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

estimating diversity via frequency ratios we wish to estimate the total number of classes in a population based on sample counts especially in the presence of high latent diversity drawing on probability theory that characterizes distributions on the integers by ratios of consecutive probabilities we construct a nonlinear regression model for the ratios of consecutive frequency counts this allows us to predict the unobserved count and hence estimate the total diversity we believe that this is the first approach to depart from the classical mixed poisson model in this problem our method is geometrically intuitive and yields good fits to data with reasonable standard errors it is especially well suited to analyzing high diversity datasets derived from next generation sequencing in microbial ecology we demonstrate the method s performance in this context and via simulation and we present a dataset for which our method outperforms all competitors

a unified treatment of integer parameter models abstract it is shown that there is an analog to the score function the derivative of the log likelihood with respect to the parameter here called the difference score which plays an important theoretical role in integer parameter models a unified treatment of integer parameter models can be obtained by recognizing that many commonly used models have a linear difference score lds it is established that many of the exact results available for exponential family models also hold for lds models in particular we find an information lower bound show the uniqueness of the maximum likelihood estimator establish monotone likelihood ratio and establish optimality properties in two sided tests it is also shown that if one has a multiparameter likelihood in which there is a conditional likelihood depending on n alone then these exact results apply there as well it is also established when and why the usual likelihood methodology will apply to a class of nonstandard models in which the natural approxima

species sampling models consistency for the number of species this paper considers species sampling models using constructions that arise from bayesian nonparametric prior distributions a discrete random measure used to generate a species sampling model can have either a countable infinite number of atoms which has been the emphasis in the recent literature or a finite number of atoms k while allowing k to be assigned a prior probability distribution on the positive integers it is the latter class of model we consider here due to the interpretation of k as the number of species we demonstrate the consistency of the posterior distribution of k as the sample size increases copyright 2013 oxford university press

estimating species richness by a poisson compound gamma model we propose a poisson compound gamma approach for species richness estimation based on the denseness and nesting properties of the gamma mixture we fix the shape parameter of each gamma component at a unified value and estimate the mixture using nonparametric maximum likelihood a least squares crossvalidation procedure is proposed for the choice of the common shape parameter the performance of the resulting estimator of n is assessed using numerical studies and genomic data

a martingale estimating equation for a capture recapture experiment in discrete time we use martingale theory and a method of moments technique to derive a class of estimators for the size of a closed population in a capture recapture experiment in discrete time where the capture probabilities on the successive capture occasions are allowed to be different the same capture probability is assumed to apply to all animals in the population on the same occasion explicit expressions are given for these estimators and their associated standard errors which involve only simple computation an optimal estimator for the population size is found asymptotic results are readily obtained by an application of a martingale central limit theorem two examples are given to compare the results

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

estimating the number of classes in a finite population abstract we use an extension of the generalized jackknife approach of gray and schucany to obtain new nonparametric estimators for the number of classes in a finite population of known size we also show that generalized jackknife estimators are closely related to certain horvitz thompson estimators to an estimator of shlosser and to estimators based on sample coverage in particular the generalized jackknife approach leads to a modification of shlosser s estimator that does not suffer from the erratic behavior of the original estimator the performance of both new and previous estimators is investigated by means of an asymptotic variance analysis and a monte carlo simulation study

estimating the size of a closed population from a single mark recapture regime by using a diffusion probability of marked individuals a population estimate model was developed to account for the unequal recapture probability of marked animals when their diffusion into the released population has yet to reach the equilibrium we assumed a spatial bias for marked individuals within the finite area of the animals territory and used a correction parameter to account for the bias mark recapture data previously collected for field colonies of the formosan subterranean termite coptotermes formosanus shiraki were applied to the new model and the results were compared with the peterson and weighted mean models

a statistical estimate of the structure of multi extremal problems the solution of ak extremal problem is defined as the set of pairs xi i i 1 k where xt isith local minimum and i is the volume of the set of attraction of this minimum a bayesian estimate ofk and 1 k is constructed

two dimensional geodesic flows having first integrals of higher degree we present a family of riemannian metrics on two sphere having the property that the geodesic flows admit first integrals which are fiberwise homogeneous polynomials of degree greater than 2 they also have the property that all geodesics are closed

