statistical models for autocorrelated count data a generalized linear mixed model is an increasingly popular choice for the modelling of correlated non normal responses in a regression setting a number of methods are currently available for fitting a generalized linear mixed model including monte carlo markov chain maximum likelihood algorithms approximate maximum likelihood pql iterative bias correction and others of interest in this paper is to compare the parameter estimation of the various methods in the modelling of a count data set the incidence of polio in the usa over the period 1970 1983 using a longlinear generalized linear mixed model with an autoregressive correlation structure despite the fact that all of these methods are considered valid modelling techniques we find that parameter estimates and standard errors differ substantially between analyses particularly in the estimation of the parameters describing the random effects distribution a small simulation study is helpful in understanding some of these differences the methods lead to reasonably similar predictions for future observations with small differences observed in some monthly counts

estimation in generalized mixed models regression models containing fixed and random effects may have a response variable which is not normally distributed the generalized mixed model includes both discrete and continuous response variables and is developed here for problems in which the regression variables enter linearly into the model best linear unbiased predictor methods are extended to maximum likelihood and residual maximum likelihood estimation procedures applications in modelling discrete response variables and in survival analysis are discussed

the extraspecial case of the k gv problem let e be an extraspecial type group and v a faithful absolutely irreducible k e module where k is a finite field let g be the normalizer in gl v of e we show that with few exceptions there exists a v v such that the restriction of v to c h v is self dual whenever h g and h v 1

variance component models with binary response interviewer variability summary interviewer variability in a binary response is an example of a problem requiring variance component estimation in a non normal family the maximum likelihood estimation procedure is derived and used to examine some binary items on a large questionnaire this raises some interesting questions about the use of unbalanced ano va methods with these data two interviewers asking the same question of the same respondent can obtain different responses such interviewer effects create problems in the design and analysis of survey questionnaires collins 1980 gives a review of these problems together with a variety of partial remedies these include pilot studies and careful design to eliminate difficulties with wording of questions and instructions on the questionnaires in spite of these precautions interviewer effects remain questionnaire items are usually categorical and occasionally binary but usually for analysis the questionnaire item is reduced to a binary response it is the variability of the population of interviewers that is of interest and not just the individual interviewers thus this variability is best modelled as a random effect rather than a fixed effect the number of interviews completed by each interviewer varies so that the design is unbalanced thus we have an unbalanced variance component model with a categorical or binary response the intraclass correlation the ratio of the interviewer variance to the sum of all variances in the model is of great interest as a measure of interviewer variability in estimating the variance components the categorical or binary nature of the response is usually ignored and the analysis carried out using analysis of variance this is not maximum likelihood but analysis of variance does have some desirable properties when the error variables are non normal the rule of thumb generally applied is that the analysis of variance is reasonably accurate as long as the proportions in each of the categories of the response are between 0 1 and 0 9 scheffe 1959 p 343 discusses the effects of non normality in a one way balanced model very little theoretical work has been

generalized linear models with random effects salamander mating revisited in recent years much effort has been devoted to extending regression methodology to non gaussian data where responses are not independent these methods for dependent responses are suitable for data from longitudinal studies or nested designs however use of these methods for crossed designs seems to have serious limitations due to the intensive computations involved because of the intractable nature of the joint distribution in this paper we cast the problem in a bayesian framework and use a monte carlo method the gibbs sampler to avoid current computational limitations the flexibility of this approach is illustrated by analyzing the interesting salamander mating data reported by mccullagh and nelder 1989 generalized linear models 2nd edition london chapman and hall

robust estimation in generalized linear mixed models generalized linear mixed models glmms are widely used to analyse non normal response data with extra variation but non robust estimators are still routinely used we propose robust methods for maximum quasi likelihood and residual maximum quasi likelihood estimation to limit the influence of outlying observations in glmms the estimation procedure parallels the development of robust estimation methods in linear mixed models but with adjustments in the dependent variable and the variance component the methods proposed are applied to three data sets and a comparison is made with the nonparametric maximum likelihood approach when applied to a set of epileptic seizure data the methods proposed have the desired effect of limiting the influence of outlying observations on the parameter estimates simulation shows that one of the residual maximum quasi likelihood proposals has a smaller bias than those of the other estimation methods we further discuss the equivalence of two glmm formulations when the response variable follows an exponential family their extensions to robust glmms and their comparative advantages in modelling are described some possible modifications of the robust glmm estimation methods are given to provide further flexibility for applying the method

