Showing papers on "Poisson distribution published in 1990"

Spline models for observational data

Abstract: This book serves well as an introduction into the more theoretical aspects of the use of spline models. It develops a theory and practice for the estimation of functions from noisy data on functionals. The simplest example is the estimation of a smooth curve, given noisy observations on a finite number of its values. Convergence properties, data based smoothing parameter selection, confidence intervals, and numerical methods are established which are appropriate to a number of problems within this framework. Methods for including side conditions and other prior information in solving ill posed inverse problems are provided. Data which involves samples of random variables with Gaussian, Poisson, binomial, and other distributions are treated in a unified optimization context. Experimental design questions, i.e., which functionals should be observed, are studied in a general context. Extensions to distributed parameter system identification problems are made by considering implicitly defined functionals.

A simple method to calculate the confidence interval of a standardized mortality ratio (SMR)

Abstract: In analyzing standardized mortality ratios (SMRs), it is of interest to calculate a confidence interval for the true SMR. The exact limits of a specific interval can be obtained by means of the Poisson distribution either within an iterative procedure or by one of the tables. The limits can be approximated in using one of various shortcut methods. In this paper, a method is described for calculating the exact limits in a simple and easy way. The method is based on the link between the chi 2 distribution and the Poisson distribution. Only a table of the chi 2 distribution is necessary.

Theoretical and Empirical Advantages of Truncated Count Data Estimators for Analysis of Deer Hunting in California

Abstract: Truncated Poisson and truncated negative binomial count data models, as well as standard count data models, OLS, nonlinear normal, and truncated nonlinear normal MLE were used to estimate demand for deer hunting in California. The truncated count data estimators and their properties are reviewed. A large sample (N = 2223) allowed random segmenting of the data into specification, estimation, and out-of-sample prediction portions. Statistics of interest are therefore unbiased by the specification search, and the prediction results allow comparison of the statistical models' robustness. The new estimators are found to be more appropriate for estimating and predicting demand and social benefits than the alternative estimators based on a variety of criteria.

Poisson Approximation and the Chen-Stein Method

Abstract: The Chen-Stein method of Poisson approximation is a powerful tool for computing an error bound when approximating probabilities using the Poisson distribution. In many cases, this bound may be given in terms of first and second moments alone. We present a background of the method and state some fundamental Poisson approximation theorems. The body of this paper is an illustration, through varied examples, of the wide applicability and utility of the Chen-Stein method. These examples include birthday coincidences, head runs in coin tosses, random graphs, maxima of normal variates and random permutations and mappings. We conclude with an application to molecular biology. The variety of examples presented here does not exhaust the range of possible applications of the Chen-Stein method.

Tests of Hypotheses in Overdispersed Poisson Regression and other Quasi-Likelihood Models

Abstract: Test statistics for evaluating the significance of added variables in a regression equation are developed for mixed Poisson models, where the structural parameter φ that determines the mean/variance relationship var(μ; φ) = μ + φ · μ 2 is estimated by the method of moments and the regression coefficients are estimated by quasi-likelihood. The formulas presented for test statistics and related estimating equations are applicable generally to quasi-likelihood models specified by an arbitrary mean value function μ(x; β), together with a variance function V(μ; φ) that contains one or more unknown parameters. Two versions of the Wald and score tests are investigated—one calculated from the usual model-based covariance matrix whose validity depends on correct specification of the variance function, and another using an “empirical” covariance matrix that has a more general asymptotic justification. Monte Carlo simulations demonstrate that the quasi-likelihood/method of moments (QL/M) procedures yield ap...

Poisson approximation for large deviations

Abstract: Upper and lower bounds are given for P(S ≤ k), 0 ≤ k ≤ ES, where S is a sum of indicator variables with a special structure, which appears, for example, in subgraph counts in random graphs. in typical cases, these bounds are close to the corresponding probabilities for a Poisson distribution with the same mean as S. There are no corresponding general bounds for P(S ≥ k), k > ES, but some partial results are given.

