integration of quantum many body problems by affine knizhnik zamolodchikov equations abstract an equivalence of the affine analogues of knizhnik zamolodchikov and dunkl equations is established for arbitrary root systems as a corollary an isomorphism is obtained from the solutions of the above kz equations in the induced representations of hecke algebras to the eigenfunctions for the corresponding matrix quantum calogero sutherland operators which generalizes the matsuo map

information theory and maximum product of spacings estimation summary a principle of least information is proposed for estimating continuous univariate distribu tions data are represented by quantiles the principle yields a reference distribution that is identical with the maximum product of spacings estimate a method to estimate continuous univariate distributions is proposed it is suitable when the distribution type is uncertain or when the data are only a small random sample in 1957 jaynes presented the maximum entropy approach to estimation jaynes 1983 which is valid for discrete random variables the product is the least biased estimate possible for the information given data in jaynes s analysis are in the form of a finite set of moments the analysis was extended to continuous random variables by kullback and leibler and axiomatized by shore and johnson 1980 it uses moments and requires an unspecified prior distribution a development of jaynes s entropy theory is presented here the analysis is derived from the kullback leibler relative entropy but uses quantiles instead of moments and gives a minimum information principle for selecting the prior i e the reference distribution p in the univariate case p is identical with the maximum product of spacings mps distribution due to cheng and amin 1983 and ranneby 1984 the mps method thus finds a rationale in information theory conversely these and more recent works titterington 1985 cheng and stephens 1989 provide information on some of the properties of the method proposed in general the mps distribution p does not satisfy the constraints of the sample rule stated here it is not a minimum information model but an approximation there is however a unique model called the posterior q that is closest to p in that it minimizes the relative entropy with respect to p and satisfies the constraints of the data the two minimum principles for p and q are unified into a principle of least information for selection of the reference distribution and determination of the posterior distribution cheng and amin 1983 considered a candidate set in the form of distributions with density p x 6 and cumulative density function p x 6 where 6 is a parameter vector they mapped the sample x and the end points of the domain of x into the unit interval 0 1 by using the transformation y p x 6 subdividing the unit interval into m spacings their mps method chooses 6 to maximize the geometric

bayesian prediction of transformed gaussian random fields abstract a model for prediction in some types of non gaussian random fields is presented it extends the work of handcock and stein to prediction in transformed gaussian random fields where the transformation is known to belong to a parametric family of monotone transformations the bayesian transformed gaussian model btg provides an alternative to trans gaussian kriging taking into account the major sources of uncertainty including uncertainty about the normalizing transformation itself in the computation of the predictive density function unlike trans gaussian kriging this approach mitigates the consequences of a misspecified transformation giving in this sense a more robust predictive inference because the mean of the predictive distribution does not exist for some commonly used families of transformations the median is used as the optimal predictor the btg model is applied in the spatial prediction of weekly rainfall amounts cross validation shows the predicting performance of the btg mo

nonsymmetric macdonald polynomials and demazure characters we establish a connection between a specialization of the nonsymmetric macdonald polynomials and the demazure characters of the corresponding affine kac moody algebra this allows us to obtain a representation theoretical interpretation of the coefficients of the expansion of the specialized symmetric macdonald polynomials in the basis formed by the irreducible characters of the associated finite lie algebra

non regular maximum likelihood problems four non regular estimation problems are reviewed and discussed one the unbounded likelihood problem involves distributions with infinite spikes for which maximum likelihood can fail to give consistent estimators a comparison is made with modified likelihood and spacings methods which do give efficient estimators in this case an application to the box cox shifted power transform is given the other three problems occur when the true parameter lies in some special subregion in one the constrained parameter problem the subregion is a boundary the other two the embedded model and the indeterminate parameters problems occur when the model takes on a special form in the subregion these last two problems have previously been investigated separately we show that they are equivalent in some situations both often arise in non linear models and we give a directed graph approach which allows for their occurrence in nested model building it is argued that many non regular problems can be handled systematically without having to resort to elaborate technical assumptions relatively uncomplicated methods may be used provided that the underlying nature of the non regularity is understood