estimating the population size for capture recapture data with unequal catchability a point estimator and its associated confidence interval for the size of a closed population are proposed under models that incorporate heterogeneity of capture probability real data sets provided in edwards and eberhardt 1967 journal of wildlife management 31 87 96 and carothers 1973 journal of animal ecology 42 125 146 are used to illustrate this method and to compare it with other estimates the performance of the proposed procedure is also investigated by means of monte carlo experiments the method is especially useful when most of the captured individuals are caught once or twice in the sample for which case the jackknife estimator usually does not work well numerical results also show that the proposed confidence interval performs satisfactorily in maintaining the nominal levels

bayesian estimation of the number of species summary this paper is concerned with a method of estimating the number of species present in a population by considering the number of species obtained in a sample a bayesian approach is used and results are based on a zero truncated negative binomial prior distribution for the number of species in the population results for the posterior model are compared with the prior information and guidelines are suggested in the choice of values for the parameters the estimation of the number of species in a population or the number of species unrepresented by a sample is an inferential problem that occurs frequently in the biological sciences much previous work has stemmed from a parametric model due to fisher corbet and williams 1943 where fisher fitted a theoretical distribution to williams data on macrolepidoptera also good 1953 and later good and toulmin 1956 derived a model based on a set of statistical hypotheses concerning the population frequencies of the individual species however the estimation of the total number of species in the population was not the prime concern of any of the work described above when such estimation has been the main aim it has usually been assumed that the species have equal relative abundances as in for example the maximum likelihood approach of driml and ullrich 1967 clearly this assumption will not be appropriate in many practical situations more recently nonequiprobable models have been derived by efron and thisted 1976 and hill 1979 efron and thisted obtained a method of estimating the number of unseen species by an euler transformation and a linear programming approach they took the number of unseen species to be the number of words shakespeare knew but did not use in print hill provided a model that enables an estimate of the number of species in the population to be made by imposing a zero truncated negative binomial prior distribution for the number of species however hill s final analysis is limited to a uniform prior distribution for the number of species it is stated that this gives a sufficiently close approximation in the present paper it is proposed both to derive and to analyse a model based on the zero truncated negative binomial prior distribution for the number of species a similar generalization of hill s model for zipf s law was given by chen 1980 the form of joint prior density for the unknown species relative abundances given by equation 2 1 is derived under the assumptions used by kempton and wedderburn 1978 they explained that when the frequency distribution of the species abundances can be approximated by a continuous distribution then in practice the gamma distribution provides a reasonable fit hence the abundances of species in the population constitute independent observations from a gamma distribution the relative abundances are then beta variates and their joint density function is as shown in 2 1

distributions on partitions on obtient une famille de distributions a deux parametres sur des partitions par melange d une variable multinomiale avec une distribution de dirichlet symetrique

on the statistical analysis of capture experiments summary a procedure is given for estimating the size of a closed population in the presence of heterogeneous capture probabilities using capture recapture data when it is possible to model the capture probabilities of individuals in the population using covariates the results include the estimation of the parameters associated with the model of the capture probabilities and the use of these estimated capture probabilities to estimate the population size confidence intervals for the population size using both the asymptotic normality of the estimator and a bootstrap procedure for small samples are given

stratified capture recapture estimation of the size of a closed population this paper considers the estimation of the size of a closed population in a stratified capturerecapture experiment using the stratified petersen lincoln model introduced by darroch 1961 biometrika 48 241 260 a generalization of darroch s model where the tagging or the recovery probabilities depend on explanatory variables possibly related to the capture mechanism is proposed a simple scoring algorithm for the estimation of the nuisance parameters i e the vectors of tagging and of recovery probabilities and the matrix for the distribution of the units in the capturerecapture area is derived this algorithm is based on a new closed form expression for the inverse of the fisher information matrix of the conditional likelihood for the estimation of the nuisance parameters large sample tests for the tagging and the recovery probabilities are obtained pooling of adjacent recovery strata with equal capture probabilities is proposed as a method for getting accurate estimators of the size of the population conditional and unconditional estimators of the size of the population are proposed analyses of data on smolt migration illustrate the methodology presented in this paper