Maximum likelihood inference of phylogenetic trees, with special reference to a poisson process model of dna substitution and to parsimony analyses

Abstract: Maximum likelihood inference is discussed, and some of its advantages and dis- advantages are noted. The application of maximum likelihood inference to phylogenetics is examined, and a simple Poisson process model of DNA substitution is used as one example. Further examples follow from the clarification of implicit models underlying traditional "par- simony" and "compatibility" analyses. From the elucidation of these models and analyses, it is seen that Poisson process analysis gives a statistically consistent estimate of phylogeny, and that parsimony methods do indeed have a maximum likelihood foundation but give potentially incorrect estimates of phylogeny. The maximum likelihood formulation provides a common framework within which these analyses are discussed and compared. (Phylogenetic inference; maximum likelihood inference; Poisson process model; parsimony analysis; DNA substitution.)

The Erdos-Renyi Law in Distribution, for Coin Tossing and Sequence Matching

Abstract: We study approximations to the distribution of counts of matches in the best matching segment of specified length when comparing two long sequences of i.i.d. letters. The key tools used are large-deviation inequalities and the Chen-Stein method of Poisson approximation. The origin of the problem in molecular biology is indicated.

On Arrivals That See Time Averages

Abstract: We investigate when Arrivals See Time Averages ASTA in a stochastic model; i.e., when the stationary distribution of an embedded sequence, obtained by observing a continuous-time stochastic process just prior to the points arrivals of an associated point process, coincides with the stationary distribution of the observed process. We also characterize the relation between the two distributions when ASTA does not hold. We introduce a Lack of Bias Assumption LBA which stipulates that, at any time, the conditional intensity of the point process, given the present state of the observed process, be independent of the state of the observed process. We show that LBA, without the Poisson assumption, is necessary and sufficient for ASTA in a stationary process framework. Consequently, LBA covers known examples of non-Poisson ASTA, such as certain flows in open Jackson queueing networks, as well as the familiar Poisson case PASTA. We also establish results to cover the case in which the process is observed just after the points, e.g., when departures see time averages. Finally, we obtain a new proof of the Arrival Theorem for product-form queueing networks.

How Well Is the Probability of Tumor Cure after Fractionated Irradiation Described by Poisson Statistics

Abstract: The probability of tumor cure in a homogeneous population of tumors exposed to fractionated radiotherapy was modeled using numerical simulations and compared with the predictions of Poisson statistics, assuming exact knowledge of the relevant tumor parameters (clonogen number, radiosensitivity, and growth kinetics). The results show that although Poisson statistics (based on exact knowledge of all parameters) accurately describes the probability of tumor cure when no proliferation occurs during treatment, it underestimates the cure rate when proliferation does occur. In practice, however, the inaccuracy is not likely to be more than about 10%. When the tumor parameters are unknown and are estimated by fitting an empirical Poisson model to tumor-cure data from a homogeneous population of proliferative tumors, the resulting estimates of tumor growth rate and radiosensitivity accurately reflect the true values, but the estimate of initial clonogen number is biased downward. A new formula that is more accurate than Poisson statistics in predicting the probability of tumor cure when proliferation occurs during treatment is discussed.

When does the surplus reach a given target

Abstract: In the classical model of risk theory a martingale method is used to compute the distributions of the first and last passage times (and of their difference) of the surplus process at a given level. As a byproduct a certain probabilistic identity related to Lagrange's formula is derived; furthermore Consul's generalized Poisson distribution is explained.

Generalized Poisson Distributions: Properties and Applications

Abstract: Presents 28 bar diagrams that illustrate the versatility of the generalized Poisson model and discusses stochastic processes leading to the generalized Poisson distribution. Examines theoretical properties that vary in difficulty, includes proofs for numerous theorems, explores confidence intervals

Monitoring Poisson Observations Using Modified Exponentially Weighted Moving Average Control Charts

Abstract: In certain production processes, it is necessary or more convenient to use counts of defects or conformance per unit of measurement to indicate whether a production process is in control or not. Counts of this kind are often well fitted by a Poisson distribution. Three modified exponentially weighted moving average (EWMA) control charts are developed in this paper for monitoring the Poisson counts. The average run length (ARL) and the probability function of the run length of these modified control charts can be computed exactly using results from the Markov Chain theory. These modified control charts are demonstrated to be generally superior than the Shewhart control chart based on ARL consideration. Tables of in-control ARLs of these modified control charts are given to assist the implementation of these modified control charts. The implementation and design of these EWMA control charts are discussed. The use of these modified EWMA control charts is illustrated with an example.

