properties of estimators of the finite population distribution function two leading estimators of the finite population distribution function are studied their large sample mean squared errors are compared under a given model neither estimator is universally better circumstances are identified where the model based estimator would do poorly despite employing the correct model

saddlepoint expansions in finite population problems summary saddlepoint approximations are derived for the sample mean in the case where samples are obtained randomly without replacement from a finite population asymptotic properties are investigated the great accuracy is demonstrated by numerical examples extensions to more general statistics are discussed

characterization of best model based predictors in survey sampling on presente des conditions necessaires et suffisantes d optimalite d un predicteur lineaire arbitraire

on the universality of crossing probabilities in two dimensional percolation six percolation models in two dimensions are studied percolation by sites and by bonds on square hexagonal and triangular lattices rectangles of widtha and heightb are superimposed on the lattices and four functions representing the probabilities of certain crossings from one interval to another on the sides are measured numerically as functions of the ratioa b in the limits set by the sample size and by the conventions and on the range of the ratioa b measured the four functions coincide for the six models we conclude that the values of the four functions can be used as coordinates of the renormalization group fixed point

cohomology of hopf algebras and the clifford s extension problem in this paper we continue our previous study of the extension problem for modules in the case of hopf galois extensions let h be a hopf algebra over a field k let arb be a faithfully flat h galois extension and let m m be a right b module we suppose that all of them are k vector spaces of finite dimension we say that there exists an a extension of m if there exists a k linear map m m m a a m such that m m is a right k a module and m m m b s m m m b m g m b g b w x in 8 for an algebraically closed field k a cocommutative hopf algebra h and a simple stable b module m we have shown that the extension problem is equivalent to the vanishing of a certain 2 cohomology class in 2 h h k and moreover the set of isomorphism classes of a extensions is 1 in one to one correspondence with h h k here k is an h module algebra with trivial action and the cohomology groups are defined as in w x 16 the purpose of this paper is to prove these results for any field not only for algebraically closed ones in the first section we shall prove some results which will help us to reduce our problem to the case of a simple ring b first of all we shall prove that the right ideal generated by the annihilator i of m is a two sided ideal of a and a right coideal so aria is a comodule algebra next we shall prove that its coinvariant subalgebra is isomorphic with bri and the extension arai bri is faithfully flat h galois moreover there exists an a extension for m if and only if there exists an arai ex

statistical inference for poisson and multinomial models for capture recapture experiments summary the classical multinomial model used for estimating the size of a closed population is compared to the highly flexible poisson models introduced by cormack 1981 the multinomial model and generalizations of it which allow for dependence between samples may be obtained from that of cormack by conditioning on the population size the maximum likelihood estimators for n the population size and 0 the vector of parameters describing the capture process are the same in both models completely general formulae for the asymptotic variances of the maximum likelihood estimates of n for both models are given the substantial differences between the variances under the two models are discussed hypotheses concerning 0 may be tested using the log likelihood ratio the procedures which result from both models are asymptotically equivalent under the null hypothesis but differ in power under the alternative

consistent systems of correlators in non semisimple conformal field theory based on the modular functor associated with a not necessarily semisimple finite non degenerate ribbon category d we present a definition of a consistent system of bulk field correlators for

the schur clifford subgroup of the brauer clifford group abstract we define a schur clifford subgroup of turull s brauer clifford group similar to the schur subgroup of the brauer group the schur clifford subgroup contains exactly the equivalence classes coming from the intended application to clifford theory of finite groups we show that the schur clifford subgroup is indeed a subgroup of the brauer clifford group as are certain naturally defined subsets we also show that this schur clifford subgroup behaves well with respect to restriction and corestriction maps between brauer clifford groups

a new estimator for the finite population distribution function summary in this paper we propose a new estimator for the finite population distribution function when there is auxiliary information available about the population the new estimator has a smaller asymptotic mean squared error than two leading estimators in the literature in fact it is in an optimised form in terms of mean squared error results are reported for simulation studies on real and artificial populations

ignorable and informative designs in survey sampling inference summary the role of the sample selection mechanism in a model based approach to finite population inference is examined when the data analyst has only partial information on the sample design then a design which is ignorable when known fully may become informative conditions under which partially known designs can be ignored are established and examined for some standard designs the results are illustrated by an example used by scott 1977

linear regression after selection abstract we study the effect upon linear regression of explicit selection on the dependent variable if the explanatory variables are multinormally distributed along with the dependent variable then the regression coefficient vector in the selected population is a scalar multiple of that in the original population