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

logistic regression in capture recapture models the effect of population heterogeneity in capture recapture or dual registration models is discussed an estimator of the unknown population size based on a logistic regression model is introduced the model allows different capture probabilities across individuals and across capture times the probabilities are estimated from the observed data using conditional maximum likelihood the resulting population estimator is shown to be consistent and asymptotically normal a variance estimator under population heterogeneity is derived the finite sample properties of the estimators are studied via simulation an application to finnish occupational disease registration data is presented

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

small sample estimation of species richness applied to forest communities summary many well known methods are available for estimating the number of species in a forest community however most existing methods result in considerable negative bias in applications where field surveys typically represent only a small fraction of sampled communities this article develops a new method based on sampling with replacement to estimate species richness via the generalized jackknife procedure the proposed estimator yields small bias and reasonably accurate interval estimation even with small samples the performance of the proposed estimator is compared with several typical estimators via simulation study using two complete census datasets from panama and malaysia

an inference procedure for a capture and recapture experiment with time dependent capture probabilities summary estimates for the size of a closed population are given for multiple recapture studies in discrete time the estimates are derived by a method of moments for martingales an estimate and associated standard error of the population size are derived for a homogeneous population when the capture probabilities are permitted to be time dependent in the discrete case corresponding results are obtained for the continuous case as well explicit expressions are given for these estimates and their standard errors only simple computation is involved

on a class of integrable systems with a cubic first integral a few years ago selivanova gave an existence proof for some integrable models in fact geodesic flows on two dimensional manifolds with a cubic first integral however the explicit form of these models hinged on the solution of a nonlinear third order ordinary differential equation which could not be obtained we show that an appropriate choice of coordinates allows for integration and gives the explicit local form for the full family of integrable systems the relevant metrics are described by a finite number of parameters and lead to a large class of models mainly on the manifolds mathbb s 2 and mathbb h 2 many of these systems are globally defined and contain as special cases integrable systems due to goryachev chaplygin dullin matveev and tsiganov

on vector subdivision in this paper we give a complete characterization of the convergence of stationary vector subdivision schemes and the regularity of the associated limit function these results extend and complete our earlier work on vector subdivision and its use in the construction of multiwavelets

nonparametric estimation of the probability of discovering a new species abstract a random sample is taken from a population consisting of an unknown number of distinct species a quantity of interest is the probability of discovering a new species when an additional draw from the population is made an estimator of this quantity was introduced by starr 1979 we prove a conjecture of starr s that the estimator is uniformly minimum variance unbiased and give various asymptotic properties of the estimator a nonparametric maximum likelihood estimator that has similar asymptotic properties is introduced a monte carlo study that suggests guidelines for choosing an estimator under various circumstances is given to amplify suppose that if we take a sample of size 1 from a population then the probability of drawing a representative of the ith species is p 1 if n draws are made with replacement the unconditional probability that species i will be observed for the first time on the n 1st draw is given simply as the term p i q n i q i 1 p i from a geometric distributio

estimating the number of classes estimating the unknown number of classes in a population has numerous important applications in a poisson mixture model the problem is reduced to estimating the odds that a class is undetected in a sample the discontinuity of the odds prevents the existence of locally unbiased and informative estimators and restricts confidence intervals to be one sided confidence intervals for the number of classes are also necessarily one sided a sequence of lower bounds to the odds is developed and used to define pseudo maximum likelihood estimators for the number of classes

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

did shakespeare write a newly discovered poem summary the consistency of the word usage in a previously unknown nine stanza poem attributed to shakespeare with that of the shakespearean canon is examined using a nonparametric empirical bayes model we consider also poems by jonson marlowe and donne as well as four poems definitely attributed to shakespeare on balance the poem is found to fit previous shakespearean usage reasonably well

estimating the number of classes via sample coverage abstract assume that a random sample is drawn from a population with unknown number of classes and possibly unequal class probabilities a nonparametric estimation technique is proposed to estimate the number of classes using the idea of sample coverage which is defined as the sum of the cell probabilities of the observed classes since expected sample coverage can be well estimated we were motivated to find its role in the estimation of the number of classes this work generalizes the result of esty to a nonparametric approach and extends darroch and ratcliff to incorporate the heterogeneity of the class probabilities the coefficient of variation of the class sizes is shown to play an important role in the recommended estimation procedures the performance of the proposed estimators is investigated by means of monte carlo simulations

