vary greatly from state to state, we offer a state-level analysis of factors that produce differences in collateral sanctions levels adopted by U.S. states. Religion, political climate, and minority threat explanations have been linked to criminal penalty severity; however, no one to date has applied these explanations to felony collateral sanctions besides felony disenfranchisement at the macro level. With zero-truncated Poisson (ZTP) procedures, this study examines whether religion, political climate, and minority threat explanations result in states’ adopting greater collateral sanctions against convicted felons. Our research reveals that conservative climate, religiosity, and racial threat, but not ethnic threat or punitiveness, significantly affect state-level collateral sanctions. States with large minority and conservative populations are more likely to have more stigmatizing collateral sanction that can affect recidivism. We find that race rather than ethnicity is an important predictor of the collateral sanctions levels that states adopt; however, public ideology in the form of conservatism and religiosity has a greater effect on collateral sanctions. Policy implications are discussed.

Functional linear models for zero-inflated count data with application to modeling hospitalizations in patients on dialysis.

We propose functional linear models for zero-inflated count data with a focus on the functional hurdle and functional zero-inflated Poisson (ZIP) models. Although the hurdle model assumes the counts come from a mixture of a degenerate distribution at zero and a zero-truncated Poisson distribution, the ZIP model considers a mixture of a degenerate distribution at zero and a standard Poisson distribution. We extend the generalized functional linear model framework with a functional predictor and multiple cross-sectional predictors to model counts generated by a mixture distribution. We propose an estimation procedure for functional hurdle and ZIP models, called penalized reconstruction, geared towards error-prone and sparsely observed longitudinal functional predictors. The approach relies on dimension reduction and pooling of information across subjects involving basis expansions and penalized maximum likelihood techniques. The developed functional hurdle model is applied to modeling hospitalizations within the first 2 years from initiation of dialysis, with a high percentage of zeros, in the Comprehensive Dialysis Study participants. Hospitalization counts are modeled as a function of sparse longitudinal measurements of serum albumin concentrations, patient demographics, and comorbidities. Simulation studies are used to study finite sample properties of the proposed method and include comparisons with an adaptation of standard principal components regression.

Weighted hurdle regression method for joint modeling of cardiovascular events likelihood and rate in the US dialysis population.

We propose a new weighted hurdle regression method for modeling count data, with particular interest in modeling cardiovascular events in patients on dialysis. Cardiovascular disease remains one of the leading causes of hospitalization and death in this population. Our aim is to jointly model the relationship/association between covariates and (i) the probability of cardiovascular events, a binary process, and (ii) the rate of events once the realization is positive-when the 'hurdle' is crossed-using a zero-truncated Poisson distribution. When the observation period or follow-up time, from the start of dialysis, varies among individuals, the estimated probability of positive cardiovascular events during the study period will be biased. Furthermore, when the model contains covariates, then the estimated relationship between the covariates and the probability of cardiovascular events will also be biased. These challenges are addressed with the proposed weighted hurdle regression method. Estimation for the weighted hurdle regression model is a weighted likelihood approach, where standard maximum likelihood estimation can be utilized. The method is illustrated with data from the United States Renal Data System. Simulation studies show the ability of proposed method to successfully adjust for differential follow-up times and incorporate the effects of covariates in the weighting.

A Combination of CUSUM Charts for Monitoring a Zero-Inflated Poisson Process

The Zero-inflated Poisson distribution (ZIP) is used to model the defects in processes with a large number of zeros. We propose a control charting procedure using a combination of two cumulative sum (CUSUM) charts to detect increases in the parameters of ZIP process, one is a conforming run length (CRL) CUSUM chart and another is a zero truncated Poisson (ZTP) CUSUM chart. The control limits of the control charts are obtained using both Markov chain-based methods and simulations. Simulation experiments show that the proposed method outperforms an existing method. Finally, a real example is presented.

Common Dolphin (Delphinus delphis) Bycatch in New Zealand Commercial Trawl Fisheries

Marine mammals are regularly reported as bycatch in commercial and artisanal fisheries, but data are often insufficient to allow assessment of these incidental mortalities. Observer coverage of the mackerel trawl fishery in New Zealand waters between 1995 and 2011 allowed evaluation of common dolphin Delphinus delphis bycatch on the North Island west coast, where this species is the most frequently caught cetacean. Observer data were used to develop a statistical model to estimate total captures and explore covariates related to captures. A two-stage Bayesian hurdle model was used, with a logistic generalised linear model predicting whether any common dolphin captures occurred on a given tow of the net, and a zero-truncated Poisson distribution to estimate the number of dolphin captures, given that there was a capture event. Over the 16-year study period, there were 119 common dolphin captures reported on 4299 observed tows. Capture events frequently involved more than one individual, with a maximum of nine common dolphin observed caught in a single tow. There was a peak of 141 estimated common dolphin captures (95% c.i.: 56 to 276; 6.27 captures per 100 tows) in 2002–03, following the marked expansion in annual effort in this fishery to over 2000 tows. Subsequently, the number of captures fluctuated although fishing effort remained relatively high. Of the observed capture events, 60% were during trawls where the top of the net (headline) was <40 m below the surface, and the model determined that this covariate best explained common dolphin captures. Increasing headline depth by 21 m would halve the probability of a dolphin capture event on a tow. While lack of abundance data prevents assessment of the impact of these mortalities on the local common dolphin population, a clear recommendation from this study is the increasing of headline depth to reduce common dolphin captures.

