Showing papers in "Metrika in 1984"

Regression-type estimators using two auxiliary variables and the model of double sampling from finite populations

Abstract: In this paper, we use two auxiliary variablesx andy to construct two regression-type estimators for the population mean of the study variabley. The efficiency of the proposed estimators is investigated under a super-population model. A numerical study is done to demonstrate the practical use of different estimation formulae and empirically demonstrate the performance of the constructed estimators.

Bayes theory: Hartigan, Springer-Verlag, New York 1983, p. 145, DM 46,-

Abstract: This is a book about the theory of Bayesian inference at a rather sophisticated mathematical level. It is based on lectures given to students who already have had a course in measure-theoretic probability, and has the rather clipped style of notes. This led me to some difficulties of comprehension, especially when typographical errors occur, as in the definition of a random variable. Against this there is no unnecessary material and space for a few human touches. The development takes as fundamental the notion of expectation, though that word is scarcely used it does not appear in the inadequate index but has a brief mention on page 17. The book begins therefore with linear, non-negative, continuous operators and the treatment has the novelty that it does not require that the total probability be one: indeed, infinity is admitted, this having the advantage that improper distributions of the Jeffreys type can be included. There is an original and interesting account of marginal and conditional distributions with impropriety. For example, in discussing a uniform distribution over pairs (i,D of integers, the sets]=l and ]------2 both have infinite probability and cannot therefore be compared; so that conditional probabilities p( i=l / ]=l) , p(i=lff------2) require separate discussion. My own view is that this feature is not needed, for although improper distributions have some interest in low dimensions (and mainly in achieving an unnecessary match between Bayesian and Fisherian ideas) they fail in high dimensions, as Hartigan shows in chapter 9, where there is an admirable account of many normal means. A lesser objection is the complexity introduced by admitting impropriety: Bayes theorem takes 14 lines to state and 20 to prove. Chapter 5 is interestingly called \"Making Probabilities\" and discusses Jaynes' maximum entropy principle, Jeffreys' invariance, and similarity as ways of constructing distributions; those produced by the first two methods are typically improper. This attitude is continued into chapter 8 where exponential families are introduced as those minimizing information subject to constraints. There is a discussion of decision theory, as distinct from inference, but there is no attempt to consider utility: all is with respect to an undefined loss function. The consideration of the different types of admissibility is very brief and the opportunity to discuss the mathematically sensitive but practically meaningful aspects of this topic is lost. Other chapters are concerned with convergence, unbiasedness and confidence, multinomials, asymptotic normality, robustness and non-parametric procedures; the last being mainly devoted to a good account of the Dirichlet process. Before all this mathematics, the book begins with a brief account of the various theories of probability: logical, empirical and subjective. At the end of the account is a fascinating discussion of why the author thinks \"there is a probability 0.05 that there will be a large scale nuclear war between the U.S. and the U.S.S.R before 2000\". This connection between mathematics and reality is most warmly to be welcomed. The merit of this book lies in the novelty of the perspective presented. It is like looking at a courtyard from some unfamiliar window in an upper turret. Things look different from up there. Some corners of the courtyard are completely obscured. (It is suprising that there is no mention at all of the likelihood principle; and only an aside reference to likelihood.) Other matters are better appreciated because of the unfamiliar aspect normal means, for example. The book does not therefore present a balanced view of Bayesian theory but does provide an interesting and valuable account of many aspects of it and should command the attention of any statistical theorist.

Classical statistical analysis based on a certain hypercomplex multivariate normal distribution

Abstract: Goodman [1963] generalized the real normal multivariate model to the complex case.Goodman [1963], andKhatri [1965] derived the sampling distribution theory underlying this model. The present paper generalizes the complex multivariate normal theory to the hypercomplex case. The hypercomplex case studied here includes Hamilton's quaternions, biquaternions, octonions, and bioctonions. It is shown that the complex case results straightforwardly generalize to the hypercomplex case.

Inferences about the parameters of a time series model with changing variance

Abstract: We consider the problem of making inferences about the parameters of a time series model when there is the possibility of a discrete variance change at an unknown time point. For this we obtain the posterior distributions of the parameters and of the variance ratio.