Models for distributions on permutations

Abstract: A parametric distribution on permutations of k objects is derived from gamma random variables. The probability of a permutation is set equal to the probability that k independent gamma random variables with common shape parameter and different scale parameters are ranked according to that permutation. This distribution is motivated by considering a competition in which k players, scoring points according to independent Poisson processes, are ranked according to the time until r points are scored. The distributions obtained in this way include the popular Luce-Plackett and Thurstone-Mosteller-Daniels ranking models. These gamma-based distributions can serve as alternatives to the null ranking model in which all permutations are equally likely. Here, the gamma models are used to estimate the probability distribution of the order of finish in a horse race when only the probability of finishing first is given for each horse. Gamma models with shape parameters larger than 1 are found to be superior to...

Analysis of multi-type recurrent events in longitudinal studies; application to a skin cancer prevention trial

Abstract: We consider the statistical modeling and analysis of replicated multi-type point process data with covariates. Such data arise when heterogeneous subjects experience repeated events or failures which may be of several distinct types. The underlying processes are modeled as nonhomogeneous mixed Poisson processes with random (subject) and fixed (covariate) effects. The method of maximum likelihood is used to obtain estimates and standard errors of the failure rate parameters and regression coefficients. Score tests and likelihood ratio statistics are used for covariate selection. A graphical test of goodness of fit of the selected model is based on generalized residuals. Measures for determining the influence of an individual observation on the estimated regression coefficients and on the score test statistic are developed. An application is described to a large ongoing randomized controlled clinical trial for the efficacy of nutritional supplements of selenium for the prevention of two types of skin cancer.

A Bayesian Approach to the Two-period Style-goods Inventory Problem with Single Replenishment and Heterogeneous Poisson Demands

Abstract: This paper presents a model of the two-period style-goods inventory problem for a firm which stocks many hundreds of distinctive items having heterogeneous Poisson demands. The model uses a Bayesian procedure for forecast and probability revisions based on an aggregation-by-items scheme. These revised forecasts are then incorporated into a model which is used to derive the optimal inventory-stocking policies which maximize expected profit during the season. The model is illustrated using an actual case study of inventory planning for unframed poster art.

A Poisson Probability Model of Entry and Market Structure with an Application to U. S. Industries during 1972-77

Abstract: The performance characteristics of an industry are closely linked to the nature of entry and exit in the industry. If entry barriers are low, the threat of potential entry can effectively constrain the ability of incumbent firms to raise price above the competitive level. On the other hand, as entry barriers rise and the probability of entry diminishes, the potential for monopolistic practices increases. Prior empirical studies of entry have focused mainly on its determinants, emphasizing industry characteristics as entry barriers. Examples include McGuckin [19], Orr [21], McDonald [17], and Duetsch [7], which analyze the number of new entrants, and Berry [4], McDonald [17; 18], and Masson and Shannon [15; 16] which focus on the market share of entering firms. Our study uses the model of Orr, and its later extension by Duetsch, as its initial reference point. Like the Orr-Duetsch studies we estimate a model for entry determinants across industries based on the number of new firms. Our contribution is two-fold. First, we analyze a new sample period, 1972-77. Second, we provide a methodological improvement over the logarithmic regression approaches of Orr and Duestch. Because the observations on entry are count data (non-negative integers), our model is developed from the premise that entry requires a statistical framework based on a discrete probability distribution. To meet this requirement, we specify and estimate an econometric model of entry based on the Poisson distribution. Our methodology is in the spirit of Hausman, Hall, and Griliches [11] who apply the Poisson distribution to count data on patent application across firms. Hausman, Hall and Griliches point out that the Poisson model offers an improved methodology for a wide range of economic applications that feature data in the form of repeated counts. This observation motivated our application of the Poisson distribution to the entry problem. The Poisson approach admits a richer analysis of the entry data than the logarithmic regression approach in two ways. First, the logarithmic specification, while computationally convenient, provides a rather incomplete description of the entry data. The log of entry is only well defined

Analyzing Frequencies of Several Types of Events: A Mixed Multinomial- Poisson Approach

Abstract: A flexible, generalized Poisson model is combined with the multinomial distribution to jointly predict households' choices among types of trips and frequency of trips. The model is compared with conventional Poisson models. The problem of a time-variant mean for frequencies is also addressed, as well as the mean-variance property of the conventional Poisson model that is avoided by use of the generalized formulation. The generalized model is found to outperform the conventional models. Copyright 1990 by MIT Press.

Frequency distribution of lymphatic filariasis microfilariae in human populations: population processes and statistical estimation.