expectation propagation for likelihood free inference many models of interest in the natural and social sciences have no closed form likelihood function which means that they cannot be treated using the usual techniques of statistical inference in the case where such models can be efficiently simulated bayesian inference is still possible thanks to the approximate bayesian computation abc algorithm although many refinements have been suggested abc inference is still far from routine abc is often excruciatingly slow due to very low acceptance rates in addition abc requires introducing a vector of summary statistics s y the choice of which is relatively arbitrary and often require some trial and error making the whole process laborious for the user we introduce in this work the ep abc algorithm which is an adaptation to the likelihood free context of the variational approximation algorithm known as expectation propagation the main advantage of ep abc is that it is faster by a few orders of magnitude than standard algorithms while producing

binary mosaics and the spatial pattern of heather an idealized model for the spatial pattern of heather in a planar region is proposed in this model the subregiorl occupied by heather is represented by the union of randomly located discs of randomly varying radii some previous applications of the model are noted techniques for parameter estimation and for testing goodness of fit are suggested these techniques are applied to data showing the incidence of heather in a lomx20m rectangle at jadraas sweden thereby demonstrating the extension to binary mosaics of methodology previously illustrated with spatial point processes

accurate and robust tests for indirect inference in this paper we propose accurate parameter and over identification tests for indirect inference under the null hypothesis the new tests are asymptotically 2 distributed with a relative error of order n 1 they exhibit better finite sample accuracy than classical tests for indirect inference which have the same asymptotic distribution but an absolute error of order n 1 2 robust versions of the tests are also provided we illustrate their accuracy in nonlinear regression poisson regression with overdispersion and diffusion models

monte carlo studies on three species two particle diffusion limited reactions abstract we studied two models of three species two particle diffusion limited reactions with and without a drift model a involves annihilations a b 0 b c 0 and c a 0 with hard core interactions between like species and assumes equal initial densities of the species the density decays according to a power law c t t the critical exponent was estimated as 0 353 6 without the drift and 0 390 4 with the maximum drift model b is a coagulation model as a b c b c a and c a b in this model we observed the critical exponent 1 2 regardless of the drift we argue that model b belongs to the same universality class as single species annihilation or coagulation diffusion limited reactions we also studied the average distance of the nearest neighbor species the average domain size and the average number of the species pairs

estimation in generalized linear models with random effects summary a conceptually very simple but general algorithm for the estimation of the fixed effects random effects and components of dispersion in generalized linear models with random effects is proposed conditions are described under which the algorithm yields approximate maximum likelihood or quasi maximum likelihood estimates of the fixed effects and dispersion components and approximate empirical bayes estimates of the random effects the algorithm is applied to two data sets to illustrate the estimation of components of dispersion and the modelling of overdispersion

estimation of the scaling parameter for a kernel type density estimate abstract an algorithm is given for estimation of the scaling parameter in the kernel type density estimate derived from the fourier integral a monte carlo study of five different densities compares the integrated square error ise for the resulting estimate to the ise for the fourier integral estimate fie with mean integrated squared error mise optimal and to the ise for the normal kernel with asymptotically mise optimal the study shows the fie with estimated compares favorably with the other estimates and demonstrates the asymptotic optimality of the fie for smooth densities

fast diffusion mean field theory for k body reactions in one dimension the authors derive an improved mean field approximation for k body annihilation reactions ka to inert for hard core diffusing particles on a line annihilating in groups of k neighbours with probability 0 q or 1 the hopping and annihilation processes are correlated to mimic chemical reactions their new mean field theory accounts for hard core particle properties and has a larger region of applicability than the standard chemical rate equation especially for large k values criteria for validity of the mean field theory and its use in phenomenological data fits are derived numerical tests are reported for k 3 4 5 6