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

nonexistence of informative unbiased estimators in singular problems in many nonparametric problems such as density estimation nonparametric regression and so on all the existing informative estimators are biased asymptotic or finite sample there has long been a suspicion that either informative unbiased estimators do not exist for such problems or they must be quite complicated in this paper we clarify the nonexistence of informative unbiased estimators in all singular problems both for fixed sample size and asymptotically this includes most problems with optimal rate of convergence slower than n1 2 we also discuss situations in regular problems where such nonexistences can occur

completely integrable 2d lagrangian systems and related integrable geodesic flows on various manifolds in this study we have formulated a theorem that generates deformations of the natural integrable conservative systems in the plane into integrable systems on riemannian and other manifolds by introducing additional parameters into their structures the relation of explicit solutions of the new and the original dynamics to the corresponding jacobi maupertuis geodesic flow is clarified for illustration we apply the result to three concrete examples of the many available integrable systems in the literature complementary integrals in those systems are polynomial in velocity with degrees 3 4 and 6 respectively as a special case of the first deformed system a new several parameter family of integrable mechanical systems and geodesic flows on s2 is constructed

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

a normal limit law for a nonparametric estimator of the coverage of a random sample the coverage of a multinomial random sample is the sum of the probabilities of the observed classes a normal limit law is rigorously proved for good s 1953 coverage estimator the result is valid under very general conditions and all terms except the coverage itself are observable nevertheless the implied confidence intervals are not much wider than those developed under restrictive assumptions such as in the classical occupancy problem the asymptotic variance is somewhat unexpected the proof utilizes a method of holst 1979

on the bayesian estimation of a closed population size in the presence of heterogeneity and model uncertainty we consider the estimation of the size of a closed population often of interest for wild animal populations using a capture recapture study the estimate of the total population size can be very sensitive to the choice of model used to fit to the data we consider a bayesian approach in which we consider all eight plausible models initially described by otis et al 1978 wildlife monographs 62 1 135 within a single framework including models containing an individual heterogeneity component we show how we are able to obtain a model averaged estimate of the total population incorporating both parameter and model uncertainty to illustrate the methodology we initially perform a simulation study and analyze two datasets where the population size is known before considering a real example relating to a population of dolphins off northeast scotland

applications and extensions of chao s moment estimator for the size of a closed population summary this article revisits chao s 1989 biometrics 45 427 438 lower bound estimator for the size of a closed population in a mark recapture experiment where the capture probabilities vary between animals model mh first an extension of the lower bound to models featuring a time effect and heterogeneity in capture probabilities mth is proposed the biases of these lower bounds are shown to be a function of the heterogeneity parameter for several loglinear models for mth small sample bias reduction techniques for chao s lower bound estimator are also derived the application of the loglinear model underlying chao s estimator when heterogeneity has been detected in the primary periods of a robust design is then investigated a test for the null hypothesis that chao s loglinear model provides unbiased abundance estimators is provided the strategy of systematically using chao s loglinear model in the primary periods of a robust design where heterogeneity has been detected is investigated in a monte carlo experiment its impact on the estimation of the population sizes and of the survival rates is evaluated in a monte carlo experiment

bayesian non parametric inference for species variety with a two parameter poisson dirichlet process prior summary a bayesian non parametric methodology has been recently proposed to deal with the issue of prediction within species sampling problems such problems concern the evaluation conditional on a sample of size n of the species variety featured by an additional sample of size m genomic applications pose the additional challenge of having to deal with large values of both n and m in such a case the computation of the bayesian non parametric estimators is cumbersome and prevents their implementation we focus on the two parameter poisson dirichlet model and provide completely explicit expressions for the corresponding estimators which can be easily evaluated for any sizes of n and m we also study the asymptotic behaviour of the number of new species conditionally on the observed sample such an asymptotic result combined with a suitable simulation scheme allows us to derive asymptotic highest posterior density intervals for the estimates of interest finally we illustrate the implementation of the proposed methodology by the analysis of five expressed sequence tags data sets