stress strength reliability analysis of system with multiple types of components using survival signature abstract in this paper we study the estimation for stress strength reliability of the system with multiple types of components based on survival signature in the situation that different types of components are subjected to different types of random stresses the maximum likelihood estimator maximum spacing estimator bootstrap p confidence interval two point estimators and generalized confidence interval using generalized pivotal quantity for system stress strength reliability are derived under the assumption that the stresses and strengths variables follow the gompertz distributions with common or unequal scale parameters additionally when the stresses and strengths variables follow the gompertz distributions with unequal scale parameters a modified generalized confidence interval for the system stress strength reliability based on the fisher z transformation is also proposed in the situation that the system is subjected to the common stress the above point estimators and confidence intervals for the system stress strength reliability are also developed monte carlo simulations are performed to compare the performance of these point estimators and confidence intervals a real data analysis is presented for an illustration of the findings

statistical inference of weibull distribution based on generalized progressively hybrid censored data abstract weibull distribution and generalized progressive hybrid censoring scheme gphcs have been used widely in reliability engineering the gphcs can improve the efficiency of statistical inference by allowing experimenters to observe a pre assigned number of failure samples within a proper testing period in this paper we consider maximum likelihood least squares ls weighted least squares wls and maximum product of spacings mps estimates for two parameter weibull distribution under the assumption that samples are generalized progressively hybrid censored the maximum likelihood estimation mle of unknown parameter is obtained using newton raphson algorithm and expectation maximization em algorithm the asymptotic normality of the mle and bootstrap method are used to construct approximate confidence intervals a parametric bootstrap goodness of fit gof testing is proposed for the weibull distribution with generalized progressively hybrid censored sample extensive monte carlo simulations are conducted to assess the proposed methods finally one real data set is analyzed for illustrative purposes

the box cox metric for nearest neighbour classification improvement abstract the nearest neighbour rule is a distribution free classification method in the literature it has been shown however that the performance of the method improves if complex often locally defined metrics are chosen in this paper we demonstrate that in many real world discrimination problems comparable or even better classification results can be obtained with a simple global metric based on the box cox transformation in three case studies this is demonstrated synthetic data the iris data and real radar data in the latter experiment a large reduction in classification error of more than a factor four is achieved copyright 1997 pattern recognition society

the c function expansion of a basic hypergeometric function associated to root systems we derive an explicit c function expansion of a basic hypergeometric function associated to root systems the basic hypergeometric function in question was constructed as an explicit series expansion in symmetric macdonald polynomials by cherednik in case the associated twisted ane root system is reduced its construction was extended to the nonreduced case by the author it is a meromorphic weyl group invariant solution of the spectral problem of the macdonald q dierence operators the c function expansion is its explicit expansion in terms of the basis of the space of meromorphic solutions of the spectral problem consisting of q analogs of the harish chandra series we express the expansion coecients in terms of a q analog of the harish chandra c function which is explicitly given as product of q gamma functions the c function expansion shows that the basic hypergeometric function formally is a q analog of the heckmanopdam hypergeometric function which in turn specializes to elementary spherical functions on noncompact riemannian symmetric spaces for special values of the parameters

on the choice of prior distribution for the box cox transformed linear model summary the noninformative prior distribution for the parameters of the linear model transformed following box cox 1964 has the non bayesian property of depending to some extent on the data an alternative choice of prior which is not outcome dependent was suggested by pericchi 1981 but it is argued here that this prior has some undesirable features an alternative family of non outcome dependent priors is suggested leading to a noninformative prior which is closer in spirit to that proposed by box cox the posterior consequences of adopting this prior are fully explored and an example discussed

forecasting records by maximum likelihood abstract a maximum likelihood method of fitting a model to a series of records is proposed using ideas from the analysis of censored data to construct a likelihood function based on observed records this method is tried out by fitting several models to series of athletics records for mile and marathon races a form of residual analysis is proposed for testing the models forecasting consequences are also considered in the case of mile records a steady linear improvement since 1931 is found the marathon data are harder to interpret with a steady improvement until 1965 with only slight improvement in world records since then in both cases the normal distribution appears at least as good as extreme value distributions for the distribution of annual best performances short term forecasts appear satisfactory but serious reservations are expressed about using regression type methods to predict long term performance limits

