the use of mixed logit models to reflect heterogeneity in capture recapture studies summary we examine issues in estimating population size n with capture recapture models when there is variable catchability among subjects we focus on a logistic normal mixed model for which the logit of the probability of capture is an additive function of a random subject and a fixed sampling occasion parameter when the probability of capture is small or the degree of heterogeneity is large the log likelihood surface is relatively flat and it is difficult to obtain much information about n we also discuss a latent class model and a log linear model that account for heterogeneity and show that the log linear model has greater scope models assuming homogeneity provide much narrower intervals for n but are usually highly overly optimistic the actual coverage probability being much lower than the nominal level

on comparison of mixture models for closed population capture recapture studies summary a mixture model is a natural choice to deal with individual heterogeneity in capture recapture studies pledger 2000 biometrics 56 434 442 2005 biometrics 61 868 876 advertised the use of the two point mixture model dorazio and royle 2003 biometrics 59 351 364 2005 biometrics 61 874 876 suggested that the beta binomial model has advantages the controversy is related to the nonidentifiability of the population size link 2003 biometrics 59 1123 1130 and certain boundary problems the total bias is decomposed into an intrinsic bias an approximation bias and an estimation bias we propose to assess the approximation bias the estimation bias and the variance with the intrinsic bias excluded when comparing different estimators the boundary problems in both models and their impacts are investigated real epidemiological and ecological examples are analyzed

estimating the species accumulation curve using mixtures as a significant tool in ecological studies the species accumulation curve or the collector s curve is the graph of the expected number of detected species as a function of sampling effort the problem of estimating the species accumulation curve based on an empirical data set arising from quadrat sampling is studied in a nonparametric binomial mixture model it will be shown that estimating the species accumulation curve not only is independent of the unknown number of species but also includes estimating the number of species as a limiting case for the purpose of interpolation moment based estimators associated with asymptotic confidence intervals are developed from several points of view a likelihood based procedure is developed for the purpose of extrapolation associated with bootstrap confidence intervals the proposed methods are illustrated by ecological data sets

capture recapture estimation using finite mixtures of arbitrary dimension reversible jump markov chain monte carlo rjmcmc methods are used to fit bayesian capture recapture models incorporating heterogeneity in individuals and samples heterogeneity in capture probabilities comes from finite mixtures and or fixed sample effects allowing for interactions estimation by rjmcmc allows automatic model selection and or model averaging priors on the parameters stabilize the estimates and produce realistic credible intervals for population size for overparameterized models in contrast to likelihood based methods to demonstrate the approach we analyze the standard snowshoe hare and cottontail rabbit data sets from ecology a reliability testing data set

nonidentifiability of population size from capture recapture data with heterogeneous detection probabilities heterogeneity in detection probabilities has long been recognized as problematic in mark recapture studies and numerous models developed to accommodate its effects individual heterogeneity is especially problematic in that reasonable alternative models may predict essentially identical observations from populations of substantially different sizes thus even with very large samples the analyst will not be able to distinguish among reasonable models of heterogeneity even though these yield quite distinct inferences about population size the problem is illustrated with models for closed and open populations

nonparametric mle under two closed capture recapture models with heterogeneity we conduct nonparametric maximum likelihood estimation under two common heterogeneous closed population capture recapture models our models specify mixture models as did previous researchers models which have a common generating distribution say f for the capture probabilities using lindsay and roeder s 1992 journal of the american statistical association 87 785 794 mixture model results and the em algorithm a nonparametric maximum likelihood estimator mle of f for any specified population size n is obtained then the nonparametric mle of the n f pair and thus for n is determined perhaps most importantly since our mle pair maximizes the likelihood under the entire nonparametric probability model it provides an excellent foundation for estimating properties of estimators conducting a goodness of fit test and performing a likelihood ratio test these are illustrated in the paper

