a thermodynamic like approach for the study of probabilistic systems abstract this paper starts by considering an arbitrary system for which the probability distribution p p 1 p 2 p n of a measurable property with n possible independent results is known by using the hartley information pseudo energy the escort probabilities of p and its shannon entropy a thermodynamic like formalism is developed including the deduction of the fundamental equation for the mean value of the pseudo energies thus the method allows a macrodescription of the system that is based on a purely mathematical procedure and involves thermodynamic like variables as a specific example we consider a bank that measures customer satisfaction by doing a statistical survey satisfied or unsatisfied customers

models of connectance in food webs a food web is a set of species that are connected by feeding interactions or trophic links in a particular habitat the connectance in a web is the proportion of possible trophic links that are actually observed three statistical models of connectance as a function of the number of species s in a web are considered linear regression of the log transformed number of links l vs log s logistic regression of the proportion of engaged links l s2 vs s and modified logistic regression based on a singly truncated binomial distribution which adjusts for the constraint that l s 1 in order for the collection of species to be a web all three models provide adequate fits to real data sets but the log log regression approach appears to reject the constant connectance hypothesis more readily than the other two methods the truncated logistic regression model may predict relatively high connectances for small webs even when the probability of a link being engaged does not depend on s for data simulated under that null hypothesis the log log and ordinary logistic regression approaches frequently rejected the constant connectance hypothesis

over dispersed age period cohort models summarywe consider inference and forecasting for aggregate data organised in a two way table with age and cohort as indices but without measures of exposure this is modelled using a poisson likelihood with an age period cohort structure for the mean while allowing for over dispersion we propose a repetitive structure that keeps the dimension of table fixed while increasing the latent exposure for this we use a class of infinitely divisible distributions which include a variety of compound poisson models and poisson mixture models this results in asymptotic f inference and t forecast distributions

multiscale modeling and estimation of poisson processes with application to photon limited imaging many important problems in engineering and science are well modeled by poisson processes in many applications it is of great interest to accurately estimate the intensities underlying observed poisson data in particular this work is motivated by photon limited imaging problems this paper studies a new bayesian approach to poisson intensity estimation based on the haar wavelet transform it is shown that the haar transform provides a very natural and powerful framework for this problem using this framework a novel multiscale bayesian prior to model intensity functions is devised the new prior leads to a simple bayesian intensity estimation procedure furthermore we characterize the correlation behavior of the new prior and show that it has 1 f spectral characteristics the new framework is applied to photon limited image estimation and its potential to improve nuclear medicine imaging is examined

bayesian inspection model with the negative binomial prior in the presence of inspection errors one of the basic assumptions in bayesian inspection models is that we have some prior knowledge about the number of defects in a certain product or software system the prior knowledge can be often described as a probability distribution e g poisson distribution in the paper we propose three conditions that should be put forth as desirable properties for a prior probability distribution of the number of defects in the product we review various prior probability distributions and test if they meet those conditions the negative binomial distribution is found to be the only one that satisfies all the desirable conditions with the negative binomial prior we analyze the effects of various parameters on the bayesian estimate of the number of undetected errors still remaining in the product

latent change in recurrent choice data this paper introduces dynamic latent class models for the analysis and interpretation of stability and change in recurrent choice data these latent class models provide a nonparametric representation of individual taste differences changes in preferences are modeled by allowing for individual level transitions from one latent class to another over time the most general model facilitates a saturated representation of class membership changes several special cases are presented to obtain a parsimonious description of latent change mechanisms an easy to implement em algorithm is derived for parameter estimation the approach is illustrated by a detailed analysis of a purchase incidence data set

asymptotic estimates for stieltjes constants a probabilistic approach let n n 0 be the sequence of stieltjes constants appearing in the laurent expansion of the riemann zeta function we obtain explicit upper bounds for n whose order of magnitude is as n tends to

best constants for the riesz projection we prove the following inequality with a sharp constant p f l p t csc p f lp t f lp t where 1 p and p lp t hp t is the riesz projection onto the hardy space hp t on the unit circle t in other words the angle between the analytic and co analytic subspaces of lp t equals p where p max p pp 1 this was conjectured in 1968 by i gohberg and n krupnik we also prove an analogous inequality in the nonperiodic case where p f f 1 r ff is the half line fourier multiplier on r similar weighted inequalities with sharp constants for lp r x 1 p 1 are obtained in the multidimensional case our results give the norm of the half space fourier multiplier on rn

decompounding an estimation problem for poisson random sums given a sample from a compound poisson distribution we consider estimation of the corresponding rate parameter and base distribution this has applications in insurance mathematics and queueing theory we propose a plug in type estimator that is based on a suitable inversion of the compounding operation asymptotic results for this estimator are obtained via a local analysis of the decompounding functional