asymptotically design unbiased predictors in survey sampling summary a probability sampling plan is suggested in this paper for drawing the sample so that the linear least squares predictor of the total of a finite population is made asymptotically design unbiased conditions are given for the sampling plan to reduce to the more familiar sample designs discussed in the literature the linear least squares predictor of the total of a finite population royall 1970 1976 derived under a superpopulation model with explanatory variables omitted is still best under the correct model provided that it is unbiased under the correct model tam 1986 the present paper specifies probability sampling schemes which render the linear least squares predictor asymptoti cally design unbiased such probability schemes enable some general form of balancing in the sample for the explanatory variables omitted from the assumed superpopulation model and provide protection of the predictor against this type of model failure conditions are also presented in this paper for the specified sample design to become the equal probability selection sample design or the optimal sample design considered by godambe 1955 1982

representation groups for the schur index fix an algebraic number field k and an irreducible complex character x of some finite group g assume k k x contains the values of x so that the simple component a 4 x k of the group algebra kg determined by 1 is central over k it is a basic consequence of the brauer witt theorem that a is a so called cyclotomic algebra see ls we exhibit here a slightly different point of view namely that of constructing a representation group for a similar to the concept used in clifford theory from class field theory it is known that the order of the class a of a in the brauer group br k equals its index m x it suffices to study the primary components of a so fix a prime p and let a denote the pcomponent of a in br k let k be the field obtained from k by adjoining the nth roots of unity if n exp g then by brauer k is a splitting field for a and g we shall see that there is a field l s k with p power degree over k such that

optimality criteria in survey sampling summary this paper attempts to fill the gap between some different and conflicting approaches in survey sampling based on a fixed population regression type model a class of optimality criteria similar to those well accepted in optimal experimental design theory is introduced the minimax and superpopulation approaches in survey sampling turn out to correspond to two extreme criteria in the proposed class this helps understand the role of randomization the strategy of simple random sampling with sample mean and the rao hartley cochran strategy are shown to be criterion robust

on estimating distribution functions and quantiles from survey data using auxiliary information summary ratio and difference estimators of a population distribution function under a general sampling design are obtained using auxiliary population information the relative mean errors and relative mean square errors of these estimators and a model based estimator of chambers dunstan 1986 are compared through a simulation study the advantages of the design based estimators over the model based estimator under model misspecifications especially for large samples are demonstrated ratio and difference estimators of a population quantile are also studied

the effect of measurement error summary measurement error causes the distribution generating data to differ from the distribution that is of substantive interest to understand the effect of measurement error on the information produced by statistical procedures it is thus necessary to understand the distortions induced by measurement error this paper sheds light on this by providing small error variance approximations to the distribution of variates subject to contamination by measurement error these provide the basis for the analysis of the properties of estimators and tests and less formal procedures when measurement error is present and the basis for the development of tests to detect measurement error and of models which incorporate its influence

logarithmic operators and logarithmic conformal field theories logarithmic operators and logarithmic conformal field theories are reviewed prominent examples considered here include c 2 and c 0 logarithmic conformal field theories c 0 logarithmic conformal field theories are especially interesting since they describe some of the critical points of a variety of longstanding problems involving a two dimensional quantum particle moving in a spatially random potential as well as critical two dimensional self avoiding random walks and percolation lack of classification of logarithmic conformal field theories remains a major impediment to progress towards finding complete solutions to these problems

extending irreducible representations of normal subgroups let h be a normal subgroup of some finite group g and let v be an irreducible right fh module where f is an arbitrary field it is one of the primary objectives of clifford theory to find necessary and sufhcient conditions so that v can be extended to an fg module an obvious necessary condition is that v has to be g invariant stable sufficiency relies upon various properties of the group extension h g w s where s g h and the module itself the theory is well developed in case v is absolutely irreducible recently isaacs 12 and dade 4 proved generalizing a theorem by gallagher that v can be always extended to an fg module provided h is a hall subgroup of g and v is g invariant the present paper resulted from an effort to understand this remarkable and somewhat surprising result assume in the sequel that v is g invariant let d end v be the opposite endomorphism ring of k basic clifford theory yields a group extension d 52 p p s such that there exists an fg module extending v if and only if 52 52 v splits over the multiplicative group d of the division algebra d see theorem 1 below in general d is noncommutative and so the usual cohomological methods seem to fail dade even remarked in 4 sceptically that none of the nice theorems about extensions of v such as those of gallagher which depend upon calculations in the cohomology groups of s with coefficients in a commutative group can be proved this way in the following situation however one can argue as usual