a bootstrap revival of some nonparametric distance tests abstract several tests based on the empirical measure have been proposed to test independence of variables goodness of fit equality of distributions rotational invariance and so forth these tests have excellent power properties but critical values are difficult if not impossible to obtain furthermore these tests usually assume that the data are real valued with continuous distributions here critical values are determined by bootstrapping and the resulting tests are shown to have the correct asymptotic level under minimal assumptions for example given data xi x i 1 xi d i 1 n it may be desired to test independence of the d components the proposed test compares the empirical measure and the product of its marginals by taking a supremum over an appropriate vapnik cervonenkis class of sets no assumptions are made on the probability distribution of the data or on the space in which it lives indeed some components may be discrete some continuous and others categorical simil

multilist population estimation with incomplete and partial stratification multilist capture recapture methods are commonly used to estimate the size of elusive populations in many situations lists are stratified by distinguishing features such as age or sex stratification has often been used to reduce biases caused by heterogeneity in the probability of list membership among members of the population however it is increasingly common to find lists that are structurally not active in all strata we develop a general method to deal with cases when not all lists are active in all strata using an expectation maximization em algorithm we use a flexible log linear modeling framework that allows for list dependencies and differential probabilities of ascertainment in each list finally we apply our method of estimating population size to two examples

the geometry of mixture likelihoods a general theory in this paper certain fundamental properties of the maximum likelihood estimator of a mixing distribution are shown to be geometric properties of the likelihood set the existence support size likelihood equations and uniqueness of the estimator are revealed to be directly related to the properties of the convex hull of the likelihood set and the support hyperplanes of that hull it is shown using geometric techniques that the estimator exists under quite general conditions with a support size no larger than the number of distinct observations analysis of the convex dual of the likelihood set leads to a dual maximization problem a convergent algorithm is described the defining equations for the estimator are compared with the usual parametric likelihood equations for finite mixtures sufficient conditions for uniqueness are given part ii will deal with a special theory for exponential family mixtures

spatially explicit maximum likelihood methods for capture recapture studies live trapping capture recapture studies of animal populations with fixed trap locations inevitably have a spatial component animals close to traps are more likely to be caught than those far away this is not addressed in conventional closed population estimates of abundance and without the spatial component rigorous estimates of density cannot be obtained we propose new flexible capture recapture models that use the capture locations to estimate animal locations and spatially referenced capture probability the models are likelihood based and hence allow use of akaike s information criterion or other likelihood based methods of model selection density is an explicit parameter and the evaluation of its dependence on spatial or temporal covariates is therefore straightforward additional nonspatial variation in capture probability may be modeled as in conventional capture recapture the method is tested by simulation using a model in which capture probability depends only on location relative to traps point estimators are found to be unbiased and standard error estimators almost unbiased the method is used to estimate the density of red eyed vireos vireo olivaceus from mist netting data from the patuxent research refuge maryland u s a estimates agree well with those from an existing spatially explicit method based on inverse prediction a variety of additional spatially explicit models are fitted these include models with temporal stratification behavioral response and heterogeneous animal home ranges

non linearity measures for a class of siso non linear systems a new measure of non linearity that is captured by two linear systems is proposed to quantify the size of the non linearity in the input output behaviour of siso non linear systems as such the controller design for the non linear system is simplified to the design of a controller for these two linear systems it is also shown that when hammerstein and wiener models are involved the controller design for these two structures are equivalent if the linear dynamic term is stable two examples a numerical example and a non linear system characterized by a time delay and a variable gain are used to demonstrate these concepts

mixture models for estimating the size of a closed population when capture rates vary among individuals nous developpons une parametrisation du melange de lois beta et binomiale qui fournit un cadre inferentiel approprie pour estimer l effectif d une population fermee lorsque les probabilites de capture ou de detection varient entre individus nous ajustons trois classes de modeles de melange beta binomiale logistique normale et modele a classes latentes d une part a des donnees de recapture de lievres d amerique pour en estimer l abondance et d autre part a des comptages d especes d oiseaux pour estimer la richesse specifique dans les deux jeux de donnees les taux de detection varient davantage entre individus lievres ou especes qu entre occasions ou lieux de capture les estimateurs de l effectif de la population et de la richesse specifique sont sensibles aux hypotheses faites sur la distribution sous jacente des probabilites individuelles de detection nous montrons par simulations que les outils habituels d evaluation de la qualite du modele comme la deviance ne sont pas fiables pour choisir entre des modeles de melanges celui qui fournit une inference correcte de l effectif de la population lorsqu elles existent des connaissances a priori sur les causes d heterogeneite des taux de detection doivent etre utilisees pour aider a choisir quelle classe de modeles de melanges utiliser pour l inference