variance of the truncated negative binomial distribution abstract citations to formulas for the moments of the truncated negative binomial distribution usually reference the paper by gurmu and trivedi 1992 however their second moments of the truncated negative binomial are incorrect we derive the correct second moments for both the left and right truncated negative binomial distribution the second moments of the truncated distributions are written in a form that shows they will converge to the second moment of the un truncated distribution when the truncated first moment approaches the un truncated first moment

macroscopic characterization of data sets by using the average absolute deviation abstract this paper describes a method for getting a synthesis of the knowledge about a given system assuming that a data set x of measurements of a variable is known i e separate data are combined in order to form a coherent whole a la thermodynamics for getting the macroscopic characterization of time series data one takes advantage of the average absolute deviation concept together with an already known thermodynamic like approach the method is applied to the macroscopic characterization of existing time series for the vostok antarctica ice cores deuterium content of the ice and temperature variation for a depth range between 0 and 3 310 meters and the age of the ice between 0 and 422 766 years before the present

two part multiple spell models for health care demand the demand for certain types of health care services depends on decisions of both the individual and the health care provider this paper studies the conditions under which it is possible to separately identify the parameters driving the two decision processes using only count data on the total demand it is found that the frequently used hurdle models may not be adequate to describe this type of demand especially when the assumption of a single illness spell per observation period is violated a test for the single spell hypothesis is developed and alternative modelling strategies are suggested including one that allows for correlated unobserved heterogeneity the results of the paper are illustrated with an application

on the best constants in the weak type inequalities for re expansion operator and hilbert transform we study the weak type inequalities for the operator i fsfc where fc and fs are the cosine and sine fourier transforms on the positive half line respectively and i is the identity operator we also derive sharp constants in related weak type estimates for i ht i hr and i hr where ht hr and hr denote the hilbert transforms on the circle on the real line and the positive halfline respectively our main tool is the weak type inequality for orthogonal martingales which is of independent interest

sharp upper and lower bounds for the moments of bernstein polynomials we give upper and lower bounds for the moments and the uniform moments of bernstein polynomials asymptotically such bounds are best possible

orthogonal polynomials in stein s method stein s method provides a way of finding approximations to the distribution say of a random variable which at the same time gives estimates of the approximation error involved in essence the method is based on a defining equation or equivalently an operator of the distribution and a related stein equation up to now it was not clear which equation to take one could think of a lot of equations we show how for a broad class of distributions there is one convenient equation we give a systematic treatment including the stein equation and its solution a key tool in stein s theory is the generator method developed by barbour and we suggest employing the generator of a markov process for the operator of the stein equation for a given distribution there may be various markov processes which fit in barbour s method and up to now it was still not clear which markov process to take to obtain good results we show how for a broad class of distributions there is a special markov process a birth and death process or a diffusion which takes a leading role in the analysis furthermore a key role is played by the classical orthogonal polynomials it turns out that the defining operator is based on a hypergeometric difference or differential equation which lies at the heart of the classical orthogonal polynomials furthermore the spectral representation of the transition probabilities of the markov process involved are in terms of orthogonal polynomials closely related to the distribution to be approximated this systematic treatment together with the introduction of orthogonal polynomials in the analysis seems to be new furthermore some earlier uncovered examples like the beta the student s t and the hypergeometric distribution are now worked out

inverse adaptive cluster sampling summary consider a population in which the variable of interest tends to be at or near zero for many of the population units but a subgroup exhibits values distinctly different from zero such a population can be described as rare in the sense that the proportion of elements having nonzero values is very small obtaining an estimate of a population parameter such as the mean or total that is nonzero is difficult under classical fixed sample size designs since there is a reasonable probability that a fixed sample size will yield all zeroes we consider inverse sampling designs that use stopping rules based on the number of rare units observed in the sample we look at two stopping rules in detail and derive unbiased estimators of the population total the estimators do not rely on knowing what proportion of the population exhibit the rare trait but instead use an estimated value hence the estimators are similar to those developed for poststratification sampling designs we also incorporate adaptive cluster sampling into the sampling design to allow for the case where the rare elements tend to cluster within the population in some manner the formulas for the variances of the estimators do not allow direct analytic comparison of the efficiency of the various designs and stopping rules so we provide the results of a small simulation study to obtain some insight into the differences among the stopping rules and sampling approaches the results indicate that a modified stopping rule that incorporates an adaptive sampling component and utilizes an initial random sample of fixed size is the best in the sense of having the smallest variance