Abstract: This paper uses simple mathematical models and statistical estimation techniques to analyse the frequency distribution of microfilariae (mf) in blood samples from human populations which are endemic for lymphatic filariasis. The theoretical analysis examines the relationship between microfilarial burdens and the prevalence of adult (macrofilarial) worms in the human host population. The main finding is that a large proportion of observed mf-negatives may be ‘true’ zeros, arising from the absence of macrofilarial infections or unmated adult worms, rather than being attributable to the blood sampling process. The corresponding mf distribution should then follow a Poisson mixture, arising from the sampling of mf positives, with an additional proportion of ‘true’ mf-zeros. This hypothesis is supported by analysis of observed Wuchereria bancrofti mf distributions from Southern India, Japan and Fiji, in which zero-truncated Poisson mixtures fit mf-positive counts more effectively than distributions including the observed zeros. The fits of two Poisson mixtures, the negative binomial and the Sichel distribution, are compared. The Sichel provides a slightly better empirical description of the mf density distribution; reasons for this improvement, and a discussion of the relative merits of the two distributions, are presented. The impact on observed mf distributions of increasing blood sampling volume and extraction efficiency are illustrated via a simple model, and directions for future work are identified.

Bayes predictive analysis of a fundamental software reliability model

Abstract: The concepts of Bayes prediction analysis are used to obtain predictive distributions of the next time to failure of software when its past failure behavior is known. The technique is applied to the Jelinski-Moranda software-reliability model, which in turn can show an improved predictive performance for some data sets even when compared with some more sophisticated software-reliability models. A Bayes software-reliability model is presented which can be applied to obtain the next time to failure PDF (probability distribution function) and CDF (cumulative distribution function) for all testing protocols. The number of initial faults and the per-fault failure rate are assumed to be s-independent and Poisson and gamma distributed respectively. For certain data sets, the technique yields better predictions than some alternative methods if the frequential likelihood and U-plot criteria are adopted. >

LAN traffic analysis and workload characterization

Abstract: The results of a LAN traffic analysis are studied in order to characterize the workload in a typical Ethernet environment and to examine the mathematical modeling assumptions that are made regarding packet arrivals, packet size distributions, and user access patterns. These results show that the packet size distribution is application dependent and may not always follow a bimodal pattern. The overall packet arrival distribution is not Poisson, as is generally assumed, and is directly related to the network load. During the heavy load hours, the packet arrival process is strictly Poisson. On the other hand, during the light load periods, the hypothesis that the arrival process is Poisson is statistically rejected. >

a Hydroclimatic Application Strategy for the Poisson Partial Duration Model

Abstract: A strategy for formulating and testing the Poisson partial duration extreme value model is presented. The procedure is demonstrated using recorded Streamflow series from a humid subtropical region of the southern United States. The observed data series are partitioned by climatic causes and tested for both the Poisson assumption and the validity of the exponential as marginal distributions. Several statistical tests are utilized in making these determinations. Some important aspects of the model as applied to humid climates are demonstrated. It was found that a majority of Streamflow series could be represented by the model and that significant differences do exist between the arrival structures of floods resulting from different climatic mechanisms. However, these differences generally do not exist in the distribution of the flood magnitudes. In addition, it is possible that model validity is restricted by drainage basin size.

Poisson approximation and dna sequence matching

Abstract: The Poisson distribution is commonly used to model the number of occurrences of independent rare events. However, many instances arise where dependence exists, for example, in counting the length of long head runs in coin tossing, or matches between two DNA sequences. The Chen-Stein method of Poisson approximation yields bounds on the error incurred when approximating the number of occurrences of possibly dependent events by a Poisson random variable of the same mean. In addition to the problems related to the motivating examples from molecular biology involving runs and matches, the method may be applied to questions as varied as calculating probabilities involving extremes of sequences of random variables and approximating the probability of general birthday coincidences.

Approximation of sums by compound Poisson distributions with respect to stop-loss distances

Abstract: The approximation of sums of independent random variables by compound Poisson distributions with respect to stop-loss distances is investigated. These distances are motivated by risk-theoretic considerations. In contrast to the usual construction of approximating compound Poisson distributions, the method suggested in this paper is to fit several moments. For two moments, this can be achieved by scale transformations. It is shown that the new approximations are more stable and improve the usual approximations by accompanying laws in examples where the probability 1 - pi that the ith summand is zero is not too large. RISK THEORY; INSURANCE MATHEMATICS