computer assisted analysis of mixtures c a man statistical algorithms summary this paper presents various algorithmic approaches for computing the maximum likelihood estimator of the mixing distribution of a one parameter family of densities and provides a unifying computeroriented concept for the statistical analysis of unobserved heterogeneity i e observations stemming from different subpopulations in a univariate sample the case with unknown number of population subgroups as well as the case with known number of population subgroups with emphasis on the first is considered in the computer package c a man computer assisted mixture analysis it includes an algorithmic menu with choices of the em algorithm the vertex exchange algorithm a combination of both as well as the vertex direction method to ensure reliable convergence a steplength menu is provided for the three latter methods each achieving monotonicity for the direction of choice c a man has the option to work with restricted support size that is the case when the number of components is known a priori in the latter case the em algorithm is used applications of mixture modelling in medical problems are discussed

nonsymmetric difference whittaker functions starting with nonsymmetric global difference spherical functions we define and calculate spinor nonsymmetric global q whittaker functions for arbitrary reduced root systems which are reproducing kernels of the daha fourier transforms of nil daha and solutions of the q toda dunkl eigenvalue problem we introduce the spinor q toda dunkl operators as limits of the difference dunkl operators in daha theory under the spinor variant of the ruijsenaars procedure their general algebraic theory any reduced root systems is the key part of this paper based on the new technique of w spinors and corresponding developments in combinatorics of affine root systems

estimation of nonlinear functionals of densities with confidence this paper introduces a class of k nearest neighbor k nn estimators called bipartite plug in bpi estimators for estimating integrals of nonlinear functions of a probability density such as shannon entropy and renyi entropy the density is assumed to be smooth have bounded support and be uniformly bounded from below on this set unlike previous k nn estimators of nonlinear density functionals the proposed estimator uses data splitting and boundary correction to achieve lower mean square error specifically we assume that t i i d samples xi rd from the density are split into two pieces of cardinality m and n respectively with m samples used for computing a k nn density estimate and the remaining n samples used for empirical estimation of the integral of the density functional by studying the statistical properties of k nn balls explicit rates for the bias and variance of the bpi estimator are derived in terms of the sample size the dimension of the samples and the underlying probability distribution based on these results it is possible to specify optimal choice of tuning parameters m t k for maximizing the rate of decrease of the mean square error the resultant optimized bpi estimator converges faster and achieves lower mean squared error than previous k nn entropy estimators in addition a central limit theorem is established for the bpi estimator that allows us to specify tight asymptotic confidence intervals

a simulation based approach to the parameter estimation for the three parameter gamma distribution abstract the gamma distribution is one of the commonly used statistical distribution in reliability while maximum likelihood has traditionally been the main method for estimation of gamma parameters hirose has proposed a continuation method to parameter estimation for the three parameter gamma distribution in this paper we propose to apply markov chain monte carlo techniques to carry out a bayesian estimation procedure using hirose s simulated data as well as two real data sets the method is indeed flexible and inference for any quantity of interest is readily available

semiparametric estimation of normal mixture densities a semiparametric method for estimating densities of normal mean mixtures is presented this consistent data driven method of estimation is based on probability spacings the estimation technique involves iteratively fixing the standard deviation of the normal kernel that serves as a smoothing parameter and then maximizing a function of the probability spacings over all mixing distributions based on the distribution of uniform spacings a distribution free goodness of fit criterion is developed to guide the selection of the smoothing parameter the result is a set of consistent estimators indexed by a range of smoothing parameters empirical process results are used to prove consistency

optimum biased coin designs for sequential clinical trials with prognostic factors patients in a clinical trial arrive sequentially and are assigned to one of t treatments this assignment should maintain a balance between the numbers receiving each treatment yet should be sufficiently random to avoid any suspicion of conscious or unconscious cheating to balance these requirements efron 1971 introduced biased coin designs for the comparison of two treatments in which allocation of the treatment is determined probabilistically but with a bias towards the underrepresented treatment one disadvantage of efron s scheme is that it does not include balance over covariates or prognostic factors which may affect the response of the patient to the treatment biased coin schemes which do force balance over both treatments and prognostic factors are given by pocock simon 1975 and efron 1980 the properties of the designs have been elucidated by numerical studies and they are now being increasingly used in clinical trials reviews of the literature and discussions of the practical implications of the designs are given by pocock 1979 and simon 1979 however the designs suffer from the disadvantage that they rely on arbitrary functions to achieve the desired balance the procedures thus lack a firm theoretical framework an alternative approach in the presence of prognostic factors begg iglewicz 1980 uses optimum design theory to suggest a deterministic design criterion which is then modified for computational convenience in this paper i use optimum design theory to provide a procedure of the biased coin type for an arbitrary number of treatments in the presence or absence of prognostic factors this has the theoretical advantage of obviating dependence on a series of arbitrary functions the necessary optimum design theory is presented in 2 and in 3 applied to biased coin experiments the simplest case that of two treatments in the absence of prognostic factors is studied in 4 the extension to three or more treatments is in 5 followed in 6 by the allowance for prognostic factors although in the biased coin designs allocation of treatments is made probabilistically the results of the paper can be used for the sequential construction of da optimum designs in which randomization is not introduced by the experimenter all the designs mentioned so far are sequential but not adaptive in the useful distinction made by pericchi 1981 they are data dependent but not outcomedependent each allocation depends on the previous allocations and prognostic factors