on the generalization of the likelihood function and the likelihood principle abstract the parametric likelihood and likelihood principle lp play a central role in parametric methodology and in the foundations of statistics the main purpose of this article is to extend the concepts of likelihood and lp to general inferential aims and models covering for example prediction and empirical bayes models the likelihood function is the joint distribution of the data and the unobserved variables of inferential interest considered as a function of the parameters and these inferential variables lp is based on this likelihood and the principles of sufficiency sp and conditionality cp are modified such that the equivalence sp and cp lp is valid generalizing birnbaum s theorem

balanced samples and robust bayesian inference in finite population sampling summary bayesian inference in finite populations uses probability models at two stages i to describe relationships among population units and ii to express uncertainty concerning the values of parameters appearing at stage i here we consider the bayes posterior distribution of the population total when a multivariate normal regression model is used at stage i with a diffuse prior distribution on the regression coefficients we study the situation where the stage i model is in error because an important regressor is omitted and we show that in balanced samples such errors do not affect the posterior distribution cases where the covariance matrix contains an unknown scale parameter or is itself misspecified are also considered

regression estimation and post stratification in factor analysis regression estimation and poststratification are methods used in survey sampling to estimate a population mean when additional information is available for some auxiliary variables the extension of these methods to factor analysis is considered these methods may be used either to improve the efficiency of estimation or to adjust for the effects of nonrandom selection the estimation procedure may be formulated in a lisrel framework

multivariate prediction from selected samples minimum variance unbiased predictors of the mean vector and covariance matrix of a finite population are obtained under the assumption that the population values obey a given linear model an example is given of stratified sampling

survey design under the regression superpopulation model abstract the construction of sample designs and estimators under a linear regression superpopulation model is considered the anticipated variance the variance of the predictor computed with respect to the sampling design and the superpopulation model is used as a criterion for evaluating probability designs and model unbiased predictors regression predictors that are model unbiased and design consistent are constructed

weighted finite population sampling to maximize entropy summary attention is drawn to a method of sampling a finite population of n units with unequal probabilities and without replacement the method was originally proposed by stern cover 1989 as a model for lotteries the method can be characterized as maximizing entropy given coverage probabilities 7ci or equivalently as having the probability of a selected sample proportional to the product of a set of weights wi we show the essential uniqueness of the wi given the 7i and describe practical geometrically convergent algorithms for computing the wi from the 7i we present two methods for stepwise selection of sampling units and corresponding schemes for removal of units that can be used in connection with sample rotation inclusion probabilities of any order can be written explicitly in closed form second order inclusion probabilities 7rij satisfy the condition 0 ij 7ic 7j which guarantees yates grundy s variance estimator to be unbiased definable for all samples and always nonnegative for any sample size

the effect of sample design on principal component analysis abstract most sample surveys are multivariate and many lend themselves to multivariate methods of analysis the most usual mode of such analysis is a standard statistical package such as bmdp or spss in which the multivariate analyses are based on the underlying assumption that the data are generated as independent observations from a common probability distribution this assumption ignores the sample selection procedure involved in the survey which leads to the following basic questions what effects can the sample design have on methods of multivariate analysis how should such effects be taken into account this article considers the case of principal component analysis and in particular the point estimation of the eigenvalues and eigenvectors of a covariance matrix it is assumed that the selection of the sample depends on the population values of auxiliary variables as for example in stratified sampling the conventional estimators based on the assumption of simple random sampling are compar

mixed orthogonal arrays for variance estimation with unequal numbers of primary selections per stratum summary for estimating the variance of nonlinear statistics in large scale complex surveys the method of balanced repeated replications has received special attention the method at present is useful in stratified clustered designs with equal number of primary selections from each stratum this paper extends the method to general sample designs with arbitrary number of selections from each stratum it is shown that mixed orthogonal arrays of strength two or equivalently equal frequency orthogonal main effect plans for asymmetrical factorials give a set of balanced subsamples useful in variance estimation

on the geometric approach to multivariate selection multivariate selection can be represented as a linear transformation in a geometric framework this approach has led to considerable simplification in the study of the effects of selection on factor analysis in this note this approach is extended to describe the effects of selection on regression analysis and to adjust for the effects of selection using the inverse of the linear transformation