estimating the number of species in a stochastic abundance model consider a stochastic abundance model in which the species arrive in the sample according to independent poisson processes where the abundance parameters of the processes follow a gamma distribution we propose a new estimator of the number of species for this model the estimator takes the form of the number of duplicated species i e species represented by two or more individuals divided by an estimated duplication fraction the duplication fraction is estimated from all frequencies including singleton information the new estimator is closely related to the sample coverage estimator presented by chao and lee 1992 journal of the american statistical association 87 210 217 we illustrate the procedure using the malayan butterfly data discussed by fisher corbet and williams 1943 journal of animal ecology 12 42 58 and a 1989 christmas bird count dataset collected in florida u s a simulation studies show that this estimator compares well with maximum likelihood estimators i e empirical bayes estimators from the bayesian viewpoint for which an iterative numerical procedure is needed and may be infeasible

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

new examples of integrable conservative systems on s 2 and the case of goryachev chaplygin in this paper we propose new examples of conservative systems on s 2 possessing cubic integrals in momenta we obtain a classification of a large class of such systems we show that this class contains new families of integrable systems as well as the classical case of goryachev chaplygin

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

stability implies convergence of cascade algorithms in sobolev space this article proves that the stability of the shifts of a refinable function vector ensures the convergence of the corresponding cascade algorithm in sobolev space to which the refinable function vector belongs an example of hermite interpolants is presented to illustrate the result

a multinomial bayesian approach to the estimation of population and vocabulary size we approach estimation of the size of a population or a vocabulary through a bayesian analysis of the multinomial distribution we view the sample as being generated from such a distribution with an unknown number of cells and unknown cell probabilities and develop a bayesian procedure to estimate the number of cells and the coverage of the sample the prior distribution of the number of cells is arbitrary given that number the cell probabilities are assumed to follow a symmetric dirichlet prior a two stage approach is developed for use when the flattening constant of the latter prior cannot be specified in advance our procedures are applied to samples of butterflies insect species and alleles to the works of shakespeare and joyce and to eldridge s sample of english words

diversity as a concept and its measurement abstract this paper puts forth the view that diversity is an average property of a community and identifies that property as species rarity an intrinsic diversity ordering of communities is defined and is shown to be equivalent to stochastic ordering also the sensitivity of an index to rare species is developed culminating in a crossing point theorem and a response theory to perturbations diversity decompositions analogous to the analysis of variance are discussed for two way classifications and mixtures the paper concludes with a brief survey of genetic diversity linguistic diversity industrial concentration and income inequality

modeling unobserved sources of heterogeneity in animal abundance using a dirichlet process prior in surveys of natural populations of animals a sampling protocol is often spatially replicated to collect a representative sample of the population in these surveys differences in abundance of animals among sample locations may induce spatial heterogeneity in the counts associated with a particular sampling protocol for some species the sources of heterogeneity in abundance may be unknown or unmeasurable leading one to specify the variation in abundance among sample locations stochastically however choosing a parametric model for the distribution of unmeasured heterogeneity is potentially subject to error and can have profound effects on predictions of abundance at unsampled locations in this article we develop an alternative approach wherein a dirichlet process prior is assumed for the distribution of latent abundances this approach allows for uncertainty in model specification and for natural clustering in the distribution of abundances in a data adaptive way we apply this approach in an analysis of counts based on removal samples of an endangered fish species the okaloosa darter results of our data analysis and simulation studies suggest that our implementation of the dirichlet process prior has several attractive features not shared by conventional fully parametric alternatives

the use of auxiliary variables in capture recapture and removal experiments the dependence of animal capture probabilities on auxiliary variables is an important practical problem which has not been considered in the development of estimation procedures for capturerecapture and removal experiments in this paper the linear logistic binary regression model is used to relate the probability of capture to continuous auxiliary variables the auxiliary variables could be environmental quantities such as air or water temperature or characteristics of individual animals such as body length or weight maximum likelihood estimators of the population parameters are considered for a variety of models which all assume a closed population testing between models is also considered the models can also be used when one auxiliary variable is a measure of the effort expended in obtaining the sample