a new upside down bathtub shaped hazard rate model for survival data analysis abstract in medical engineering besides demography and other applied disciplines it is pronounced in some applications that the hazard rate of the data initially increased to a pick in the beginning age declined abruptly till it stabilized in statistics literature such hazard rate is known as the upside down bathtub shaped hazard rate and propound in the various survival studies in this paper we proposed a transmuted inverse rayleigh distribution which possesses the upside down bathtub shape for its hazard rate the fundamental properties such as mean variance mean deviation order statistics renyi entropy and stress strength reliability of the proposed model are explored here further three methods of estimation namely maximum likelihood least squares and maximum product spacings methods are used for estimating the unknown parameters of the transmuted inverse rayleigh distribution and compared through the simulation study finally the applicability of the proposed distribution is shown for a set of real data representing the times between failures of the secondary reactor pumps

a bayesian transformation model for wavelet shrinkage wavelet shrinkage estimators in general make the additive normal noise assumption and disregard the nonlinear nature of contamination we develop bayesian wavelet shrinkage estimators based on the power transformations in the linear model to accommodate a broad class of noise models in image processing applications we intend to admit under one roof the widespread additive model the product models common in imaging such as in synthetic aperture radar sar imagery as well as noise that may exist amid these two extremes tactful prior elicitation in this model such as the simultaneous assignment of mixture priors for wavelet coefficients and the transformation imparts flexibility and ample insight into the underlying structure the model permits estimation with unknown noise structure for reasonably unimodal and well behaved on the tails distributions wherein it can outperform common shrinkage estimators extensions with multiple transformations and markov random field priors are also considered for adaptation to local variations in contamination modern markov chain monte carlo mcmc bayesian computation has been used for simulations and several examples are reported

inference from grouped data in three parameter weibull models with applications to breakdown voltage experiments an important objective of breakdown voltage experiments is to estimate either the threshold below which no breakdown occurs or other voltage levels below which breakdown occurs with a small probability a commonly used statistical technique in these studies is to fit a three parameter weibull distribution to the experimental data difficulties in applying the method of maximum likelihood to three parameter weibull models due to nonregularity have led to a variety of alternative approaches in the literature in practice however voltage breakdown and other failure time data are usually discretized or rounded off for grouped data from the weibull model there are still difficulties in applying likelihood methods when the sample size is moderate because of the so called embedded model problem after a brief review of this problem and related likelihood theory we show how the difficulties can be resolved and how likelihood based confidence intervals for the weibull parameters and quantiles can be constru

bayesian likelihood methods for estimating the end point of a distribution summary we consider maximum likelihood methods for estimating the end point of a distribution the likelihood function is modified by a prior distribution that is imposed on the location parameter the prior is explicit and meaningful and has a general form that adapts itself to different settings results on convergence rates and limiting distributions are given in particular it is shown that the limiting distribution is non normal in non regular cases parametric bootstrap techniques are suggested for quantifying the accuracy of the estimator we illustrate performance by applying the method to multiparameter weibull and gamma distributions