Best unbiased estimators for the parameters of a two-parameter Pareto distribution

Abstract: For a two-parameter Pareto distributionMalik [1970] has shown that the maximum likelihood estimators of the parameters are jointly sufficient. In this article the maximum likelihood estimators are shown to be jointly complete. Furthermore, unbiased estimators for the two parameters are obtained and are shown to be functions of the jointly complete sufficient statistics, thereby establishing them as the best unblased estimators of the two parameters.

Upper bound of the speed of convergence of moment density Estimators for stationary point processes

Abstract: The speed of convergence of moment density estimators for stationary point processes is studied. Under relevant assumptions the order of magnitude for its upper bound is the same as in the i.i.d. case, when the process is Brillinger-mixing. The case of convariance density estimators is also considered.

A characterization of the exponential distribution by higher order gap

Abstract: SupposeX is a non-negative random variable with an absolutely continuous (with respect to Lebesgue measure) distribution functionF (x) and the corresponding probability density functionf(x). LetX 1,X 2,...,X n be a random sample of sizen fromF andX i,n is thei-th smallest order statistics. We define thej-th order gapg i,j(n) asg i,j(n)=X i+j,n−Xi,n′ 1≤i

On a Morgenstern-type bivariate gamma distribution

Abstract: In this paper a Morgenstern-type bivariate gamma distribution has been studied. Its moment generating function has been derived. The distribution of the product and quotient are derived in terms of the modified Bessel function. The results for the independent case follow as special cases. Further the regression function has been analysed, in terms of its deviation from linear regression function.

Konkrete optimale Versuchspläne für ein lineares Modell mit einem qualitativen und zwei quantitativen Einflußfaktoren

Abstract: The problem of characterizing exactU- orD-optimal designs for a linear model with one qualitative and two quantitative factors of influence is treated. For three parameter-functionals of special interest easily applicable characterizations of optimal designs are given within the class of designs with fixed total sample size as well as in the class of designs with fixed treatment marginals. Necessary conditions for optimal designs are derived if no useful characterizations could be found.

Minimum variance unbiased estimation for the zero class truncated bivariate poisson and logarithmic series distributions

Abstract: The minimum variance unbiased estimation for the zero class truncated bivariate Poisson and logarithmic series distributions is discussed The distribution of a suitable complete and sufficient statistic is obtained in each case These distributions, as well as the corresponding minimum variance unbiased estimators have been expressed in terms of certain polynomials with coefficients Stirling numbers

Über die Anzahl von Zufallspunkten mit typ-gleichem nächsten Nachbarn und einen multivariaten Zwei-Stichproben-Test

Abstract: For independents-variate samplesX1, ...,Xm i.i.d.f. (.),Y1, ...,Yn i.i.d. g. (.), where the densitiesf (.),g (.) are assumed to be continuous on their respective sets of positivity, consider the numberTm,n of pointsZ of the pooled sample (which are either of ‘typeX’ or of ‘typeY’) such that the nearest neighbor ofZ is of the same type asZ. We show that, as\(m,n \to \infty ,\frac{m}{{m + n}} \to \tau ,0< \tau< 1,f(.) \equiv g(.),\lim Var\frac{1}{{\sqrt {m + n} }}T_{m,n} = \sigma ^2 (\tau ,s)\), independently of (.). An omnibus test for the two sample problem ‘f(.)≡g(.) orf(.)≢g(.)?’ may be obtained by rejecting the hypothesisf(.)≡g(.) for large values ofTm,n.

Optimale Planung eines Kovarianzanalyse- und eines Intraclass Regressions-Experiments

Abstract: The modeling and design of general experiments with qualitative and quantitative factors of influence is presented and discussed. A complete and ready to apply characterization of concrete optimal designs and their operational characteristics are given for the two basic models of analysis of covariance and of intra class regression under different experiment constraints.

Rank tests for the 2×2 split plot design

Abstract: In the 2×2 split plot design there are given rank tests for the classical hypotheses of the main effects and interaction.

Are robust estimation methods useful in the structural errors-in-variables model?

Abstract: In this paper it is investigated whether robust estimation procedures for the parameters of a regression model are also applicable when the observations are generated by the errors-in-variables model. Specifically, attention is paid to bounded-influence estimators, i.e. estimators that are constructed in such a way that the influence of a single observation on the outcome of the estimator is bounded. Both the classical errors-in-variables model and models with contaminated observational errors are considered.