fitting prospective regression models to case control data summary we consider fitting prospective regression models to data obtained by case control or response selective sampling from a finite population with known population totals in each response category maximum likelihood estimation is developed and compared with two pseudo likelihood approaches the relative efficiencies of the methods are explored in the special case of estimating the parameters of the proportional odds model for ordinal responses for such applications the method hsieh manski mcfadden 1985 call conditional maximum likelihood is shown to be essentially as efficient as maximum likelihood the latter is considerably more difficult to implement in contrast the use of a weighted estimate of the prospective likelihood can lead to a substantial loss of efficiency

the weighted residual technique for estimating the variance of the general regression estimator of the finite population total summary the paper deals with design based estimation of the variance of the general regression estimator of the finite population total the usual taylor linearization variance estimator is an expression in the design weighted regression residuals in many applications the resulting expression is counterintuitive from a model based standpoint the improved variance estimator in this paper attaches another simple weight called g weight to each individual residual this new variance estimator i gives valid design based confidence intervals ii is nearly unbiased uiider a suitably chosen regression model and iii works well for conditional inference examples are given

the analysis of categorical data from complex sample surveys chi squared tests for goodness of fit and independence in two way tables abstract the effect of stratification and clustering on the asymptotic distributions of standard pearson chi squared test statistics for goodness of fit simple hypothesis and independence in a two way contingency table denoted as x 2 and xi 2 respectively is investigated it is shown that both x 2 and xi 2 are asymptotically distributed as weighted sums of independent 1 2 random variables the weights are then related to the familiar design effects deffs used by survey samplers a simple correction to x 2 which requires only the knowledge of variance estimates or deffs for individual cells in the goodness of fit problem is proposed and empirical results on the performance of corrected x 2 provided empirical work on xi 2 indicated that the distortion of nominal significance level is substantially smaller with xi 2 than with x 2 some results under simple models for clustering are also given

optimality of royall s predictor under a gaussian superpopulation model summary this note shows that under the assumption of a gaussian superpopulation model with a general symmetric and positive definite covariance matrix the model based predictor of royall 1976 is the unbiased minimum mean squared error predictor of the total of a given finite population this somewhat anticipated result is a generalization of recent published work in which it was derived under more restrictive assumptions

asking sensitive questions indirectly summary we propose a new method for conducting surveys on sensitive topics which does not require direct answers from the respondents it overcomes some of the difficulties associated with traditional randomized response techniques and is applicable to both qualitative and quantitative characteristics data collected using the proposed method are distributed according to a mixture distribution and the problem reduces to that of estimating a mixing proportion or a mixing distribution the method can be extended easily to allow multiple trials this enables us to improve the efficiency of our estimator without sacrificing respondent protection and at no increase in sampling cost

conformal algebras of two dimensional disordered systems we discuss the structure of two dimensional conformal field theories at a central charge c 0 describing critical disordered systems polymers and percolation we construct a novel extension of the c 0 virasoro algebra characterized by a number b measuring the effective number of massless degrees of freedom and by a logarithmic partner of the stress tensor it is argued to be present at a generic random critical point lacking super kac moody or other higher symmetries and is a tool to describe and classify such theories interestingly this algebra is not only consistent with but indeed naturally accommodates in general an underlying global supersymmetry polymers and percolation realize this algebra unexpectedly we find that the c 0 kac table of the degenerate fields contains two distinct theories with b 5 6 and b 5 8 which we conjecture to correspond to percolation and polymers respectively a given kac table field can be degenerate only in one of them remarkably we also find this algebra and thereby an ensuing hidden supersymmetry realized at general replica averaged critical points for which we derive an explicit formula for b

representation groups of characters the subject we are concerned with has its roots in the work of schur who introduced the notion of a representation group darstellungs gruppe these groups have the property to lift linearize all projective representations of some finite group over an algebraically closed field 16 fong 8 j noticed that schur s method applies in context with clif ford s work exhibiting some single invariant stable character or module the familiar proof for the famous fong swan theorem gives the model for this kind of representation groups see serre 17 or feit 7 throughout we fix a finite group g with normal subgroup h and quotient group s g h we also fix an absolutely irreducible frobenius character x of h and a field f containing the values of x we assume that x is g invariant as passage from the inertia group to the whole group usually is easy to handle this is not too restrictive it is a necessary con dition for x to be extendible to g