wavelet domain filtering for photon imaging systems many imaging systems rely on photon detection as the basis of image formation one of the major sources of error in these systems is poisson noise due to the quantum nature of the photon detection process unlike additive gaussian white noise the variance of poisson noise is proportional to the underlying signal intensity and consequently separating signal from noise is a very difficult task in this paper we perform a novel gedankenexperiment to devise a new wavelet domain filtering procedure for noise removal in photon imaging systems the filter adapts to both the signal and the noise and balances the trade off between noise removal and excessive smoothing of image details designed using the statistical method of cross validation the filter is simultaneously optimal in a small sample predictive sum of squares sense and asymptotically optimal in the mean square error sense the filtering procedure has a simple interpretation as a joint edge detection estimation process moreover we derive an efficient algorithm for performing the filtering that has the same order of complexity as the fast wavelet transform itself the performance of the new filter is assessed with simulated data experiments and tested with actual nuclear medicine imagery

on modified chebyshev polynomials abstract two elegant representations are derived for the modified chebyshev polynomials discussed by witula and slota r witula d slota on modified chebyshev polynomials j math anal appl 324 2006 321 343

generalized nonlinear yule models with the aim of considering models related to random graphs growth exhibiting persistent memory we propose a fractional nonlinear modification of the classical yule model often studied in the context of macroevolution here the model is analyzed and interpreted in the framework of the development of networks such as the world wide web nonlinearity is introduced by replacing the linear birth process governing the growth of the in links of each specific webpage with a fractional nonlinear birth process with completely general birth rates among the main results we derive the explicit distribution of the number of in links of a webpage chosen uniformly at random recognizing the contribution to the asymptotics and the finite time correction the mean value of the latter distribution is also calculated explicitly in the most general case furthermore in order to show the usefulness of our results we particularize them in the case of specific birth rates giving rise to a saturating behaviour a property that is often observed in nature the further specialization to the non fractional case allows us to extend the yule model accounting for a nonlinear growth

on moments of classical orthogonal polynomials abstract in this work using the inversion coefficients and some connection coefficients between some polynomial sets we give explicit representations of the moments of all the orthogonal polynomials belonging to the askey wilson scheme generating functions for some of these moments are also provided

wavelet based rician noise removal for magnetic resonance imaging it is well known that magnetic resonance magnitude image data obey a rician distribution unlike additive gaussian noise rician noise is signal dependent and separating signal from noise is a difficult task rician noise is especially problematic in low signal to noise ratio snr regimes where it not only causes random fluctuations but also introduces a signal dependent bias to the data that reduces image contrast this paper studies wavelet domain filtering methods for rician noise removal we present a novel wavelet domain filter that adapts to variations in both the signal and the noise

run related probability functions applied to sampling inspection run related distributions are frequently encountered in sampling inspection in the application of stopping rules for different phases of sampling in many cases the generating function of these distributions is known nevertheless it is too complicated to derive the pdf using the ordinary method and only the expectation and variance are obtained we present an approach based on partial fraction expansion to derive the exact pdf of several run related variables from their generating function using efficient algorithms for finding roots of polynomials and simple recursive formulas we complete the method presented by feller we derive and present exact probability functions of some run related variables an application to sampling inspection of attribute data in batches and for continuous production is suggested

on a ramanujan equation connected with the median of the gamma distribution in this paper we consider the sequence n n 0 solving the ramanujan equation e n 2 n k 0 n k k n n n n 1 n 0 1 the three main achievements are the following we introduce a continuous time extension 0 t of on and show its close connections with the medians an of the n 1 1 distributions and the charlier polynomials we give upper and lower bounds for both 0 t and an in particular for on which are sharper than other known estimates finally we show and at the same time complete two conjectures by chen and rubin referring to the sequence of medians n n 1

eaton s markov chain its conjugate partner and mathscr p admissibility suppose that x is a random variable with density f x and that x is a proper posterior corresponding to an improper prior the prior is called p admissible if the generalized bayes estimator of every bounded function of is almost admissible under squared error loss eaton showed that recurrence of the markov chain with transition density r x f x dx is a sufficient condition for p admissibility of we show that eaton s markov chain is recurrent if and only if its conjugate partner with transition density r y x f y x d is recurrent this provides a new method of establishing p admissibility often one of these two markov chains corresponds to a standard stochastic process for which there are known results on recurrence and transience for example when x is poisson and an improper gamma prior is placed on the markov chain defined by r y x is equivalent to a branching process with immigration we use this type of argument to establish p admissibility of some priors when f is a negative binomial mass function and when f is a gamma density with known shape