a bayesian approach for inverse modeling data assimilation and conditional simulation of spatial random fields 1 this paper addresses the inverse problem in spatially variable fields such as hydraulic conductivity in groundwater aquifers or rainfall intensity in hydrology common to all these problems is the existence of a complex pattern of spatial variability of the target variables and observations the multiple sources of data available for characterizing the fields the complex relations between the observed and target variables and the multiple scales and frequencies of the observations the method of anchored distributions mad that we propose here is a general bayesian method of inverse modeling of spatial random fields that addresses this complexity the central elements of mad are a modular classification of all relevant data and a new concept called anchors data types are classified by the way they relate to the target variable as either local or nonlocal and as either direct or indirect anchors are devices for localization of data they are used to convert nonlocal indirect data into local distributions of the target variables the target of the inversion is the derivation of the joint distribution of the anchors and structural parameters conditional to all measurements regardless of scale or frequency of measurement the structural parameters describe large scale trends of the target variable fields whereas the anchors capture local inhomogeneities following inversion the joint distribution of anchors and structural parameters is used for generating random fields of the target variable s that are conditioned on the nonlocal indirect data through their anchor representation we demonstrate mad through a detailed case study that assimilates point measurements of the conductivity with head measurements from natural gradient flow the resulting statistical distributions of the parameters are non gaussian similarly the moments of the estimates of the hydraulic head are non gaussian we provide an extended discussion of mad vis a vis other inversion methods including maximum likelihood and maximum a posteriori with an emphasis on the differences between mad and the pilot points method

factorized weight functions vs factorized scattering starting from an extensive class of factorized weight functions w p on the n dimensional torus we construct an orthonormal base of symmetric n variable polynomials for l 2 s w p dp via lexicographic ordering of the monomial symmetric functions free boson states and the gram schmidt procedure we show that the dominant asymptotics of these polynomials is factorized as a corollary we obtain a large class of quantum integrable soliton systems on the symmetric subspace of l 2 n the class of weight functions contains in particular the weight function yielding macdonald polynomials for that special case the quantum soliton system can be viewed as the dual relativistic calogero sutherland system

on the beta transformation family several flexible one parameter classes of transformations have been proposed for random variables taking values in 0 11 this article studies a transformation family based on the incomplete beta function that unlike other families proposed includes the arcsin squareroot transformation the logit transformation and the identity transformation applications to regression correlation analysis and response surface analysis are presented a new method of comparing members of different transformation families is derived and connections with generalized linear models are discussed the article concentrates on continuous data rather than binomial data

extremal part of the pbw filtration and nonsymmetric macdonald polynomials abstract given a reduced irreducible root system the corresponding nil daha is used to calculate the extremal coefficients of nonsymmetric macdonald polynomials in the limit t and for antidominant weights which is an important ingredient of the new theory of nonsymmetric q whittaker function these coefficients are pure q powers and their degrees are expected to coincide in the untwisted setting with the extremal degrees of the so called pbw filtration in the corresponding finite dimensional irreducible representations of the simple lie algebras for any root systems this is a particular case of a general conjecture in terms of the level one demazure modules we prove this coincidence for all lie algebras of classical type and for g 2 and also establish the relations of our extremal degrees to minimal q degrees of the extremal terms of the kostant q partition function they coincide with the latter only for some root systems

new integer representations as the sum of three cubes we describe a new algorithm for finding integer solutions to x 3 y 3 z 3 k for specific values of k we use this to find representations for values of k for which no solution was previously known including k 30 and k 52

quantum k theory on flag manifolds finite difference toda lattices and quantum groups let x y be homogeneous coordinates on cp a degree d holomorphic map cp 1 cp is uniquely determined up to a constant scalar factor by n 1 relatively prime degree d binary forms f0 x y fn x y omitting the condition that the forms are relatively prime we compactify the space of degree d holomorphic maps cp 1 cp to a complex projective space of dimension n 1 d 1 1 we denote this compactification of the space of maps by cp d this construction defines the compactification d r i 1cp ni 1 di of the space of degree d d1 dr maps from cp 1 to composing degree d holomorphic maps from cp 1 to the flag manifold x with the plucker embedding we embed the space of such maps into d the closure qmd of this space in d is often referred to as the drinfeld s compactification of the space of degree d maps from cp 1 to x and will be called the space of quasimaps following 14 it is a generally speaking singular irreducible projective variety of complex dimension dimx 2d1 2dr the flag manifold is a homogeneous space of the group slr 1 c and of its maximal compact subgroup sur 1 the action of these groups on x lifts naturally to the spaces d and qmd in addition to this action the spaces d and the subspaces qmd carry the circle action induced by the rotation of cp 1 defined by x y 7 x ey thus the product group g s sur 1 and its complex version gc c slr 1 c act on the quasimap spaces we will see later that qmd have g equivariant desingularizations qmd

estimation reference ranges and goodness of fit for the three parameter log normal distribution the three parameter log normal distribution 3pl is an appropriate model for many of the continuous variables encountered in medicine it is shown how to obtain different types of estimate and approximate sometimes conservative confidence intervals for the parameters of the 3pl and for certain functions of them particularly in the calculation of reference ranges of clinical measurements a simple non iterative estimate of the shift parameter is described the shapiro wilk test of non normality is modified to allow it to be used for testing for departure from the 3pl its power is compared with that of other well known tests the methods are illustrated using several data sets