on conditioning for model based inference in survey sampling summary a discontinuity of sampling distribution inference between a prediction problem and an estimation problem in model based survey sampling is discussed the problem is resolved by conditioning on an appropriate ancillary within a curved exponential family framework the ignorability of the sampling design for inference is discussed

clifford theory for mackey algebras we develop a clifford theory for mackey algebras for simple mackey functors using their classification we prove mackey algebra versions of clifford s theorem and the clifford correspondence let r g be the mackey algebra of a finite group g over a commutative unital ring r a nd let 1 n be the unity of r n where n is a normal subgroup of g observing that 1n r g 1n is a crossed product of g n over r n a number of results concerning group graded algebras are extended to the context of mackey algebras including fong s theorem green s indecomposibility theorem and some reduction and extension techniques for indecomposable mackey functors

on clifford theory with galois action abstract let g be a finite group n a normal subgroup of g and irr n let f be a subfield of the complex numbers and assume that the galois orbit of over f is invariant in g we show that there is another triple g 1 n 1 1 of the same form such that the character theories of g over and of g 1 over 1 are essentially the same over the field f and such that the following holds g 1 has a cyclic normal subgroup c contained in n 1 such that 1 n 1 for some linear character of c and such that n 1 c is isomorphic to the abelian galois group of the field extension f f 1 more precisely having the same character theory means that both triples yield the same element of the brauer clifford group brcliff g f defined by a turull

models for nonresponse in sample surveys abstract the literature on the analysis of incomplete data using models is reviewed in the context of nonresponse in sample surveys the modeling approach provides a large body of methods for handling unit and item nonresponse some of which cannot be derived from the randomization theory of inference for surveys key concepts from the literature on incomplete data such as factorizations of the likelihood for special data patterns the em algorithm for general data patterns and ignorability of the response mechanism are discussed within the survey context model based procedures are related to common methods for handling nonresponse in surveys such as weighting or imputation of means within subclasses of the population

adjusting for nonresponse bias using logistic regression summary a method of adjustment for nonresponse is considered no information about the nonrespondents is required but the respondents are assumed to answer all questions of interest one or more call backs are assumed the probabilities of response are represented by a logistic regression model in which variables in the survey are explanatory variables the probabilities are estimated using a modified conditional maximum likelihood approach based on the respondents the estimated probabilities are used in a horvitzthompson type estimator to reduce bias in the estimation of sample moments simulations suggest that the method reduces the mean squared error of estimation when response probabilities depend strongly on variables in the survey

variance estimation under two phase sampling with application to imputation for missing data summary ratio estimation under two phase simple random sampling is studied a new linearisation variance estimator that makes more complete use of the sample data than a standard one is proposed a jackknife variance estimator and its linearised version are also obtained unconditional and conditional repeated sampling properties of these variance estimators are studied through simulation applications to mass imputation under two phase sampling and deterministic imputation for missing data are also given

biometrika centenary sample surveys summary despite karl pearson s stated aim that the evolutionist has to become in the widest sense of the words a registrar general for all forms of life the papers published in biometrika on sample surveys have been mainly theoretical and do not start to appear until the 1940s there has been little work on sampling in the life sciences with the exception of capture recapture methods the major areas have been in variable probability designs the foundations of inference for finite populations analytic surveys and the role of selection mechanisms in inference in recent years the work on survey analysis has been the outcome of an interaction between sample selection mechanisms and mainstream theory with interesting consequences for both areas

optimal controlled sampling designs optimal controlled sampling designs that minimize the probability of selecting a nonpreferred sample are obtained these plans satisfy either of the following two requirements i match specified joint inclusion probabilities ij associated with an uncontrolled sampling plan having certain desirable properties ii match specified first order inclusion probabilities and provide stable nonnegative unbiased variance estimators limitations of other approaches using methods of experimental designs are demonstrated

regression analysis based on stratified samples summary jewell 1985 described an iterative method of estimating parameters in the linear regression model from data collected from stratified samples of the dependent variable with no parametric assumptions on the residual distribution the present paper considers an alternative iterative technique which is generally superior in terms of efficiency often by a considerable amount the method is extended to cover stratified sampling on a design variable which is correlated with the dependent variable but is not included as an independent variable regression analysis is often used in the analysis of survey data in situations where complex sampling designs are employed we consider estimation of the p x 1 vector of parameters f3 in the assumed population model e y i x x 3jx under stratified sampling if sample selection is based on values of the independent variables standard regression methods apply however if selection is based on values of the dependent variable y or on a design variable z correlated with the dependent variable but not included as an independent variable ordinary least squares regression estimates are in general asymptotically biased for examples and discussion of previous research see holt smith winter 1980 nathan holt 1980 hausman wise 1982 and jewell