sharp logarithmic inequalities for riesz transforms abstract let d be a given positive integer and let r j j 1 d denote the collection of riesz transforms on r d for any k 2 we determine the optimal constant l such that the following holds for any locally integrable borel function f on r d any borel subset a of r d and any j 1 2 d we have a r j f x d x k r d f x d x a l here t t 1 log t 1 t for t 0 the proof is based on probabilistic techniques and the existence of certain special harmonic functions as a by product we obtain related sharp estimates for the so called re expansion operator an important object in some problems of mathematical physics

an estimator of species overlap using a modified beta binomial model a new species overlap measure is defined as the probability that given a randomly selected species is present in at least one of the two communities it is present in both communities this measure can be thought of as a community analog of the jaccard index for species overlap between two samples standard maximum likelihood methods are used to estimate the parameters of a modified beta binomial distribution this leads directly to estimates of the community overlap measure in an example from decapod communities on coral reefs it is shown that this new estimate of community species overlap is 50 larger than the commonly used jaccard measure of species overlap

estimating overdispersion when fitting a generalized linear model to sparse data summary we consider the problem of fitting a generalized linear model to overdispersed data focussing on a quasilikelihood approach in which the variance is assumed to be proportional to that specified by the model and the constant of proportionality is used to obtain appropriate standard errors and model comparisons it is common practice to base an estimate of on pearson s lack of fit statistic with or without farrington s modification we propose a new estimator that has a smaller variance subject to a condition on the third moment of the response variable we conjecture that this condition is likely to be achieved for the important special cases of count and binomial data we illustrate the benefits of the new estimator using simulations for both count and binomial data

fragment length distributions and collision probabilities for aflp markers aflp is a dna fingerprinting technique frequently used in plant and animal sciences a drawback of the technique is the occurrence of multiple dna fragments of the same length in a single aflp lane which we name a collision in this article we quantify the problem the well known birthday problem plays a role calculation of collision probabilities requires a fragment length distribution fld we discuss three ways to estimate the fld based on theoretical considerations on in silico determination using dna sequence data from arabidopsis thaliana or on direct estimation from aflp data in the latter case we use a generalized linear model with monotone smoothing of the fragment length probabilities collision probabilities are calculated from two perspectives assuming known fragment counts and assuming known band counts we compare results for a number of fld s ranging from uniform to highly skewed the conclusion is that collisions occur often with higher probabilities for higher numbers of bands for more skewed distributions and to a lesser extent for smaller scoring ranges for a typical plant genome an aflp with 19 bands is likely to contain the first collision practical implications of collisions are discussed aflp examples from lettuce and chicory are used for illustration

irregular trellis for the near capacity unary error correction coding of symbol values from an infinite set irregular joint source and channel coding jscc scheme is proposed which we refer to as the irregular unary error correction iruec code this code operates on the basis of a single irregular trellis instead of employing a set of separate regular trellises as in previous irregular trellis based codes our irregular trellis is designed with consideration of the uec free distance which we characterize for the first time in this paper we conceive the serial concatenation of the proposed iruec code with an irregular unity rate code irurc code and propose a new extrinsic information transfer exit chart matching algorithm for parametrizing these codes this facilitates the creation of a narrow exit tunnel at a low e text b n 0 value and provides near capacity operation owing to this our scheme is found to offer a low symbol error ratio ser which is within 0 4 db of the discrete input continuous output memoryless channel dcmc capacity bound in a particular practical scenario where gray mapped quaternary phase shift keying qpsk modulation is employed for transmission over an uncorrelated narrowband rayleigh fading channel with an effective throughput of 0 508 text bit text s 1 text hz 1 furthermore the proposed iruec irurc scheme offers a ser performance gain of 0 8 db compared to the best of several regular and irregular separate source and channel coding sscc benchmarkers which is achieved without any increase in transmission energy bandwidth transmit duration or decoding complexity

power laws for family sizes in a duplication model qian luscombe and gerstein j molecular biol 313 2001 673 681 introduced a model of the diversification of protein folds in a genome that we may formulate as follows consider a multitype yule process starting with one individual in which there are no deaths and each individual gives birth to a new individual at rate 1 when a new individual is born it has the same type as its parent with probability 1 r and is a new type different from all previously observed types with probability r we refer to individuals with the same type as families and provide an approximation to the joint distribution of family sizes when the population size reaches n we also show that if 1 s n 1 r then the number of families of size at least s is approximately cns 1 1 r while if n 1 r s the distribution decays more rapidly than any power