On estimating the variance of a generalized Laplace distribution

Abstract: When |X| k has a gamma distribution with parametersb −k andk −1,Jakuszenkow [1979] has, for the loss function $$(\hat \theta - \theta )^2 \theta ^{ - 2} $$ , considered the best multiple of ΣX 2 as an estimator ofb 2 and shown that it is Lehmann unbiased. In this paper, the best multiple of (Σ|X i |k)2/k is shown to be Lehmann unbiased, admissible and better than Jakuszenkow's estimator.

The invertibility of sampled and aggregated ARMA models

Abstract: Necessary and sufficient conditions for a sampled (resp. aggregated) stationary ARMA process to be invertible are derived. It is shown that a sampled (resp. aggregated) invertible stationary ARMA process is always invertible.

Optimale Zweifachblockpläne bei Seriell Korrelierten Fehlern

Abstract: Williams-designs are proved to be optimal within the class of latin 4×4-squares in rowcolumn-models, where serial correlations are fitted with an autoregressive scheme of first order.

Wrapped distributions and measurement errors

Abstract: For a random variableX and α>0 letUα≔nα (X)−X, wherenα(x)=n∈Z iffx∈(nα−α/2,nα+α/2]. Random variables of this type are important in the theory of measurement errors. We derive formulas for the distribution ofUα and apply them to the case X∼N(θ,σ2). General conditions for the unimodality ofUα are given. The correlation of the measurement errorsX−E (X) andUα (X) is seen to beO (αj) withj depending on the smoothness and asymptotic behavior of the density ofX. This gives a precise sense to the assertion that scale errors upwards and downwards are averagely well-balanced. In the normal case the density ofUα is shown to be constant up to\(o (e^{ - C\alpha ^{ - 2} } )\), as α→0.

Some aspects of random permutation models in finite population sampling theory

Abstract: We first consider Neyman's optimum allocation of sample size to strata in the light of available auxiliary information for which a suitable random permutation model is assumed. For a special case of this model the allocation of the sample size reduces to the same as when a certain superpopulation regression model is assumed. Motivated by this, more generally, we discuss some optimality results under random permutation models and compare them with the corresponding results when a superpopulation regression model is assumed.

The large-sample power of the 195-1195-1195-1test for multidimensional contingency tables

Abstract: The asymptotic power of the frequency χ2 test depends on a noncentrality parameter, λ,Mitra [1958] offered a general expression for λ, which is rather difficult to apply. This work provides simplified formulae for λ in various models associated with multidimensional contingency tables.

Minimal sufficient statistics for the group divisible partially balanced incomplete block design (GD-PBIBD) with interaction under an Eisenhart Model II

Abstract: In this paper, an Eisenmhart Model II with interaction for a GD-PBIB design withp replicates per cell is considered. Specifically the Model Yijl=µ+βi+τj+(βτ)ij+eijl is assumed, wherei=1, 2, ...,b; j=1, 2, ...,t andl=0, 1, 2, ...p sij wheresij=1, if treatmentj appears in blocki, 0, otherwise.

Estimation of symmetric functions of a finite population

Abstract: Estimation of symmetric functions under the assumptions of noninformativeness of labels in a finite population of distinguishable units has been examined. The primitive strategies of srs, sample mean, sample variance etc. are found to play important roles.

A local limit theorem for attraction to the standard normal law: The case of infinite variance

Abstract: Let ≏Xn≃ be a sequence of independent random variables each having a common dfF. Suppose thatF belongs to the domain of attraction of the standard normal law. Here we show that if the cf ofX1 is absolutely integrable in ther-th power for some integerr≧1, then for all largen, the df of the normalized sumZn ofX1,X2, ...,Xn is absolutely continuous with a pdffn such that asn→∞, $$\mathop {\sup }\limits_{ - \infty< x< \infty } (1 + |x|)^\beta |f_n (x) - \phi (x)| = o(1)$$ for every β<2 or β≦2 according asEX12=∞ orEX12<∞, π being the standard normal pdf.

On Brewer's class of robust sampling designs for large-scale surveys

Abstract: In this note we observe that Brewer's (1979) result concerning the asymptotically design unbiased strategy which has minimum expected variance under a super population model can be established in a more general setting.