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

a goodness of fit test using moran s statistic with estimated parameters summary moran s log spacings statistic for testing goodness of fit is shown to have the same asymptotic distribution when parameters must be estimated from the sample as it does when parameters are known provided the estimation is done efficiently for small samples a chi squared approximation is given and shown to be accurate when estimation is carried out by minimizing moran s statistic itself regarded as a function of the unknown parameters a numerical example is given to show the usefulness of moran s statistic

embedded models in three parameter distributions and their estimation summary two distinct problems can arise in maximum likelihood estimation in certain threeparameter problems one problem associated with the fitting of a threshold parameter occurs if the fitted distribution has to be very positively skewed this problem is well known however there is a second and unrelated difficulty which occurs quite often in practice and which arises when the distribution is not at all skew this problem is not so well understood it is shown in this paper that within the three parameter model there is an embedded twoparameter special case which corresponds to infinite parameter values in the original model the problem arises when this embedded model is the best fit to the data the problem is shown to be easily resolved by first carrying out a check to see whether the embedded model should be fitted instead of the three parameter model formal tests for this are discussed the gamma inverse gaussian log normal and weibull distributions are examples where the problem occurs numerical examples are provided for illustration

a generalization of the beta distribution with applications abstract this paper introduces a five parameter beta distribution gb which nests the generalized beta and gamma distributions and includes more than thirty distributions as limiting or special cases the generalized beta leads to an exponential generalized beta egb distribution which includes generalized forms of the logistics exponential gompertz and gumbell distributions and the normal as special cases the egb family of distributiosn provides the basis for partially adaptive estimation of econometric models with possibly skewed and leptokurtic error distributions applications of the models to investigating the distribution of income stock returns and in regression analysis are considered

a bayesian approach to transformations to normality summary the analysis of transformation of observations in the linear model with normal errors proposed by box cox 1964 is considered a different choice of noninformative unnormed prior is advocated which is not outcome dependent this new selection of prior leads to a formal identity between likelihood and bayesian inference both for the estimation of the best transformation to normality and for the presence of homoscedasticity and additivity under this transformation extension to a related problem is mentioned

rogers ramanujan type identities and nil daha abstract using the daha fourier transform of q hermite polynomials multiplied by level one theta functions we obtain expansions of products of any number of such theta functions in terms of the q hermite polynomials an ample family of modular functions satisfying rogers ramanujan type identities for arbitrary reduced twisted affine root systems is obtained as an application a relation to rogers dilogarithm and nahm s conjecture is discussed the q hermite polynomials are closely related to the demazure level one characters in the twisted case sanderson ion which connects our formulas to tensor products of level one integrable kac moody modules their coset theory and the level rank duality

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

estimating parameters in continuous univariate distributions with a shifted origin summary a general method of estimating parameters in continuous univariate distributions is proposed it is especially suited to cases where one of the parameters is an unknown shifted origin this occurs for example in the three parameter lognormal gamma and weibull models for such distributions it is known that maximum likelihood ml estimation can break down because the likelihood is unbounded and this can lead to inconsistent estimators properties of the proposed method are described in particular it is shown to give consistent estimators with asymptotic efficiency equal to ml estimators when these exist moreover it gives consistent asymptotically efficient estimators in situations where ml fails examples are given including numerical ones showing the advantages of the method

on searching for solutions of the diophantine equation x 3 y 3 z 3 n we propose a new search algorithm to solve the equation x 3 y 3 z 3 n for a fixed value of n 0 by parametrizing x min x y z this algorithm obtains y and z if they exist by solving a quadratic equation derived from divisors of x 3 n by using several efficient number theoretic sieves the new algorithm is much faster on average than previous straightforward algorithms we performed a computer search for 51 values of n below 1000 except n 4 mod9 for which no solution has previously been found we found eight new integer solutions for n 75 435 444 501 600 618 912 and 969 in the range of x 2 10 7