interpreting parameters in the logistic regression model with random effects summary logistic regression with random effects is used to study the relationship between explanatory variables and a binary outcome in cases with nonindependent outcomes in this paper we examine in detail the interpretation of both fixed effects and random effects parameters as heterogeneity measures the random effects parameters included in the model are not easily interpreted we discuss different alternative measures of heterogeneity and suggest using a median odds ratio measure that is a function of the original random effects parameters the measure allows a simple interpretation in terms of well known odds ratios that greatly facilitates communication between the data analyst and the subject matter researcher three examples from different subject areas mainly taken from our own experience serve to motivate and illustrate different aspects of parameter interpretation in these models

a mixture model for interval censored time to response quantal assay data various mixture models are proposed and fitted to a set of time to response quantal assay data the models provide a natural extension to the time dependent case of the classical threshold model for quantal assay data the approach deals simply with interval censoring and provides flexible families of distributions for both the mixing probability and the cumulative distribution function for time to response for data of the kind considered in this paper mixture models may provide a more suitable description than other models from survival analysis such as the proportional hazards model

a simple method for the adjustment of profile likelihoods summary we propose a simple adjustment for profile likelihoods the aim of the adjustment is to alleviate some of the problems inherent in the use of profile likelihoods such as bias inconsistency and overoptimistic variance estimates the adjustment is applied to the profile loglikelihood score function at each parameter value so that its mean is zero and its variance is the negative expected derivative matrix of the adjusted score function for cases in which explicit calculation of the adjustments is difficult we give two methods to simplify their computation an automatic simulation method that requires as input only the profile loglikelihood and its first few derivatives first order asymptotic expressions some examples are provided and a comparison is made with the conditional profile log likelihood of cox and reid

multilevel logistic regression modelling with correlated random effects application to the smoking cessation for youth study a multilevel logistic regression model is presented for the analysis of clustered and repeated binary response data at the subject level serial dependence is expected between repeated measures recorded on the same individual at the cluster level correlations of observations within the same subgroup are present due to the inherent hierarchical setting two random components are therefore incorporated explicitly within the linear predictor to account for the simultaneous heterogeneity and autoregressive structure application to analyse a set of longitudinal data from an adolescent smoking cessation intervention that motivated this study is illustrated

a mixture model combining logistic regression with proportional hazards regression summary a model is proposed for the analysis of censored data which combines a logistic formulation for the probability of occurrence of an event with a proportional hazards specification for the time of occurrence of the event the proposed model is a semiparametric generalization of a parametric model due to farewell 1982 estimates of the regression parameters are obtained by maximizing a monte carlo approximation of a marginal likelihood and the em algorithm is used to estimate the baseline survivor function we present some simulation results to verify the validity of the suggested estimation procedure it appears that the semiparametric estimates are reasonably efficient with acceptable bias whereas the parametric estimates can be highly dependent on the parametric assumptions

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

bias reduction of approximate maximum likelihood estimates for heritability in threshold models approximate maximum likelihood estimates such as penalized quasi likelihood and iterated reweighted reml estimates for heritability or intraclass correlation in threshold models for binary data can be seriously biased two approaches to correct for this bias are studied with emphasis on animal breeding models for binary data scope for reduction of bias of heritability estimates is found to be slim because of the commonly large number of fixed effects in animal breeding models minimal dimensions for the data are identified such that bias and root mean squared error are of modest size and useful inference on heritability or intraclass correlation is feasible

standard errors of prediction in generalized linear mixed models abstract the unconditional mean squared error of prediction umsep is widely used as a measure of prediction variance for inferences concerning linear combinations of fixed and random effects in the classical normal theory mixed model but the umsep is inappropriate for generalized linear mixed models where the conditional variance of the random effects depends on the data when the random effects describe variation between independent small domains and domain specific prediction is of interest we propose a conditional mean squared error of prediction cmsep as a general measure of prediction variance the cmsep is shown to be the sum of the conditional variance and a positive correction that accounts for the sampling variability of parameter estimates we derive a second order correct estimate of the cmsep that consists of three components a a plug in estimate of the conditional variance b a plug in estimate of a taylor series approximation to the correction term and c a bootstrap estimate of