on chao s unequal probability sampling plan summary we give necessary and sufficient conditions for the second order inclusion probabilities to be strictly positive under chao s 1982 sampling plan for some order of arrangement of the population units we also prove that the variance of the horvitz thompson estimator of the population total based on this sampling plan is smaller than the variance of the hansen hurwitz estimator based on a probability proportional to size with replacement sample involving the same number of draws

adjusted chi squared tests for cross classified tables of survey data summary holt scott ewings 1980 propose a modified pearson x2 statistic for testing independence in a two way table of survey data it is shown that the multiplier of x2 needed for this modification is a weighted sum of cell and marginal cell design effects a similar adjustment of the x2 and g2 statistics is possible whenever log linear models with closed form estimates are fitted to a cross classified table of survey data

optimal estimation of finite population total under a general correlated model restricting attention to fixed size sampling designs and linear unbiased estimators of a finite population total we give methods for finding estimators with minimum model expected variance and the optimal strategy under a general correlated superpopulation model some techniques popular in the theory of optimal experiments help in the derivation several earlier optimality results are deduced as special cases

estimation of variance of the ratio estimator summary a general class of estimators of the variance of the ratio estimator is considered which includes two standard estimators vo and v2 and approximates another estimator vh suggested by royall eberhardt 1975 asymptotic expansions for the variances and biases of the proposed estimators are obtained based on this we obtain optimal variance estimators in the class and compare the relative merits of three estimators vo v1 and v2 without any model assumption under a simple regression model a more definite comparison of vo v1 and v2 is made in terms of variance and bias some key word8 asymptotic expansion bias optimal variance estimator ratio estimator superpopulation variance

the effect of changes of location on least squares estimators for samples stratified on the dependent variable summary the consistent least squares estimator proposed by jewell 1985 for linear regression with data arising from stratified samples lacks invariance with respect to a change of location of the dependent variable simulation suggests that its efficiency is reduced when the origin of the dependent variable is far from its mean this estimator can be made invariant to a change of location by specifying that the dependent variable is measured from the estimated population mean another similar invariant estimator is also suggested

a general purpose unequal probability sampling plan summary we present a general purpose unequal probability without replacement sampling plan with fixed sample size in contrast to existing such plans our scheme keeps the sample size fixed and lets the population units enter the sample one at a time through a carefully designed random mechanism consequently all high order inclusion probabilities can be easily computed

estimation of rare errors using expert judgement summary cox snell 1979 investigate inference about the total population error when most observations are zero their approach is parametric and assumes that errors are positive empirical evidence indicates however that negative errors are not uncommon and that the error distribution is unlikely to be modelled by any standard parametric distribution we propose a more general nonparametric approach that uses ferguson s dirichlet process our model allows the investigator to specify a wide range of prior prediction ranging from a standard parametric distribution to observations without any formal structure numerical examples are presented for demonstration of this flexibility in modelling comparisons with parametric models are discussed cox snell 1979 investigate inference on the total population error where most of the individual error values are zero the problem of analysing data containing many zeros appears in several areas aitchison 1955 lachenbruch 1976 owen derouen 1980 but is most prevalent in statistical auditing neter 1986 see also a report in preparation from the national research council in the present paper we propose a nonparametric bayesian model that allows the investigator maximum flexibility to incor porate his prediction of the error distribution based on expert judgement consider a finite population of n items and let yi be the recorded value of the ith item in the population the yi are assumed to be positive the population total recorded amount y e yi is known to the investigator the recorded value of each item however may be subject to an error we let di be the error associated with yi and set zi dilyi the fractional error of the ith item an upper bound for the total population error d l di is desired that is based on the observed errors among a sample of n items in auditing practice sampled items are often selected with probability proportional to their recorded values because those with larger recorded values are considered relatively more important we assume that the same sampling design is applied with replacement to simplify the analysis the statistical analyses in these situations are difficult and nonstandard because most of the items in the population are error free this means that the sample contains only a few items with errors and it is not uncommon for the sample data to be all zeros for this problem cox snell 1979 propose an approximating infinite population model the sample population is then defined by selecting items from the original population with probability proportional to their recorded values the sample is con sidered as a simple random sample from this infinite sampled population let p denote