the minimum capture proportion for reliable estimation in capture recapture models summary we derive estimates of the minimum capture proportion required to obtain a reliable estimate of the population size for several continuous and discrete time capture recapture models the models con sidered are mo mt mb m i mu and mtb m the notation of otis et al 1978 wildlife monograph 62 1 135 numerical results with simulation studies are given and two real examples for the model mh are also considered potential applications of these results are suggested

open capture recapture models with heterogeneity ii jolly seber model summary estimation of abundance is important in both open and closed population capture recapture analysis but unmodeled heterogeneity of capture probability leads to negative bias in abundance estimates this article defines and develops a suite of open population capture recapture models using finite mixtures to model heterogeneity of capture and survival probabilities model comparisons and parameter estimation use likelihood based methods a real example is analyzed and simulations are used to check the main features of the heterogeneous models especially the quality of estimation of abundance survival recruitment and turnover the two major advances in this article are the provision of realistic abundance estimates that take account of heterogenetiy of capture and an appraisal of the amount of overestimation of survival arising from conditioning on the first capture when heterogeneity of survival is present

on population size estimators in the poisson mixture model summary estimating population sizes via capture recapture experiments has enormous applications the poisson mixture model can be adopted for those applications with a single list in which individuals appear one or more times we compare several nonparametric estimators including the chao estimator the zelterman estimator two jackknife estimators and the bootstrap estimator the target parameter of the chao estimator is a lower bound of the population size those of the other four estimators are not lower bounds and they may produce lower confidence limits for the population size with poor coverage probabilities a simulation study is reported and two examples are investigated

residual diagnostics for mixture models abstract a sample is commonly modeled by a mixture distribution if the observations follow a common distribution but the parameter of interest differs between observations for example we observe the lengths but not the ages of a sample offish it may be reasonable to assume that length is normally distributed about an unknown mean that depends on the age of the fish provided there is more than one age class in the sample then the data are distributed as a mixture of normals in this article we assume that the data are a random sample from a mixture of exponential family distributions and that for each observation the parameter of interest is sampled independently from an unknown mixing distribution q the adequacy of a fitted mixture model can be assessed by examining residuals based on the ratio of the observed to expected fit residuals based on the homogeneity model in which q is a one point distribution display a convexity property when the data follow a mixture model this becomes the basis fo

nonparametric estimation of species richness two procedures the jackknife and the bootstrap are discussed as methods for estimating the number of species by the sampling of quadrats explicit formulas for both procedures are presented and evaluated under a model with a random distribution of individuals the jackknife and bootstrap are shown to reduce the bias although they underestimate the actual number of species if there is a large number of rare species and the number of quadrats sampled is small when a small number of quadrats is sampled the jackknife is shown to give better estimates when the number of quadrats is large the jackknife tends to overestimate the number of species and the bootstrap performs better

universality in two dimensional cellular structures evolving by cell division and disappearance the dynamics of two dimensional cellular networks is written in terms of coupled population equations which describe how the population of s sided cells is affected by cell division and disappearance in these equations the effect of the rest of the foam on the disappearing or dividing cell is treated as a local mean field under not too restrictive conditions the equilibrium distribution p s of cells satisfies a linear difference equation of order two or higher the population equations are asymptotically integrable the asymptotic integrability implies a universal distribution p s approximately cs kzs for large values of s which is also the boltzmann distribution associated with the maximum entropy inference asymptotic integrability of the population equations is absent in a global mean field approximation the importance of short range topological information to control the evolution of foams is thus confirmed