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

a hybrid pairwise likelihood method a modification to the pairwise likelihood method is proposed which aims to improve the estimation of the marginal distribution parameters this is achieved by replacing the pairwise likelihood score equations for estimating such parameters by the optimal linear combinations of the marginal score functions a further advantage of the proposed estimator of marginal parameters over pairwise likelihood is that it is robust to misspecification of the bivariate distributions as long as the univariate marginal distributions are correctly specified while alternating logistic regression can be seen as a special case of the proposed method it is shown that an existing generalization of alternating logistic regression applicable to ordinal data is not the same as and is inferior to the proposed method because it replaces certain conditional densities by pseudodensities that assume working independence the fitting of the multivariate negative binomial distribution is another scenario involving intractable likelihood that calls for the use of pairwise likelihood methods and the superiority of the modified method is demonstrated in a simulation study two examples based on the analyses of salamander mating and patient controlled analgesia data demonstrate the usefulness of the proposed method the possibility of combining optimally the pairwise rather than marginal scores is also considered and its difficulty and potential are discussed

a type of restricted maximum likelihood estimator of variance components in generalised linear mixed models summary the maximum likelihood estimator of the variance components in a linear model can be biased downwards restricted maximum likelihood reml corrects this problem by using the likelihood of a set of residual contrasts and is generally considered superior however this original restricted maximum likelihood definition does not directly extend beyond linear models we propose a reml type estimator for generalised linear mixed models by correcting the bias in the profile score function of the variance components the proposed estimator has the same consistency properties as the maximum likelihood estimator if the number of parameters in the mean and variance components models remains fixed however the estimator of the variance components has a smaller finite sample bias a simulation study with a logistic mixed model shows that the proposed estimator is effective in correcting the downward bias in the maximum likelihood estimator

character correspondences in solvable groups our concern in this paper is with the ordinary complex number character theory of solvable groups if the lengths of some of the earlier papers on this subject l 3 8 are a reliable indication this is a surprisingly complicated and nontrivial theory nevertheless some of the important results are considerably less deep than one would be led to believe by the number of pages involved this paper contains no new theorems and few new ideas it is addressed to the reader who while familiar with character theory has not mastered the literature on characters of solvable groups its purpose is to give such a reader a feeling for the subject and to provide him with accessible proofs for a few of the main results although we certainly do not attempt to prove the strongest or most general results possible or known nevertheless the paper does contain a few items which have application outside of solvable groups in particular we have included in section 4 a brief exposition of tensor induction although our main result could have been proved using an ad hoc argument in place of this general technique the author is now convinced after considerable initial resistance that tensor induction is valuable and deserves further exposure naturally a key idea in the study of characters of solvable groups is to consider the relationships between characters and chief sections in particular suppose that k l is an abelian chief section of g and let 9 e irr k be invariant in g then according to the going down theorem theorem 6 18 of 9 one of the following must occur

consistent estimators in generalized linear mixed models abstract a simple method based on simulated moments is proposed for estimating the fixed effects and variance components in a generalized linear mixed model glmm it is shown that the method is not only computationally attractive but also leads to consistent estimators on the other hand simulation shows that the method can be quite inefficient

monte carlo methods of inference for implicit statistical models a prescribed statistical model is a parametric specification of the distribution of a random vector whilst an implicit statistical model is one defined at a more fundamental level in terms of a generating stochastic mechanism this paper develops methods of inference which can be used for implicit statistical models whose distribution theory is intractable the kernel method of probability density estimation is advocated for estimating a log likelihood from simulations of such a model the development and testing of an algorithm for maximizing this estimated log likelihood function is described an illustrative example involving a stochastic model for quantal response assays is given possible applications of the maximization algorithm to ad hoc methods of parameter estimation are noted briefly and illustrated by an example involving a model for the spatial pattern of displaced amacrine cells in the retina of a rabbit

asymptotically unbiased estimation in generalized linear models with random effects summary obtaining estimates that are nearly unbiased has proven to be difficult when random effects are incorporated into a generalized linear model in this paper we propose a general method of adjusting any conveniently defined initial estimates to result in estimates which are asymptotically unbiased and consistent the method is motivated by iterative bias correction and can be applied in principle to any parametric model a simulation based approach of implementing the method is described and the relationship of the method proposed with other sampling based methods is discussed results from a small scale simulation study show that the method proposed can lead to estimates which are nearly unbiased even for the variance components while the standard errors are only slightly inflated a new analysis of the famous salamander mating data is described which reveals previously undetected between animal variation among the male salamanders and results in better prediction of mating outcomes