maximum likelihood estimation for mixed continuous and categorical data with missing values summary maximum likelihood procedures for analysing mixed continuous and categorical data with missing values are presented the general location model of olkin tate 1961 and extensions introduced by krzanowski 1980 1982 form the basis for our methods maximum likelihood estimation with incomplete data is achieved by an application of the em algorithm dempster laird rubin 1977 special cases of the algorithm include orchard woodbury s 1972 algorithm for incomplete normal samples fuchs s 1982 algorithms for log linear modelling of partially classified contingency tables and day s 1969 algorithm for multivariate normal mixtures applications include a imputation of missing values b logistic regression and discriminant analysis with missing predictors and unclassified observations c linear regression with missing continuous and categorical predictors and d parametric cluster analysis with incomplete data methods are illustrated using data from the st louis risk research project some key word8 cluster analysis discriminant analysis em algorithm incomplete data linear regression logistic regression log linear model mixture model

hopf galois extensions and an exact sequence for h picard groups let h be a hopf algebra and a an h galois extension we investigate h morita autoequivalences of a introduce the concept of h picard group and we establish an exact sequence linking the h picard group of a and the picard group of acoh

defects of irreducible characters ofp soluble groups abstract we prove a refinement of the p soluble case of robinson s conjectural local characterization of the defect of an irreducible character

schur indices and schur groups w x schur groups were introduced in 8 as a certain kind of smallest groups admitting characters with nontrivial schur index indeed every finite group g having an irreducible character x with m x 1 involves such schur groups they occur as terminal steps by repeated application of the brauer witt theorem over local fields of characteristic zero there is a connection with the schur subgroup s k of the brauer group br k of such a field k this s k has been studied in detail by w x w x yamada 12 and quite recently by adem and milgram 1 from the character theoretic point of view it appears to be of interest to give explicit generators for s k in terms of schur groups see section 6 in the course w x of this investigation the authors noticed that there must be a gap in 8 concerning the exceptional dyadic schur groups yielding nontrivial 2 lo cal index it turned out that the definition of the schur groups of type qd q was too restrictive the correct definition will be given in section 5 below we take the opportunity to discuss these terminal q reduc2 tions once again and the class of 2 groups involved theorem 4 and 5 the schur groups admit faithful irreducible characters which are induced from linear characters of cyclic normal subgroups for the sake of clarity we begin our present work by making some general comments on schur indices in group extensions this should be compared with turull w x 11 our approach is perhaps more elementary and we just demonstrate w x how to use the results on stable clifford theory 7 more details may be w x found in the recent thesis 5 of the first named author

unequal probability sampling without replacement through a splitting method summary a very general class of sampling methods without replacement and with unequal probabilities is proposed it consists of splitting the inclusion probability vector into several new inclusion probability vectors one of these vectors is chosen randomly thus the initial problem is reduced to another sampling problem with unequal probabilities this splitting is then repeated on these new vectors of inclusion probabilities at each step the sampling problem is reduced to a simpler problem the simplicity of this technique allows one to generate easily new sampling procedures with unequal probabilities the splitting method also generalises well known methods such as the midzuno method the elimination procedure and the chao procedure next a sufficient condition is given in order that a splitting method satisfies the sen yates grundy condition finally it is shown that the elimination procedure satisfies the gabler sufficient condition

an elimination procedure for unequal probability sampling without replacement summary an exact without replacement sampling procedure with fixed sample size and unequal inclusion probabilities is presented the proposed method eliminates a unit of the population at each of the n n steps of the algorithm in order finally to retain n units a simple expression is given for the joint inclusion probabilities

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

partial exchangeability and survey sampling inference summary the role of the sampling mechanism for bayesian inference on symmetric functions of a finite population is examined under partial design and partial label information a new definition of partial exchangeability is proposed together with a corresponding property of the sampling mechanism these properties provide conditions under which certain symmetries in priors or designs can protect against lack of symmetry in others in the sense of scott smith 1973

asymptotic minimaxity of the ratio strategy summary the use of simple random sampling together with ratio estimation is of fundamental importance in survey sampling practice from a theoretical point of view the ratio estimator has been justified by bayesian methods ericson 1969 and by superpopulation methods royall 1970 but these methods do not yield a rigorous justification of simple random sampling at the same time we use the minimax principle in an asymptotic version to justify both simple random sampling and ratio estimation