estimating the number of faults in a system abstract we consider a reliability system with an unknown number of faults which is observed for a fixed length of time failures are assumed to follow an exponential distribution with an unknown mean and we concentrate on estimation of the number of faults v the likelihood function considered as a function of v has its maximum at one of the boundary points the observed number of faults or infinity with substantial probability likelihood interval estimates and improved point estimates are discussed the harmonic mean of the endpoints of the likelihood interval turns out to be a reasonable point estimate we comment on some extensions of this problem suggested in the software reliability literature

a class of latent markov models for capture recapture data allowing for time heterogeneity and behavior effects we propose an extension of the latent class model for the analysis of capture recapture data which allows us to take into account the effect of a capture on the behavior of a subject with respect to future captures the approach is based on the assumption that the variable indexing the latent class of a subject follows a markov chain with transition probabilities depending on the previous capture history several constraints are allowed on these transition probabilities and on the parameters of the conditional distribution of the capture configuration given the latent process we also allow for the presence of discrete explanatory variables which may affect the parameters of the latent process to estimate the resulting models we rely on the conditional maximum likelihood approach and for this aim we outline an em algorithm we also give some simple rules for point and interval estimation of the population size the approach is illustrated by applying it to two data sets concerning small mammal populations

on identifiability in capture recapture models we study the issue of identifiability of mixture models in the context of capture recapture abundance estimation for closed populations such models are used to take account of individual heterogeneity in capture probabilities but their validity was recently questioned by link 2003 biometrics 59 1123 1130 on the basis of their nonidentifiability we give a general criterion for identifiability of the mixing distribution and apply it to establish identifiability within families of mixing distributions that are commonly used in this context including finite and beta mixtures our analysis covers binomial and geometrically distributed outcomes in an example we highlight the difference between the identifiability issue considered here and that in classical binomial mixture models

mixture regression models for closed population capture recapture data in capture recapture studies the use of individual covariates has been recommended to get stable population estimates however some residual heterogeneity might still exist and ignoring such heterogeneity could lead to underestimating the population size n in this work we explore two new models with capture probabilities depending on both covariates and unobserved random effects to estimate the size of a population inference techniques including horvitz thompson estimate and confidence intervals for the population size are derived the selection of a particular model is carried out using the akaike information criterion aic first we extend the random effect model of darroch et al 1993 journal of american statistical association 88 1137 1148 to handle unit level covariates and discuss its limitations the second approach is a generalization of the traditional zero truncated binomial model that includes a random effect to account for an unobserved heterogeneity this approach provides useful tools for inference about n since key quantities such as moments likelihood functions and estimates of n and their standard errors have closed form expressions several models for the unobserved heterogeneity are available and the marginal capture probability is expressed using the logit and the complementary log log link functions the sensitivity of the inference to the specification of a model is also investigated through simulations a numerical example is presented we compare the performance of the proposed estimator with that obtained under model mh of huggins 1989 biometrika 76 130 140

equivalence of truncated count mixture distributions and mixtures of truncated count distributions summary this article is about modeling count data with zero truncation a parametric count density family is considered the truncated mixture of densities from this family is different from the mixture of truncated densities from the same family whereas the former model is more natural to formulate and to interpret the latter model is theoretically easier to treat it is shown that for any mixing distribution leading to a truncated mixture a usually different mixing distribution can be found so that the associated mixture of truncated densities equals the truncated mixture and vice versa this implies that the likelihood surfaces for both situations agree and in this sense both models are equivalent zero truncated count data models are used frequently in the capture recapture setting to estimate population size and it can be shown that the two horvitz thompson estimators associated with the two models agree in particular it is possible to achieve strong results for mixtures of truncated poisson densities including reliable global construction of the unique npmle nonparametric maximum likelihood estimator of the mixing distribution implying a unique estimator for the population size the benefit of these results lies in the fact that it is valid to work with the mixture of truncated count densities which is less appealing for the practitioner but theoretically easier mixtures of truncated count densities form a convex linear model for which a developed theory exists including global maximum likelihood theory as well as algorithmic approaches once the problem has been solved in this class it might readily be transformed back to the original problem by means of an explicitly given mapping applications of these ideas are given particularly in the case of the truncated poisson family