capture recapture estimation with samples of size one using frequency data summary one way in which a capture recapture experiment can be designed is to count each capture occasion as a separate sample and tally the number of individuals caught once twice and so on this paper discusses the estimators currently used for this type of experiment and suggests some new ones each of the estimators is categorized by its derivation following this their properties are discussed particularly their bias and variance with the aim of choosing one or two of the estimators which are the best to use especially in handling a heterogeneous population it is found that the bias adjusted estimator of chao 1989 is the best to use when the number of captures is relatively small and that the estimator of darroch ratcliff 1980 should be used otherwise

convergence of subdivision schemes and smoothness of limit functions abstract starting with vector k k z l p z the subdivision scheme generates a sequence s a n n 1 of vectors by the subdivision operator s a k j z j a k 2 j k z subdivision schemes play an important role in computer graphics and wavelet analysis it is very interesting to understand under what conditions the sequence s a n n 1 converges to a l p function in an appropriate sense this problem has been studied extensively in this paper we consider the convergence of subdivision scheme in sobolev spaces with the tool of joint spectral radius firstly the conditions under which the sequence s a n n 1 converges to a w p k function in an appropriate sense are given then we show that the subdivision scheme converges for any initial vector in w p k r provided that it does for one nonzero vector in that space moreover if the shifts of the refinable function are stable the smoothness of the limit function corresponding to the vector is also independent of where the smoothness of a given function is measured by the generalized lipschitz space

on a family of integrable systems on s2 with a cubic integral of motion we discuss a family of integrable systems on the sphere s2 with an additional third order integral in momenta this family contains the coryachev chaplygin top the goryachev system the system recently discovered by dullin and matveev and two new integrable systems on the non physical sphere with zero radius all these systems are isomorphic to each other

log linear models for capture recapture summary log linear models are developed for capture recapture experiments and their advantages and disadvantages discussed ways in which they can be extended sometimes with only partial success to open populations subpopulations trap dependence and long chains of recapture periods are presented the use of residual patterns and analysis of subsets of data to identify behavioural patterns and acceptable models is emphasised and illustrated with two examples 1 capture recapture the biological problem with which we are concerned is that of estimating the parameters of a free living population of unknown size from a sequence of s samples in each of which any individual either is or is not observed the models in this paper require that each animal be marked naturally or artificially in such a way that we can observe the numbers of animals which have any particular pattern of being or not being observed through the sequence of samples there are t 2s 1 different observable patterns which we shall refer to as capture histories although there is no requirement for physical capture and denote each pattern by a sequence of l s and 2 s representing respectively seen or not seen in a particular sample a general pattern of capture history will be denoted a with ra the number of individuals observed to have this pattern thus for example with s 3 samples and a 121 r 2 is the number of animals seen in the first and third samples but not in the second the pattern with all 2 s is by definition unobservable in general the population will be subject to demographic change by birth or immigration and by death or emigration the simplest assumption for the response of an animal to the observer one which underlies the classical petersen schnabel or jolly seber estimators is that every animal alive in the population at a sampling time has the same probability of being observed this probability may vary from one sample to another or be constant over all or a number of samples if this assumption fails it may do so for either or both of two reasons the first is that of trap dependence the probability of an individual being observed in a sample depending on its past history of capture the second is that of heterogeneity between individuals who because of differences in their behaviour or in where they live inherently have different probabilities of being observed in any sample the basic idea of heterogeneity refers to properties of individuals which remain constant over time but individual animals may also have time varying behaviour the aspect most commonly considered being temporary departure of an individual from the population various excellent texts describe theoretical and practical aspects of capture recapture notably the comprehensive one by seber 1982 the introduction by begon 1979 the

modified martingale estimation for recapture experiments with hetrogeneous capture probabilities summary an estimation method of lloyd yip 1991 based on martingales and the assumption of beta distributed capture probabilities is modified to improve its performance for all but highly heterogeneous populations this estimator is compared to the jackknife estimator by simulation under both beta and non beta conditions some expressions for standard error are developed but found to behave poorly for realistic sample sizes bootstrap estimates of standard error appear superior but do not lead to accurate confidence intervals because of the difficulties in bias estimation

simple capture recapture models permitting unequal catchability and variable sampling effort we consider two capture recapture models that imply that the logit of the probability of capture is an additive function of an animal catchability parameter and a parameter reflecting the sampling effort the models are special cases of the rasch model and satisfy the property of quasi symmetry one model is log linear and the other is a latent class model for the log linear model point and interval estimates of the population size are easily obtained using standard software such as glim