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

large sample properties of matching estimators for average treatment effects matching estimators for average treatment effects are widely used in evaluation research despite the fact that their large sample properties have not been established in many cases the absence of formal results in this area may be partly due to the fact that standard asymptotic expansions do not apply to matching estimators with a fixed number of matches because such estimators are highly nonsmooth functionals of the data in this article we develop new methods for analyzing the large sample properties of matching estimators and establish a number of new results we focus on matching with replacement with a fixed number of matches first we show that matching estimators are not n 1 2 consistent in general and describe conditions under which matching estimators do attain n 1 2 consistency second we show that even in settings where matching estimators are n 1 2 consistent simple matching estimators with a fixed number of matches do not attain the semiparametric efficiency bound third we provide a consistent estimator for the large sample variance that does not require consistent nonparametric estimation of unknown functions software for implementing these methods is available in matlab stata and r

sensitivity of score tests for zero inflation in count data in many biomedical applications count data have a large proportion of zeros and the zero inflated poisson regression zip model may be appropriate a popular score test for zero inflation comparing the zip model to a standard poisson regression model was given by van den broek similarly for count data that exhibit extra zeros and are simultaneously overdispersed a score test for testing the zip model against a zero inflated negative binomial alternative was proposed by ridout hinde and demetrio however these test statistics are sensitive to anomalous cases in the data and incorrect inferences concerning the choice of model may be drawn in this paper diagnostic measures are derived to assess the influence of observations on the score statistics two examples that motivated the application of zero inflated regression models are considered to illustrate the importance of sensitivity analysis of the zero inflation tests

real and spurious contributions for the shannon renyi and tsallis entropies abstract a two parameter probability distribution is constructed by dilatation or contraction of the escort probability distribution this transformation involves a physical probability distribution p associated with the system under study and an almost arbitrary reference probability distribution p in contrast to the shannon and renyi entropies the tsallis entropy does not decompose as the sum of the physical contribution due to p and the reference or spurious part owing to p for solving this problem a slight modification to the relation between tsallis and renyi entropies must be introduced the procedure in this paper gives rise to a nonconventional one parameter shannon entropy and to two parameter renyi and tsallis entropies associated with p it also contributes to clarify the meaning and role of the escort probabilities set

finding efficient recursions for risk aggregation by computer algebra we derive recursions for the probability distribution of random sums by computer algebra unlike the well known panjer type recursions they are of finite order and thus allow for computation in linear time this efficiency is bought by the assumption that the probability generating function of the claim size be algebraic the probability generating function of the claim number is supposed to be from the rather general class of d finite functions

saddlepoint approximations of the distribution of the person parameter in the two parameter logistic model large sample theory states the asymptotic normality of the maximum likelihood estimator of the person parameter in the two parameter logistic 2pl model in short tests however the assumption of normality can be grossly wrong as a consequence intended coverage rates may be exceeded and confidence intervals are revealed to be overly conservative methods belonging to the higher order theory more specifically saddlepoint approximations are a convenient way to deal with small sample problems confidence bounds obtained by these means hold the approximate confidence level for a broad range of the person parameter moreover an approximation to the exact distribution permits to compute median unbiased estimates mue that are as likely to overestimate as to underestimate the true person parameter additionally in small samples these mue are less mean biased than the often used maximum likelihood estimator

uniform asymptotics of some abel sums arising in coding theory we derive uniform asymptotic expressions of some abel sums appearing in some problems in coding theory and indicate the usefulness of these sums in other fields like empirical processes machine maintenance analysis of algorithms probabilistic number theory queuing models etc

order statistics of ransac and their practical application for statistical analysis purposes ransac is usually treated as a bernoulli process each hypothesis is a bernoulli trial with the outcome outlier free contaminated a run is a sequence of such trials however this model only covers the special case where all outlier free hypotheses are equally good e g generated from noise free data in this paper we explore a more general model which obviates the noise free data assumption we consider ransac a random process returning the best hypothesis delta 1 1 among a number of hypotheses drawn from a finite set theta we employ the rank of delta 1 1 within theta for the statistical characterisation of the output present a closed form expression for its exact probability mass function and demonstrate that beta distribution is a good approximation thereof this characterisation leads to two novel termination criteria which indicate the number of iterations to come arbitrarily close to the global minimum in theta with a specified probability we also establish the conditions defining when a ransac process is statistically equivalent to a cascade of shorter ransac processes these conditions justify a ransac scheme with dedicated stages to handle the outliers and the noise separately we demonstrate the validity of the developed theory via monte carlo simulations and real data experiments on a number of common geometry estimation problems we conclude that a two stage ransac process offers similar performance guarantees at a much lower cost than the equivalent one stage process and that a cascaded set up has a better performance than lo ransac without the added complexity of a nested ransac implementation

