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

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

numerical determination of hitting time distributions from their laplace transforms simple cases abstract an important problem in probabilistic modeling consists of the determination of the distribution function of a positive random variable in particular a hitting time from its laplace transform in this note we use two versions of the method of maximum entropy to invert the laplace transform using a few of its values on the real line or equivalently a few fractional moments of the exponential of the variable to compare the performance of the methods we consider some simple one dimensional examples in which the laplace transform can be computed by solving a differential equation exactly and comparisons can be made

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

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

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

occupancy distributions arising in sampling from gibbs poisson abundance models estimating the number n of unseen species from a k sample displaying only p k distinct sampled species has received attention for long it requires a model of species abundance together with a sampling model we start with a discrete model of iid stochastic species abundances each with gibbs poisson distribution a k sample drawn from the n species abundances vector is the one obtained while conditioning it on summing to k we discuss the sampling formulae species occupancy distributions frequency of frequencies in this context we then develop some aspects of the estimation of n problem from the size k of the sample and the observed value of pn k the number of distinct sampled species

volumes of vector fields on spheres in this paper we study the problem what is the unit vector field of smallest volume on an odd dimensional sphere we exhibit on each sphere a unit vector field with singularity which has exceptionally small volume on spheres of dimension greater than four we conjecture that this volume is the infimum for volumes of bona fide unit vector fields and is only achieved by the singular vector field we generalize the construction of the singular vector field to give a family of cycles in stiefel manifolds each of which is a smooth manifold except for one singular point except for some low dimensional cases the tangent cones at these singular points are volume minimizing and half of the cones are nonorientable thus we obtain a new family of nonorientable volume minimizing cones

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

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

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

predicting the conditional probability of discovering a new class consider a population comprising disjoint classes an important problem arising from various fields is prediction of the random conditional probability of discovering a new class the asymptotic normality of the discovery probability is established in a poisson model where the number of individuals from each class is a poisson process with a class specific rate a new derivation is presented for the well known good toulmin predictor as a moment based estimator for the asymptotic limit of the discovery probability the good toulmin predictor is also shown to be a nonparametric empirical bayes estimator for the expectation of the discovery probability given the rates of the poisson processes and an approximation to an unbiased estimator only for the identifiable part of the expectation of the discovery probability in a multinomial model the properties of the moment based estimator are investigated so that confidence and prediction intervals can be constructed the good toulmin predictor and the discovery

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

a generalization of the eckart young mirsky matrix approximation theorem abstract the eckart young mirsky theorem solves the problem of approximating a matrix by one of lower rank however the approximation generally differs from the original in all its elements in this paper it is shown how to obtain a best approximation of lower rank in which a specified set of columns of the matrix remains fixed the paper concludes with some applications of the generalization

a maximum entropy method for inverting laplace transforms of probability density functions summary this papers presents a maximum entropy method for inverting laplace transforms of density functions of positive random variables the maximum entropy density is very flexible and can assume a variety of different shapes accurate approximations to the true density can be obtained even when only a few transform values are available numerical evidence is provided for gamma lognormal inverse gaussian and pareto distributions

numerical inversion of laplace transform on the real line of probability density functions a method for the numerical inversion of laplace transform on the real line of a heavy tailed probability density function is presented the method assumes as known a finite set of values of the laplace transform and chooses the analytical form of the approximant maximizing shannon entropy so that positivity of the approximant itself is guaranteed the problem resorts to a finite generalized moment problem and a stability analysis is also provided some numerical results are illustrated

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

the two parameter poisson dirichlet distribution derived from a stable subordinator the two parameter poisson dirichlet distribution denoted mathsf pd alpha theta is a probability distribution on the set of decreasing positive sequences with sum 1 the usual poisson dirichlet distribution with a single parameter theta introduced by kingman is mathsf pd 0 theta known properties of mathsf pd 0 theta including the markov chain description due to vershik shmidt and ignatov are generalized to the two parameter case the size biased random permutation of mathsf pd alpha theta is a simple residual allocation model proposed by engen in the context of species diversity and rediscovered by perman and the authors in the study of excursions of brownian motion and bessel processes for 0 alpha 1 mathsf pd alpha 0 is the asymptotic distribution of ranked lengths of excursions of a markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index alpha formulae in this case trace back to work of darling lamperti and wendel in the 1950s and 1960s the distribution of ranked lengths of excursions of a one dimensional brownian motion is mathsf pd 1 2 0 and the corresponding distribution for a brownian bredge is mathsf pd 1 2 1 2 the mathsf pd alpha 0 and mathsf pd alpha alpha distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable markov process whose zero set is the range of a stable subordinator of index alpha

gibbs distributions for random partitions generated by a fragmentation process in this paper we study random partitions of 1 n where every cluster of size j can be in any of wj possible internal states the gibbs n k w distribution is obtained by sampling uniformly among such partitions with k clusters we provide conditions on the weight sequence w allowing construction of a partition valued random process where at step k the state has the gibbs n k w distribution so the partition is subject to irreversible fragmentation as time evolves for a particular one parameter family of weight sequences wj the time reversed process is the discrete marcus lushnikov coalescent process with affine collision rate ki j a b i j for some real numbers a and b under further restrictions on a and b the fragmentation process can be realized by conditioning a galton watson tree with suitable offspring distribution to have n nodes and cutting the edges of this tree by random sampling of edges without replacement to partition the tree into a collection of subtrees suitable offspring distributions include the binomial negative binomial and poisson distributions

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

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

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

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

exchangeable and partially exchangeable random partitions call a random partition of the positive integerspartially exchangeable if for each finite sequence of positive integersn1 nk the probability that the partition breaks the firstn1 nk integers intok particular classes of sizesn1 nk in order of their first elements has the same valuep n1 nk for every possible choice of classes subject to the sizes constraint a random partition is exchangeable iff it is partially exchangeable for a symmetric functionp n1 nk a representation is given for partially exchangeable random partitions which provides a useful variation of kingman s representation in the exchangeable case results are illustrated by the two parameter generalization of ewens partition structure