A Bayesian model for censored positive count data in evaluating breast cancer progression

Basic science researchers transplant human cancer tissues from patients with ductal carcinoma in situ (DCIS) to animals and observe the progression of the disease. Successful transplants show invasion of human tissues across mammary ducts in animal fat pads and cause DCIS-like lesions in one or more ducts. In this work, we consider data from a recent publication of breast cancer research where positive counts of affected ducts may be subject to censoring. We fit the data with zero-truncated Poisson (ZTP) models with an informative prior of gamma. Due to the zero-truncation and right censoring, posterior distributions may not be conventional gamma and are estimated through Markov chain Monte Carlo and grid approximation. For each of the two cell lines, we fit a model with group-specific parameters for DCIS subtypes classified by the cell surface biomarkers, and another model with a homogeneous parameter across groups. Models are compared by the Deviance Information Criterion (DIC). For the chosen prior parameter values, Bayes estimates are comparative to the maximum likelihood estimates, and the DIC favors the simpler model in both cell lines.

On Poisson-stopped-sums that are mixed Poisson

Maceda (1948) characterized the mixed Poisson distributions that are Poisson-stopped-sum distributions based on the mixing distribution. In an alternative characterization of the same set of distributions here the Poisson-stopped-sum distributions that are mixed Poisson distributions is proved to be the set of Poisson-stopped-sums of either a mixture of zero-truncated Poisson distributions or a zero-modification of it.

Measurement Error Effect on the Power of Control Chart for the Ratio of Two Poisson Distributions

It is widely recognized that measurement errors often exist in practice and may considerably affect the performance of control charts in some cases. Measurement error variability has uncertainty which may be originated from several sources. Recently, Chakraborty and Khurshid [3] studied the effects of measurement error on control charts for zero-truncated Poisson distribution. In this paper, we study the effect of the two sources of variability on the power characteristics of control charts and obtain the values of average run length (ARL) for the binomial distribution based on the ratio of two Poisson distributions as studied by Sahai and Khurshid [20]. Expression of the power of the control chart under the standardized normal variable for the binomial distribution is also derived when the underlying distribution is the ratio of two Poisson distributions.

Mixture Models for Estimating the Number of Drug Users in Thailand 2005-2007

It is difficult to measure the sizes of illegal drug user populations directly by using the survey method because of many “hidden drug addicts” and the difficulty of receiving a true response. Systematic and routine information on treatment episodes of drug users is adopted to estimate the population size in this study. Mixture models of zero-truncated Poisson distributions using the nonparametric maximum likelihood estimators (NPMLE) by means of capture-recapture repeated count data were used to project the number of drug users. The method was applied to surveillance data of drug users identified by treatment episodes in over 1140 health treatment centers in Thailand from the Bureau of Health Service System Development, Ministry of Public Health. We presented how this mixture model could be utilized to construct the unobserved frequency of drug users with no treatment episode and further estimated the total population size of drug users in the country from 2005 to 2007. The result of simulation was confirmed that mixture model is suitable when population is large. By means of mixture models, the estimations for the number of drug users were fitted with excellent goodness-of-fit values and we were also compared to the conventional Chao estimates. The NPMLE for the total number of drug users in Thailand 2005, 2006, and 2007 were 184,045 (95% CI: 181,297-86,793), 230,665 (95% CI: 226,611-234,719), 299,670 (95% CI: 294,217-305,123), respectively, also 125,265 (95% CI: 123,092-127,142), 166,287 (95% CI: 163,222-169,352), 228,898 (95% CI: 224,766 - 233,030) for the number of methamphetamine (Yaba) users, and 11,559 (95% CI: 10,234-12,884), 11,333 (95% CI: 9276-13,390), 8953 (95% CI: 7878-10,028) for the number of heroin users, respectively. The numbers of marijuana, kratom-plant, opium, and inhalant users were underestimated because their symptoms were mild and not severe enough to remedy in health treatment centers which led to the smaller size of the total number of drug users. The well-estimated sizes of heroin and methamphetamine addicts are high reliable because they are based on clearly evident count with a severe addiction problem to health treatment centers. The estimation by means of mixture models can be recommended to monitor drug demand trend and drug health service routinely; it is easy to calculate via the available programs MIXTP based on request.