bayesian analysis of the differences of count data paired count data usually arise in medicine when before and after treatment measurements are considered in the present paper we assume that the correlated paired count data follow a bivariate poisson distribution in order to derive the distribution of their difference the derived distribution is shown to be the same as the one derived for the difference of the independent poisson variables thus recasting interest on the distribution introduced by skellam using this distribution we remove correlation which naturally exists in paired data and we improve the quality of our inference by using exact distributions instead of normal approximations the zero inflated version is considered to account for an excess of zero counts bayesian estimation and hypothesis testing for the models considered are discussed an example from dental epidemiology is used to illustrate the proposed methodology

statistical approaches to establishing vaccine safety in a vaccine safety trial the primary interest is to demonstrate that the vaccine is sufficiently safe rejecting the null hypothesis that the relative risk of an adverse event attributable to the new vaccine is above a prespecified value greater than one we evaluate the exact probability of type i error of the likelihood score test with sample size determined by normal approximation by enumeration of the binomial outcomes in the rejection region and show that it exceeds the nominal level in the case of rare adverse events we recommend the poisson approximation as an alternative and develop the corresponding conditional and unconditional tests we give sample size and power calculations for these tests we also propose optimal randomization strategies which either i minimize the total number of adverse cases or ii minimize the expected number of subjects when the vaccine is unsafe we illustrate the proposed methods using a hypothetical vaccine safety study copyright 2002 john wiley sons ltd

quantifying evenly distributed states in exclusion and nonexclusion processes spatial point data sets generated from a wide range of physical systems and mathematical models can be analyzed by counting the number of objects in equally sized bins we find that the bin counts are related to the p olya distribution new measures are developed which indicate whether or not a spatial data set generated from an exclusion process is at its most evenly distributed state the complete spatial randomness csr state to this end we define an index in terms of the variance between the bin counts limiting values of the index are determined when objects have access to the entire domain and when there are subregions of the domain that are inaccessible to objects using three case studies lagrangian fluid particles in chaotic laminar flows cellular automata agents in discrete models and biological cells within colonies we calculate the indexes and verify that our theoretical csr limit accurately predicts the state of the system these measures should prove useful in many biological applications

new stopping criterion for genetic algorithms abstract genetic algorithms have been successfully applied in a wide variety of problems although widely used there are few theoretical guidelines for determining when to terminate the search one result by aytug and koehler provides a loose bound on the number of ga generations needed to see all populations and hence an optimal solution with a specified probability in this paper we derive a tighter bound this new bound is on the number of iterations required to achieve a level of confidence to guarantee that a genetic algorithm has seen all strings and hence an optimal solution

on the failure of the bootstrap for matching estimators matching estimators are widely used in empirical economics for the evaluation of programs or treatments researchers using matching methods often apply the bootstrap to calculate the standard errors however no formal justification has been provided for the use of the bootstrap in this setting in this article we show that the standard bootstrap is in general not valid for matching estimators even in the simple case with a single continuous covariate where the estimator is root n consistent and asymptotically normally distributed with zero asymptotic bias valid inferential methods in this setting are the analytic asymptotic variance estimator of abadie and imbens 2006a as well as certain modifications of the standard bootstrap like the subsampling methods in politis and romano 1994

negative binomial process count and mixture modeling the seemingly disjoint problems of count and mixture modeling are united under the negative binomial nb process a gamma process is employed to model the rate measure of a poisson process whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to an nb process for count modeling a draw from the nb process consists of a poisson distributed finite number of distinct atoms each of which is associated with a logarithmic distributed number of data samples we reveal relationships between various count and mixture modeling distributions and construct a poisson logarithmic bivariate distribution that connects the nb and chinese restaurant table distributions fundamental properties of the models are developed and we derive efficient bayesian inference it is shown that with augmentation and normalization the nb process and gamma nb process can be reduced to the dirichlet process and hierarchical dirichlet process respectively these relationships highlight theoretical structural and computational advantages of the nb process a variety of nb processes including the beta geometric beta nb marked beta nb marked gamma nb and zero inflated nb processes with distinct sharing mechanisms are also constructed these models are applied to topic modeling with connections made to existing algorithms under poisson factor analysis example results show the importance of inferring both the nb dispersion and probability parameters