estimating the total number of distinct species using presence and absence data summary consider the problem of estimating the total number of distinct species in some specified region under investigation suppose the region is divided into n disjoint subregions or quadrats of equal area a sample of size n quadrats is chosen n 1 the other is to place at random n quadrats of equal area and fixed shape in the region of investigation in both cases the n quadrats in the sample are assumed to be disjoint and are totally observed so in fact when quadrat sampling is used a random sample of space is taken instead of a random sample of individuals it has been observed that species are often present with some natural clumping which creates dependence between quadrats and within quadrats heltshe and forrester 1983 and smith and van belle 1984 introduced some estimators for s when quadrat sampling is used see also burnham and overton 1979 the estimators they proposed are nonparametric and were developed by using jackknife and bootstrap arguments for a particular example palmer 1990 compared several estimators and showed that the first order jackknife estimator is preferred within this group in this paper we introduce some empirical bayes estimators for s when quadrat sampling is used by imposing a grid on the region the probabilistic model used to derive these estimators is basically a version of the efron and thisted 1976 model adapted to the case of quadrat

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

a bayesian analysis of multiple recapture sampling for a closed population summary this paper considers from a bayesian viewpoint inferences about the size of a closed animal population from data obtained by a multiple recapture sampling scheme the method developed enables prior information about the population size and the catch probabilities to be utilized to produce considerable improvements in certain cases on ordinary maximum likelihood methods several ways of expressing such prior information are explored and a practical example of the uses of these ways is given the main result of the paper is an approximation to the posterior distribution of sample size that exhibits the contributions made by the likelihood and the prior ideas the multiple recapture sampling scheme involves taking samples from a population of animals at each stage counting the number of marked animals in the sample marking the previously unmarked animals and returning the sample to the population the literature is reviewed by cormack 1968 and the papers most relevant to our problem are those of chapman 1952 and darroch 1958 in these papers maximum likelihood estimates for the population parameters and variances for these estimates are given our approach differs from previous ones by introducing prior information about the population size and about the propensities of the animals to be captured called catch probabilities by some previous authors the incorporation of these propensities is a feature not shared by many previous approaches darroch 1958 found their introduction into his model does not change the maximum likelihood estimates of the population size our approach will show that strong prior knowledge about the catch probabilities can greatly affect inference about the population size and it is in this respect that the greatest difference from previous approaches lies although it is recognized that models of an open population incorporating death and immigration such as those of jolly 1965 and seber 1965 are more practically realistic we feel that our method which deals only with a closed population is worth investigating as a step towards providing a bayesian treatment of the open population problem

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

estimating the number of species a review how many kinds are there suppose that a population is partitioned into c classes in many situations interest focuses not on estimation of the relative sizes of the classes but on estimation of c itself for example biologists and ecologists may be interested in estimating the number of species in a population of plants or animals numismatists may be concemed with estimating the number of dies used to produce an ancient coin issue and linguists may be interested in estimating the size of an author s vocabulary in this article we review the problem of statistical estimation of c many approaches have been proposed some purely data analytic and others based in sampling theory in the latter case numerous variations have been considered the population may be finite or infinite if finite samples may be taken with replacement multinomial sampling or without replacement hypergeometric sampling or by bernoulli sampling if infinite sampling may be multinomial or bernoulli or the sample may be th

low risk population size estimates in the presence of capture heterogeneity summary population estimation methods are used for estimating the size of a population from samples of individuals in many applications the probability of being observed in the sample varies across individuals resulting in sampling bias we show that in this setting estimators of the population size have high and sometimes infinite risk leading to large uncertainty in the population size as an alternative we propose estimating the population of individuals with observation probability exceeding a small threshold we show that estimators of this quantity have lower risk than estimators of the total population size the proposed approach is shown empirically to result in large reductions in mean squared error in a common model for capture recapture population estimation with heterogeneous capture probabilities