robustified maximum likelihood estimation in generalized partial linear mixed model for longitudinal data summary in this article we study the robust estimation of both mean and variance components in generalized partial linear mixed models based on the construction of robustified likelihood function under some regularity conditions the asymptotic properties of the proposed robust estimators are shown some simulations are carried out to investigate the performance of the proposed robust estimators just as expected the proposed robust estimators perform better than those resulting from robust estimating equations involving conditional expectation like sinha 2004 journal of the american statistical association99 451 460 and qin and zhu 2007 journal of multivariate analysis98 1658 1683 in the end the proposed robust method is illustrated by the analysis of a real data set

observations on a stochastic model for quantal assay data ajustement de 2 modeles stochastiques pour decrire les donnees de dosage quantitatifs incorporant le temps de reponse a des jeux de donnees reels et simules application aux modeles mecanicistes

solving non linear estimation equations summary in this paper we consider the numerical computation of a vector parameter estimate 0 which is a root of a system of unbiased non linear estimation equations a sequence 0 r is constructed which converges with a probability approaching 1 to 0 from any starting value 0 0 unlike newton type algorithms the 0 r do not involve matrices of partial derivatives furthermore it can be shown that 0 r 0 is of the order op n 2 so that great accuracy is achieved in very few iterations when the 0 r do not have closed forms an approximation is suggested an alternative and usually simpler way is also found for deriving analytically the expected value of the matrix of partial derivatives in the asymptotic covariance matrix of 0 the efficiency of the proposed algorithm is demonstrated numerically and compared with that of the em and the newton raphson algorithms applications to some well known statistical problems including missing data analysis variance component models familial correlations and errors in variables models are also briefly discussed

maximizing generalized linear mixed model likelihoods with an automated monte carlo em algorithm summary two new implementations of the em algorithm are proposed for maximum likelihood fitting of generalized linear mixed models both methods use random independent and identically distributed sampling to construct monte carlo approximations at the e step one approach involves generating random samples from the exact conditional distribution of the random effects given the data by rejection sampling using the marginal distribution as a candidate the second method uses a multivariate t importance sampling approximation in many applications the two methods are complementary rejection sampling is more efficient when sample sizes are small whereas importance sampling is better with larger sample sizes monte carlo approximation using random samples allows the monte carlo error at each iteration to be assessed by using standard central limit theory combined with taylor series methods specifically we construct a sandwich variance estimate for the maximizer at each approximate e step this suggests a rule for automatically increasing the monte carlo sample size after iterations in which the true em step is swamped by monte carlo error in contrast techniques for assessing monte carlo error have not been developed for use with alternative implementations of monte carlo em algorithms utilizing markov chain monte carlo e step approximations three different data sets including the infamous salamander data of mccullagh and nelder are used to illustrate the techniques and to compare them with the alternatives the results show that the methods proposed can be considerably more efficient than those based on markov chain monte carlo algorithms however the methods proposed may break down when the intractable integrals in the likelihood function are of high dimension

three species diffusion limited reaction with continuous density decay exponents we introduce a model of three species two particle diffusion limited reactions or b or c and or a with three persistence parameters survival probabilities in reaction of the hopping particle we consider isotropic and anisotropic diffusion hopping with a drift in one dimension we find that the particle density decays as a power law for certain choices of the persistence parameter values in the anisotropic case on one symmetric line in the parameter space the decay exponent is monotonically varying between the values close to 1 3 and 1 2 on another less symmetric line the exponent is constant for most parameter values the density does not follow a power law we also calculated various characteristic exponents for the distance of nearest particles and domain structure our results support the recently proposed possibility that one dimensional diffusion limited reactions with a drift do not fall within a limited number of distinct universality classes

consistency of likelihood estimation for gibbs point processes strong consistency of the maximum likelihood estimator mle for parametric gibbs point process models is established the setting is very general it includes pairwise pair potentials finite and infinite multibody interactions and geometrical interactions where the range can be finite or infinite the gibbs interaction may depend linearly or non linearly on the parameters a particular case being hardcore parameters and interaction range parameters as important examples we deduce the consistency of the mle for all parameters of the strauss model the hardcore strauss model the lennard jones model and the area interaction model