likelihood analyses of overdispersed poisson models written here as lagps o a the resulting estimated predictive probability mass functions being shown in scollnik s figure 3 the first set of priors was gamma 2 0 25 for 0 where the gamma oa d distribution has density o3xk c exp 3x f o and uniform o 1 for a the second set had the same gamma distribution for 0 and for a it had the probability density exp 0 5 1 1 a 2 1a 3 corresponding to a negative exponential distribution with rate 0 5 for the index of dispersion 1 the mass functions are remarkably similar raising the question as to how different the results for standard maximum likelihood ml would be janardan kerster and schaeffer 1979 fitted these data using moments obtaining an excellent fit the social behavior of the isopods would plausibly lead to a deficiency in the single bug class and a corresponding inflation of the other classes perhaps the simplest model for this is obtained by taking a basic poisson a distribution applying a factor of w to the unit class and scaling all others by the corresponding factor to produce a probability distribution abbreviated as ps1 w a cole 1946 had proposed some rather more complicated ad hoc models these two models lagps and psi when fitted by ml all the non bayesian fits described below use ml to the quoted data give results shown in the third and fourth columns of table 1 the fit for lagps 0 a is obviously very good with a chi squared value of 13 4 with 13 d f grouping to at least unity from the smallest class upwards if necessary repeatedly and a p value of 0 41 a negative binomial fit is very similar the ps1 w a fit is equally obviously very bad chi squared of 386 with 7 d f so that this simplest model is not acceptable moreover the estimate of w the factor by which the unit class is supposed to be reduced is 2 5 with a standard error of 0 6 which is not at all consistent with the model in context the alternative very similar model for which all the transferred probability from the unit class is placed in the zero class fits almost as badly for lagps comparisons of the ml estimates with the two sets of bayesian procedure estimates are shown in table 2 extracted from scollnik s figure 3 using the headings of mean and sd for

a simple soluble model of discrete sequential fragmentation a simple model of discrete sequential fragmentation consists in first breaking a unit segment into pieces of equal length a fragment is then selected at random among all fragments and broken in turn into q pieces of equal length and so on ad infinitum fragments of size are thus produced after f fragmentation events the distribution of the ensemble averaged number of fragments of size s is calculated exactly in terms of signless stirling numbers of the first kind and shown to tend asymptotically to a poisson distribution with parameter an asymptotic lognormal distribution is thus found for the distribution of l

measuring operational risk using a mean scaled individual risk model building on a new theory of parametric risk models initiated in hurlimann blatter der deutschen gesellschaft fur versicherungsmathematik xxiii 3 1998 251 it is shown how mean scaled individual risk models can be constructed the approximate computation of their distributions and related quantities can be done in the author s 1990 mathematical framework of generalized pseudo compound poisson distributions which are characterized by integral equations of volterra type the results by dhaene and de pril insurance mathematics and economics 14 1994 181 are extended to this more general class of models the method is applied to the construction of a mean scaled operational risk model

estimation of the shape parameter k of the negative binomial distribution using approximate bayesian methods an estimator of the shape parameter of the negative binomial distribution is introduced most of the available estimators of this parameter such as the method of moment and the maximum likelihood estimators mme and mle give inadmissible values when the mean of the observed sample is larger than its variance in this case the mme gives values while the mle is infinite such observed sample in not unlikely when the mean of the population is close to its variance the suggested estimator always has valid values and turned out to be more efficient for most values of the parameter space family size data is used for illustration of the procedure

nonparametric regression in exponential families most results in nonparametric regression theory are developed only for the case of additive noise in such a setting many smoothing techniques including wavelet thresholding methods have been developed and shown to be highly adaptive in this paper we consider nonparametric regression in exponential families with the main focus on the natural exponential families with a quadratic variance function which include for example poisson regression binomial regression and gamma regression we propose a unified approach of using a mean matching variance stabilizing transformation to turn the relatively complicated problem of nonparametric regression in exponential families into a standard homoscedastic gaussian regression problem then in principle any good nonparametric gaussian regression procedure can be applied to the transformed data to illustrate our general methodology in this paper we use wavelet block thresholding to construct the final estimators of the regression function the procedures are easily implementable both theoretical and numerical properties of the estimators are investigated the estimators are shown to enjoy a high degree of adaptivity and spatial adaptivity with near optimal asymptotic performance over a wide range of besov spaces the estimators also perform well numerically

frequency of frequencies distributions and size dependent exchangeable random partitions abstractmotivated by the fundamental problem of modeling the frequency of frequencies fof distribution this paper introduces the concept of a cluster structure to define a probability function that governs the joint distribution of a random count and its exchangeable random partitions a cluster structure naturally arising from a completely random measure mixed poisson process allows the probability distribution of the random partitions of a subset of a population to be dependent on the population size a distinct and motivated feature that makes it more flexible than a partition structure this allows it to model an entire fof distribution whose structural properties change as the population size varies a fof vector can be simulated by drawing an infinite number of poisson random variables or by a stick breaking construction with a finite random number of steps a generalized negative binomial process model is proposed to generate a cluster structure where in the prior the number of clusters is f

probabilistic derivation of some generating functions for the laguerre polynomials abstract a well known generating function of the classical laguerre polynomials was recently rederived probabilistically by lee in this paper some other presumably new generating functions for the laguerre polynomials are derived by means of probabilistic considerations a direct analytical proof of each of these generating functions is also presented for the sake of completeness