precision of double sampling estimators for comparing two probabilities summary it is assumed that both error prone and error free observations can be made on a binary response the second type being more expensive in double sampling error prone observations are made on all sample units and for a random subsample error free observations are made as well tenenbein 1970 studied allocation of units in double sampling for estimating a single probability we consider double sampling for comparing two probabilities for different cost ratios for error prone versus error free observations and for fixed total cost we assess how the large sample variance of the log odds ratio estimator varies with the proportion of units subject to both error prone and error free observation we determine the optimal double sampling proportion and the possible gain from using double sampling versus single sampling in which only error free observations are made

estimating distribution functions from survey data summary a simple method for estimating population distribution functions and associated quantiles from sample survey data is described and some asymptotic theory for it presented the method assumes a model based approach to survey estimation and allows auxiliary population information to be directly incorporated into the estimation process monte carlo results comparing the proposed method with conventional design based methods are given these suggest that the model based approach offers significant gains when the auxiliary population information is linearly related to the survey variables of interest

extending irreducible modules an important part of clifford s theory does not work properly for arbitrary fields r to see why we must recall the main steps of this theory we are interested in extensions to rh modules of a fixed h invariant irreducible rn module a where n is a normal subgroup of a finite group h we first pass to the factor algebra u rh 3 modulo the ideal 3 3 rh rh3 generated by the annihilator 3 of 8 in rn if sth is given its natural grading by the factor group g h n see 6 2 below then 3 is a graded ideal so that u is a strongly g graded ring in the sense of 3 or section 1 below whose l component u rn d is a finite dimensional simple ralgebra with i as its irreducible module it follows that the extensions of 8 to rh modules are just the pull backs of the extensions of 2i to 2l modules via the natural epimorphism of rh onto u when 52 is algebraically closed or more generally when i is split over r then 2i is the kronecker product

performing likelihood ratio tests with multiply imputed data sets summary existing procedures for obtaining significance levels from multiply imputed data either i require access to the completed data point estimates and variance covariance matrices which may not be available in practice when the dimensionality of the estimand is high or ii directly combine p values with less satisfactory results taking advantage of the well known relationship between the wald and log likelihood ratio test statistics we propose a complete data log likelihood ratio based procedure it is shown that for any number of multiple imputations the proposed procedure is equivalent in large samples to the existing procedure based on the point estimates and the variance covariance matrices yet it only requires the point estimates and evaluations of the complete data log likelihood ratio statistic as a function of these estimates and the completed data the proposed procedure therefore is especially attractive with highly multiparameter incomplete data problems since it does not involve the computation of any matrices

a kernel method for estimating finite population distribution functions using auxiliary information summary this paper considers the use of auxiliary information to improve the estimation of the distribution function of a finite population a method is proposed which combines the known distribution of the auxiliary variable with a kernel estimate of the conditional distribution of the survey variable given the value of the auxiliary variable the resulting estimator compares favourably with existing estimators in terms of efficiency conditional behaviour and robustness the proposed method also has the advantage of producing estimates which are bona fide distribution functions this is not the case for many calibration type estimators suggested in the literature finally the proposed method is applicable under any probability sampling scheme

clifford theory of simple modules clifford theory primarily is concerned with the representations of normal subgroups and quotient groups of finite groups the basic contribution being due to a h clifford 3 modern developments founded on the concepts of group graded rings and crossed products have shown close relations with commutative and non commutative galois theory this is most conspicuous in stable clifford theory of simple modules suppose n w h g is an extension of finite groups and v is a simple f n module where f is any field since passage from the inertia group to the whole group usually is easy to handle we may and do assume that v is g invariant or h invariant stable then the annihilator i ann v of v generates a homogeneous ideal z of f h and a f h z is a fully g graded algebra with simple l component r f n z general clifford theory is concerned with such g graded algebras a and their modules cf dade 7 lo assume the l component r a is simple artinian as above then the centre k z r has in a natural way the structure of a g field the main object of study is the endomorphism ring e end v of the a module induced from the simple r module v this is a fully g graded algebra as well generalizing wedderburn s theorem we show that a z m e as g graded algebras where d is the dimension of v over the division algebra d end v theorem 2 1 in particular k z d has the same g structure with respect to e and a and e are morita equivalent in some strong sense now assume that dim r d2m2 is finite and that the schur index m m v is relatively prime to the order of the automorphism group g c k of k induced by g then the graded structure of a e can be described by some distinguished cohomology class w v e h g k more precisely w oa v determines a crossed product sz of k with g