A Bayesian zero-truncated approach for analysing capture–recapture count data from classical scrapie surveillance in France

Capture–recapture (CR) methods are used to study populations that are monitored with imperfect observation processes. They have recently been applied to the monitoring of animal diseases to evaluate the number of infected units that remain undetected by the surveillance system. This paper proposes three Bayesian models to estimate the total number of scrapie-infected holdings in France from CR count data obtained from the French classical scrapie surveillance programme. We fitted two zero-truncated Poisson (ZTP) models (with and without holding size as a covariate) and a zero-truncated negative binomial (ZTNB) model to the 2006 national surveillance count dataset. We detected a large amount of heterogeneity in the count data, making the use of the simple ZTP model inappropriate. However, including holding size as a covariate did not bring any significant improvement over the simple ZTP model. The ZTNB model proved to be the best model, giving an estimation of 535 (CI95% 401–796) infected and detectable sheep holdings in 2006, although only 141 were effectively detected, resulting in a holding-level prevalence of 4.4‰ (CI95% 3.2–6.3) and a sensitivity of holding-level surveillance of 26% (CI95% 18–35). The main limitation of the present study was the small amount of data collected during the surveillance programme. It was therefore not possible to build complex models that would allow depicting more accurately the epidemiological and detection processes that generate the surveillance data. We discuss the perspectives of capture–recapture count models in the context of animal disease surveillance.

Use of the Ratio Plot in Capture–Recapture Estimation

Statistical graphics are a fundamental, yet often overlooked, set of components in the repertoire of data analytic tools. Graphs are quick and efficient, yet simple instruments for preliminary exploration of a dataset to understand its structure and to provide insight into influential aspects of inference such as departures from assumptions and latent patterns. In this article, we present and assess a graphical device for choosing a method for estimating population size in capture–recapture studies of closed populations. The basic concept is derived from a homogeneous Poisson distribution where the ratios of neighboring Poisson probabilities multiplied by the value of the larger neighbor count are constant. This property extends to the zero-truncated Poisson distribution, which is of fundamental importance in capture–recapture studies. In practice, however, this distributional property is often violated. The graphical device developed here, the ratio plot, can be used for assessing specific departures from a Poisson distribution. For example, simple contaminations of an otherwise homogeneous Poisson model can be easily detected and a robust estimator for the population size can be suggested. Several robust estimators are developed and a simulation study is provided to give some guidance on which one should be used in practice. More systematic departures can also easily be detected using the ratio plot. In this article, the focus is on Gamma-mixtures of the Poisson distribution that leads to a linear pattern (called structured heterogeneity) in the ratio plot. More generally, the article shows that the ratio plot is monotone for arbitrary mixtures of power series densities. This article has online supplementary materials.

On zero-truncating and mixing Poisson distributions

The distributions that result from zero-truncating mixed Poisson (ZTMP) distributions and those obtained from mixing zero-truncated Poisson (MZTP) distributions are characterised based on their probability generating functions. One consequence is that every ZTMP distribution is an MZTP distribution, but not vice versa. These characterisations also indicate that the size-biased version of a Poisson mixture and, under certain regularity conditions, the shifted version of a Poisson mixture are neither ZTMP distributions nor MZTP distributions.

On zero-truncating and mixing Poisson distributions

The distributions that result from zero-truncating mixed Poisson (ZTMP) distributions and those obtained from mixing zero-truncated Poisson (MZTP) distributions are characterised based on their probability generating functions. One consequence is that every ZTMP distribution is an MZTP distribution, but not vice versa. These characterisations also indicate that the size-biased version of a Poisson mixture and, under certain regularity conditions, the shifted version of a Poisson mixture are neither ZTMP distributions nor MZTP distributions.

A Comparison of Population Size Estimators under the Truncated Count Model With and Without Allowance for Contaminations

The purpose of the study is to estimate the population size under a homogeneous truncated count model and under model contaminations via the Horvitz-Thompson approach on the basis of a count capture-recapture experiment. The proposed estimator is based on a mixture of zero-truncated Poisson distributions. The benefit of using the proposed model is statistical inference of the long-tailed or skewed distributions and the concavity of the likelihood function with strong results available on the nonparametric maximum likelihood estimator (NPMLE). The results of comparisons, for finding the appropriate estimator among McKendrick's, Mantel-Haenszel's, Zelterman's, Chao's, the maximum likelihood, and the proposed methods in a simulation study, reveal that under model contaminations the proposed estimator provides the best choice according to its smallest bias and smallest mean square error for a situation of sufficiently large population sizes and the further results show that the proposed estimator performs well even for a homogeneous situation. The empirical examples, containing the cholera epidemic in India based on homogeneity and the heroin user data in Bangkok 2002 based on heterogeneity, are fitted with an excellent goodness-of-fit of the models and the confidence interval estimations may also be of considerable interest.

Goodness of fit for the zero-truncated Poisson distribution

The zero-truncated Poisson distribution is an important and appropriate model for many applications. Here, we give and assess several tests of fit for this model.