On construction of admissible quadratic estimators

A random linear model for spatially located sensors measured intensity of a source of signals in discrete instants of time is considered. A basis of a quadratic subspace useful in quadratic estimation of a function of model parameters is given. Received: December 1999     

On estimating the median from survey data using multiple auxiliary information

A new method to derive confidence intervals for medians in a finite population is presented. This method uses multi-auxiliary information through a multivariate regression type estimator of the population distribution function. A simulation study based on four real populations shows its behaviour versus other known methods.     

Variance estimation for unequal probability sampling

In this paper, we discuss the optimality of the variance estimator of the Horvitz-Thompson estimator proposed by Kott (1988) in the class of model-unbiased quadratic estimators. We also propose some improved estimators over Kott's estimator in the class of general quadratic estimators. Received: February 1999     

Designs for estimating an extremal point of quadratic regression models in a hyperball

This paper is devoted to studying optimal designs for estimating an extremal point of a multivariate quadratic regression model in the unit hyperball. The problem of estimating an extremal point is reduced to that of estimating certain parameters of a corresponding nonlinear (in parameters) regression model. For this reduced problem truncated locally D-optimal designs are found in an explicit form. The result is a generalization of the results of Fedorov and Müller (1997) for onedimensional quadratic regression function in the unit segment. Received February 2002     

On the strong uniform consistency of a new kernel density estimator

Summary A new multivariate kernel probability density estimator is introduced and its strong uniform consistency is proved under certain regularity conditions. This result is then applied particularly to a kernel estimator whose mean vector and covariance matrix areμ _n andV _n, respectively, whereμ _n is an unspecified estimator of the mean vector andV _n, up to a multiplicative constant, the sample covariance matrix of the probability density to be estimated, respectively. Work supported by the Natural Sciences and Engineering Research Council of Canada and by the Fonds F.C.A.R. of the Province of Quebec.     

On the interpretation of cointegration in the linear–quadratic inventory model

The traditional formulation of the linear–quadratic inventory model with unit roots predicts cointegration between inventories and sales. That formulation implies that marginal production costs and the marginal benefits of inventories are both tending to ∞, and the cointegrating coefficient reflects the optimal trade-off between these competing factors. This paper suggests a reformulation of the problem in which marginal production costs and marginal inventory benefits are both stationary and in which the cointegrating coefficient is the same as the value that characterizes the target inventory level in the cost function.     

Smooth estimators for estimating order restricted scale parameters of two gamma distributions

We consider the problem of component-wise estimation of ordered scale parameters of two gamma populations, when it is known apriori which population corresponds to each ordered parameter. Under the scale equivariant squared error loss function, smooth estimators that improve upon the best scale equivariant estimators are derived. These smooth estimators are shown to be generalized Bayes with respect to a non-informative prior. Finally, using Monte Carlo simulations, these improved smooth estimators are compared with the best scale equivariant estimators, their non-smooth improvements obtained in Vijayasree, Misra & Singh (1995), and the restricted maximum likelihood estimators. Acknowledgments. Authors are thankful to a referee for suggestions leading to improved presentation.     

On divergence and convergence of sums of nonnegative random variables

Abstract  If X ₁, X ₂,… are exponentially distributed random variables thenσ^∞_{k= 1} X_k=∞ with probability 1 iff σ^∞_{k= 1} EX_k=∞. This result, which is basic for a criterion in the theory of Markov jump processes for ruling out explosions (infinitely many transitions within a finite time) is usually proved under the assumption of independence (see FREEDMAN (1971), p. 153–154 or BREI-MAN (1968), p. 337–338), but is shown in this note to hold without any assumption on the joint distribution. More generally, it is investigated when sums of nonnegative random variables with given marginal distributions converge or diverge whatever are their joint distributions.     

Nonparametric covariance estimation in multivariate distributions

The estimation problem of the unknown covariance matrix of a multivariate distribution with the known mean is studied under a matrix-valued quadratic loss function. The conditions on the sample sizes for the best unbiased estimator to have a smaller risk than the sample covariance matrix is established. The former estimator is completely (without exceptional sets of Lebesgue measure zero) characterized by its expectation in the class of all multivariate distributions with zero mean and finite fourth moments. Received: November 1998     

Strong uniform convergence for the estimator of the regression function under ϕ-mixing conditions

Suppose the observations (X _i,Y _i), i=1,…, n, are ϕ-mixing. The strong uniform convergence and convergence rate for the estimator of the regression function was studied by serveral authors, e.g. G. Collomb (1984), L. Gy?rfi et al. (1989). But the optimal convergence rates are not reached unless the Y _i are bounded or the E exp (a|Y _i|) are bounded for some a>0. Compared with the i.i.d. case the convergence of the Nadaraya-Watson estimator under ϕ-mixing variables needs strong moment conditions. In this paper we study the strong uniform convergence and convergence rate for the improved kernel estimator of the regression function which has been suggested by Cheng P. (1983). Compared with Theorem A in Y. P. Mack and B. Silverman (1982) or Theorem 3.3.1 in L. Gy?rfi et al. (1989), we prove the convergence for this kind of estimators under weaker moment conditions. The optimal convergence rate for the improved kernel estimator is attained under almost the same conditions of Theorem 3.3.2 in L. Gy?rfi et al. (1989). Received: September 1999     

On the Fisher information in record values

In this article, we establish some general results concerning the comparison of the amount of the Fisher information contained in n record values with the Fisher information contained in n iid observations from the original distribution. Some common distributions are classified according to this criterion. We also propose some methods of estimation based on record values. The results may be of interest in some life testing problems. Received: September 1999