generalized linear models with random effects a gibbs sampling approach abstract generalized linear models have unified the approach to regression for a wide variety of discrete continuous and censored response variables that can be assumed to be independent across experimental units in applications such as longitudinal studies genetic studies of families and survey sampling observations may be obtained in clusters responses from the same cluster cannot be assumed to be independent with linear models correlation has been effectively modeled by assuming there are cluster specific random effects that derive from an underlying mixing distribution extensions of generalized linear models to include random effects has thus far been hampered by the need for numerical integration to evaluate likelihoods in this article we cast the generalized linear random effects model in a bayesian framework and use a monte carlo method the gibbs sampler to overcome the current computational limitations the resulting algorithm is flexible to easily accommodate changes in the number

maximum likelihood estimation for probit linear mixed models with correlated random effects the probit normal model for binary data mcculloch 1994 journal of the american statistical association 89 330 335 is extended to allow correlated random effects to obtain maximum likelihood estimates we use the em algorithm with its m step greatly simplified under the assumption of a probit link and its e step made feasible by gibbs sampling standard errors are calculated by inverting a monte carlo approximation of the information matrix rather than via the sem algorithm a method is also suggested that accounts for the monte carlo variation explicitly as an illustration we present a new analysis of the famous salamander mating data unlike previous analyses we find it necessary to introduce different variance components for different species of animals finally we consider models with correlated errors as well as correlated random effects

upper bounds for the number of conjugacy classes of a finite group abstract for a finite group g let k g denote the number of conjugacy classes of g we prove that a simple group of lie type of untwisted rank l over the field of q elements has at most 6 q l conjugacy classes using this estimate we show that for completely reducible subgroups g of gl n q we have k g q 10 n confirming a conjecture of kovacs and robinson for finite groups g with f g a p group we prove that k g cp a where p a is the order of a sylow p subgroup of g and c is a constant for groups with o p g 1 we obtain that k g g p this latter result confirms a conjecture of iranzo navarro and monasor we also improve various earlier results concerning conjugacy classes of permutation groups and linear groups as a by product we show that any finite group g has a soluble subgroup s and a nilpotent subgroup n such that k g s and k g n 3

estimating functions in indirect inference summary there are models for which the evaluation of the likelihood is infeasible in practice for these models the metropolis hastings acceptance probability cannot be easily computed this is the case for instance when only departure times from a g g 1 queue are observed and inference on the arrival and service distributions are required indirect inference is a method to estimate a parameter in models whose likelihood function does not have an analytical closed form but from which random samples can be drawn for fixed values of first an auxiliary model is chosen whose parameter can be directly estimated next the parameters in the auxiliary model are estimated for the original data leading to an estimate the parameter is also estimated by using several sampled data sets simulated from the original model for different values of the original parameter finally the parameter which leads to the best match to is chosen as the indirect inference estimate we analyse which properties an auxiliary model should have to give satisfactory indirect inference we look at the situation where the data are summarized in a vector statistic t and the auxiliary model is chosen so that inference on is drawn from t only under appropriate assumptions the asymptotic covariance matrix of the indirect estimators is proportional to the asymptotic covariance matrix of t and componentwise inversely proportional to the square of the derivative with respect to of the expected value of t we discuss how these results can be used in selecting good estimating functions we apply our findings to the queuing problem

on bandwidth variation in kernel estimates a square root law we consider kernel estimation of a smooth density f at a point but depart from the usual approach in admitting an adaptive dependence of the sharpness of the kernels on the underlying density proportionally varying the bandwidths like f 1 2 at the contributing readings lowers the bias to a vanishing fraction of the usual value and makes for performance seen in well known estimators that force moment conditions on the kernel and so sacrifice positivity of the curve estimate issues of equivariance and variance stabilitization are treated

likelihood analysis of spatial point patterns summary the likelihood procedure for estimating the pairwise interaction potential function is developed for statistical analysis of homogeneous spatial point patterns approximation methods of the normalizing factor of gibbs canonical distribution are discussed both to estimate a scale parameter and to measure the softness or hardness of repulsive interactions the approximations are useful up to a considerably high density the validity of our procedure is demonstrated by some computer experi ments some real data are analysed