computational aspects of n mixture models the n mixture model is widely used to estimate the abundance of a population in the presence of unknown detection probability from only a set of counts subject to spatial and temporal replication royle 2004 biometrics 60 105 115 we explain and exploit the equivalence of n mixture and multivariate poisson and negative binomial models which provides powerful new approaches for fitting these models we show that particularly when detection probability and the number of sampling occasions are small infinite estimates of abundance can arise we propose a sample covariance as a diagnostic for this event and demonstrate its good performance in the poisson case infinite estimates may be missed in practice due to numerical optimization procedures terminating at arbitrarily large values it is shown that the use of a bound k for an infinite summation in the n mixture likelihood can result in underestimation of abundance so that default values of k in computer packages should be avoided instead we propose a simple automatic way to choose k the methods are illustrated by analysis of data on hermann s tortoise testudo hermanni

inference for sichel s compound poisson distribution abstract in a long tailed compound poisson distribution introduced by sichel the probabilities can only readily be calculated if one of the parameters has half integer values that is m 1 2 m integer the maximum likelihood estimation of this parameter is compared with an approximate method derived from a grid of half integer values it is found that the approximate method which avoids excessive special computation yields inferences indistinguishable from those based on maximum likelihood and on the score statistic for testing values of thus a major numerical impediment to use of sichel s distribution is removed

interval estimation for mark recapture studies of closed populations textbooks continue to recommend the use of an asymptotic normal distribution to provide an interval estimate for the unknown size n of a closed population studied by a mark recapture experiment or multiple record system a likelihood interval approach is proposed and its implementation demonstrated for a range of models for such studies including all main effect and interaction models for incomplete contingency tables

stopping rules and estimation for recapture debugging with unequal failure rates summary a theoretical optimal stopping rule based on the minimization of the testing cost and expected penalty due to the unremoved bugs is derived without the equal failure rate assumption under nayak s 1988 recapture debugging procedure two adaptive stopping rules suitable for practical use are constructed and tested by simulation the estimation of the number of undetected bugs is also considered via remaining failure rate estimation

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 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

parameter orthogonality for a family of discrete distributions abstract the standard contagious distributions see douglas 1980 have been used in such varied fields as biology and automobile insurance often to model various physical phenomena as well as provide a good fit to count data when other models are inadequate unfortunately the parameterizations often used when working with these distributions normally lead to extremely high correlations of the maximum likelihood estimators mle s this tends to lead to mathematical complexities and causes difficulty or even errors in their interpretation furthermore numerical difficulties may arise when using numerical procedures to locate the estimates some of these difficulties were discussed by douglas 1980 pp 171 204 205 who suggested that a reparameterization to reduce or even eliminate such correlation is desirable if the mle s are asymptotically uncorrelated the parameterization is orthogonal philpot 1964 derived an orthogonal parameterization for the neyman type a distribution stein zucchini an

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

a generalization of chao s estimator for covariate information this note generalizes chao s estimator of population size for closed capture recapture studies if covariates are available chao s estimator was developed under unobserved heterogeneity in which case it represents a lower bound of the population size if observed heterogeneity is available in form of covariates we show how this information can be used to reduce the bias of chao s estimator the key element in this development is the understanding and placement of chao s estimator in a truncated poisson likelihood it is shown that a truncated poisson likelihood with log link with all counts truncated besides ones and twos is equivalent to a binomial likelihood with logit link this enables the development of a generalized chao estimator as the estimated expected value of the frequency of zero counts under a truncated all counts truncated except ones and twos poisson regression model if the regression model accounts for the heterogeneity entirely the generalized chao estimator is asymptotically unbiased a simulation study illustrates the potential in gain of bias reduction comparisons of the generalized chao estimator with the homogeneous zero truncated poisson regression approach are supplied as well the method is applied to a surveillance study on the completeness of farm submissions in great britain

estimating population size for capture recapture data when capture probabilities vary by time and individual animal there have been no estimators of population size associated with the capture recapture model when the capture probabilities vary by time and individual animal this work proposes a nonparametric estimation technique that is appropriate for such a model using the idea of sample coverage which is defined as the proportion of the total individual capture probabilities of the captured animals a simulation study was carried out to show the performance of the proposed estimation procedure numerical results indicate that it generally works satisfactorily when the coefficient of variation of the individual capture probabilities is relatively large an example is also given for illustration