a recursion and a combinatorial formula for jack polynomials the jack polynomials j x form a remarkable class of symmetric polynomials they are indexed by a partition and depend on a parameter one of their properties is that several classical families of symmetric functions can be obtained by specializing e g the monomial symmetric functions m the elementary functions e 0 the schur functions s 1 and nally the two classes of zonal polynomials 2 1 2 the jack polynomials can be de ned in various ways e g a as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b as simultaneous eigenfunctions of certain di erential operators the sekiguchi debiard operators recently opdam o constructed a similar family f x of nonsymmetric polynomials the index runs now through all compositions nn they are de ned in a completely similar fashion e g the sekiguchi debiard operators are being replaced by the cherednik di erential re ection operators see sect 3 it is becoming more and more clear that these polynomials are as important as their symmetric counterparts the purpose of this paper is to add to the existing characterizations of jack polynomials two further ones c a recursion formula among the f together with two formulas to obtain j from them d combinatorial formulas of both j and f in terms of certain generalized tableaux

maximum likelihood estimation of parameters in the inverse gaussian distribution with unknown origin maximum likelihood estimation is applied to the three parameter inverse gaussian distribution which includes an unknown shifted origin parameter it is well known that for similar distributions in which the origin is unknown such as the lognormal gamma and weibull distributions maximum likelihood estimation can break down in these latter cases the likelihood function is unbounded and this leads to inconsistent estimators or estimators not asymptotically normal it is shown that in the case of the inverse gaussian distribution this difticulty does not arise the likelihood remains bounded and maximum likelihood estimation yields a consistent estimator with the usual asymptotic normality properties a simple iterative method is suggested for the estimation procedure numerical examples are given in which the estimates in the inverse gaussian model are compared with those of the lognormal and weibull distributions

a new extension of weibull distribution properties and different methods of estimation abstract the weibull distribution has been generalized by many authors in recent years here we introduce a new generalization of the weibull distribution called alpha logarithmic transformed weibull distribution that provides better fits than some of its known generalizations the proposed distribution contains weibull exponential logarithmic transformed exponential and logarithmic transformed weibull distributions as special cases our main focus is the estimation from frequentist point of view of the unknown parameters along with some mathematical properties of the new model the proposed distribution accommodates monotonically increasing decreasing bathtub and unimodal and then bathtub shape hazard rates so it turns out to be quite flexible for analyzing non negative real life data we briefly describe different frequentist approaches namely maximum likelihood estimators percentile based estimators least squares estimators weighted least squares estimators maximum product of spacings estimators and compare them using extensive numerical simulations monte carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples the potentiality of the distribution is analyzed by means of two real data sets

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

density estimation with confidence sets exemplified by superclusters and voids in the galaxies abstract a method is presented for forming both a point estimate and a confidence set of semiparametric densities the final product is a three dimensional figure that displays a selection of density estimates for a plausible range of smoothing parameters the boundaries of the smoothing parameter are determined by a nonparametric goodness of fit test that is based on the sample spacings for each value of the smoothing parameter our estimator is selected by choosing the normal mixture that maximizes a function of the sample spacings a point estimate is selected from this confidence set by using the method of cross validation an algorithm to find the mixing distribution that maximizes the spacings functional is presented these methods are illustrated with a data set from the astronomy literature the measurements are velocities at which galaxies in the corona borealis region are moving away from our galaxy if the galaxies are clustered the velocity density will be multimodal with clusters correspond

generalized weyl modules and nonsymmetric q whittaker functions abstract we introduce generalized global weyl modules and relate their graded characters to nonsymmetric macdonald polynomials and nonsymmetric q whittaker functions in particular we show that the series part of the nonsymmetric q whittaker function is a generating function for the graded characters of generalized global weyl modules

corrected maximum likelihood in non regular problems on suggere que l utilisation du produit des densites pour la fonction de vraisemblance est une mauvaise application de la methode du maximum de vraisemblance dans des cas non reguliers

measures of location of skew distributions obtained through box cox transformations abstract box cox power transformations are used to transform a set of observations to symmetry a measure of location of the original data is obtained by applying the inverse transformation to the center of the transformed data its properties are discussed and it is compared with other measures of location