unified maximum likelihood estimates for closed capture recapture models using mixtures summary agresti 1994 biometrics50 494 500 and norris and pollock 1996a biometrics52 639 649 suggested using methods of finite mixtures to partition the animals in a closed capture recapture experiment into two or more groups with relatively homogeneous capture probabilities this enabled them to fit the models mh mbh norris and pollock and mth agresti of otis et al 1978 wildlife monographs62 1 135 in this article finite mixture partitions of animals and or samples are used to give a unified linear logistic framework for fitting all eight models of otis et al by maximum likelihood likelihood ratio tests are available for model comparisons for many data sets a simple dichotomy of animals is enough to substantially correct for heterogeneity induced bias in the estimation of population size although there is the option of fitting more than two groups if the data warrant it

a new bayesian method for nonparametric capture recapture models in presence of heterogeneity summary the intrinsic heterogeneity of individuals is a potential source of bias in estimation procedures for capture recapture models to account for this heterogeneity in the model a hierarchical structure has been proposed whereby the probabilities that each animal is caught on a single occasion are modelled as independent draws from a common unknown distribution f however there is general agreement that modelling f by a simple parametric curve may lead to unsatisfactory results here we propose an alternative bayesian approach that relies on a different parameterisation which imposes no assumption on the shape of f but drives the problem back to a finite dimensional setting our approach avoids some identifiability issues related to such a recapture model while allowing for a formal bayesian default analysis results of analyses of computer simulations and of real data show that the method performs well

a penalized nonparametric maximum likelihood approach to species richness estimation we propose a class of penalized nonparametric maximum likelihood estimators npmles for the species richness problem we use a penalty term on the likelihood because likelihood estimators that lack it have an extreme instability problem the estimators are constructed using a conditional likelihood that is simpler than the full likelihood we show that the full likelihood npmle solution given by norris and pollock can be found with great accuracy by using an appropriate penalty term on the conditional likelihood so it is an element of our class of estimators a simple and fast algorithm for the penalized npmle is developed it can be used to greatly speed up computation of the unconditional npmle it can also be used to find profile mixture likelihoods based on our goal of attaining high stability while retaining sensitivity we propose an adaptive quadratic penalty function a systematic simulation study using a wide range of scenarios establishes the success of this method relative to its competi

a flexible ratio regression approach for zero truncated capture recapture counts capture recapture methods are used to estimate the size of a population of interest which is only partially observed in such studies each member of the population carries a count of the number of times it has been identified during the observational period in real life applications only positive counts are recorded and we get a truncated at zero observed distribution we need to use the truncated count distribution to estimate the number of unobserved units we consider ratios of neighboring count probabilities estimated by ratios of observed frequencies regardless of whether we have a zero truncated or an untruncated distribution rocchetti et al 2011 have shown that for densities in the katz family these ratios can be modeled by a regression approach and rocchetti et al 2014 have specialized the approach to the beta binomial distribution once the regression model has been estimated the unobserved frequency of zero counts can be simply derived the guiding principle is that it is often easier to find an appropriate regression model than a proper model for the count distribution however a full analysis of the connection between the regression model and the associated count distribution has been missing in this manuscript we fill the gap and show that the regression model approach leads under general conditions to a valid count distribution we also consider a wider class of regression models based on fractional polynomials the proposed approach is illustrated by analyzing various empirical applications and by means of a simulation study

the performance of mixture models in heterogeneous closed population capture recapture summarydorazio and royle 2003 biometrics59 351 364 investigated the behavior of three mixture models for closed population capture recapture analysis in the presence of individual heterogeneity of capture probability their simulations were from the beta binomial distribution with analyses from the beta binomial the logit normal and the finite mixture latent class models in this response simulations from many different distributions give a broader picture of the relative value of the beta binomial and the finite mixture models and provide some preliminary insights into the situations in which these models are useful