quantiles in nonrandom samples and observational studies abstract in a nonrandom sample or a nonrandomized comparison of treated and control groups methods are developed for displaying the sensitivity of confidence intervals for population quantiles to the magnitude of the departure from random selection quantiles of a single distribution and for a shift in two distributions are both discussed these confidence intervals are based on a new inequality for the distribution of the number of successes in a nonrandom sample or nonrandomized comparison the inequality has additional applications in particular it greatly accelerates the computations in a sensitivity analysis for the mantel haenszel statistic in a 2 2 s contingency table examples include a nonrandom collection of blood samples from a town with comparatively high levels of radon gas an observational study of chromosome damage from mercury and a study of the drug allopurinol as a cause of skin rash the proof of the inequality makes use of savage s lattice for ranks and holley s inequality wh

renyi and tsallis entropies for incomplete or overcomplete systems of events abstract in this article shannon renyi and tsallis entropies are considered for a system of events characterized by an arbitrary probability distribution p that can be incomplete complete or overcomplete after a suitable transformation that leads to the escort probabilities of p these can be written as the canonical probability distribution for a set of pseudo energies hartley information e n ln p n and a dimensionless parameter q that plays the role of thermodynamics several relations between the entropies are presented including the analysis of compound systems the method is illustrated with an example

the compound poisson distribution and return times in dynamical systems previously it has been shown that some classes of mixing dynamical systems have limiting return times distributions that are almost everywhere poissonian here we study the behaviour of return times at periodic points and show that the limiting distribution is a compound poissonian distribution we also derive error terms for the convergence to the limiting distribution we also prove a very general theorem that can be used to establish compound poisson distributions in many other settings

sharp bounds on the entropy of the poisson law and related quantities one of the difficulties in calculating the capacity of certain poisson channels is that h the entropy of the poisson distribution with mean is not available in a simple form in this paper we derive upper and lower bounds for h that are asymptotically tight and easy to compute the derivation of such bounds involves only simple probabilistic and analytic tools this complements the asymptotic expansions of knessl 1998 jacquet and szpankowski 1999 and flajolet 1999 the same method yields tight bounds on the relative entropy d n p between a binomial and a poisson thus refining the work of harremoe s and ruzankin 2004 bounds on the entropy of the binomial also follow easily

statistical distributions connected to motion of substance in a channel of a network abstract we discuss a model of motion of substance through the nodes of a channel of a network the channel can be modeled by a chain of urns where each urn can exchange substance with the neighboring urns in addition the urns can exchange substance with the network nodes and the extension of the model with respect to the previous research is that we consider also the possibility for exchange of substance among the urns and the environment of the network we consider stationary regime of motion of substance through a finite channel stationary regime of exchange of substance along the chain of urns and obtain a class of statistical distributions for the substance in the channel our attention is concentrated on this class of distributions and we show that for the case of finite channel the class of distributions contains as particular cases truncated versions of the families of distributions of katz ord kemp etc several characteristics of the obtained class of distributions are calculated the theory for the case of chain containing infinite number of urns is presented in the appendix

statistical analysis for geometric distribution based on records in this paper based on records from the geometric distribution non bayesian and bayesian estimators for the unknown parameter are obtained the empirical bayes estimate is derived using the squared error loss function the admissibility of estimators are discussed also the problem of predicting the future record values either point or interval based on the past record values observed is considered from bayesian and non bayesian approaches

differential subordination and strong differential subordination for continuous time martingales and related sharp inequalities let x and y be two continuous time martingales if quadratic variation of x minus that of y is a nondecreasing and nonnegative function of time we say that y is differentially subordinate to x and prove that y p p 1 x p for 1 p where p p v q and q is the conjugate of p this inequality contains burkholder s l p inequality for stochastic integrals which implies that the above inequality is sharp we also extend his concept of strong differential subordination and several other of his inequalities and sharpen an inequality of banuelos

a compound negative binomial distribution with mutative termination conditions based on a change point abstract in this paper an extended negative binomial distribution called nbm is proposed by considering mutative termination conditions based on a change point if the condition a is satisfied before the occurrence of the change point the trial is terminated according to condition a otherwise if the condition a does not happen and the change point is satisfied the termination condition of the trial changes from condition a to condition b we consider the conditions under which the resulting distribution can be degenerated to the existing negative binomial distributions and other new negative binomial distributions the finite markov chain imbedding approach is employed to derive the new negative binomial distribution nbm and to obtain the related probabilistic indexes furthermore we study the distribution of the compound negative binomial distribution with mutative termination conditions based on a change point cnbm by means of phase type representations

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators