Shrinkage estimators of the location parameter for certain spherically symmetric distributions

We consider estimation of a location vector for particular subclasses of spherically symmetric distributions in the presence of a known or unknown scale parameter. Specifically, for these spherically symmetric distributions we obtain slightly more general conditions and larger classes of estimators than Brandwein and Strawderman (1991,Ann. Statist.,19, 1639–1650) under which estimators of the formX +ag(X) dominateX for quadratic loss, concave functions of quadratic loss and general quadratic loss.Research supported by NSF grant DMS-88-22622     

Fuzzy subobjects in a category and the theory of C-sets

In this paper we shall define the concept of a fuzzy subobject of an object in arbitrary categories. This concept is generated by the representation theorem of fuzzy sets. By using fuzzy subobjects one can include most of the fuzzy concepts defined in the literature, such as: fuzzy groups, fuzzy relations and fuzzy convex sets. In the second part of the paper we shall define a new concept; that of a $\text{C}$-set. This concept will generalize that of a fuzzy set and we shall also prove that $\text{C}$-sets can be represented by some sets of functors. More precisely, $\text{C}$-sets form a category which can be represented by a category of functors. The utility of $\text{C}$-sets resides in the fact that one can replace “ordering” by the more general concept of a morphism in category. The new representation of $\text{C}$-sets is weaker than that of fuzzy sets.     

Toward a general theory of fuzzy variables

A general framework for a theory is presented that encompasses both statistical uncertainty, which falls within the province of probability theory, and nonstatistical uncertainty, which relates to the concept of a fuzzy set and possibility theory [L. A. Zadeh, J. Fuzzy Sets1 (1978), 3–28]. The concept of a fuzzy integral is used to define the expected value of a random variable. Properties of the fuzzy expectation are stated and a mean-value theorem for the fuzzy integral is proved. Comparisons between the fuzzy and the Lebesgue integral are presented. After a new concept of dependence is formulated, various convergence concepts are defined and their relationships are studied by using a Chebyshev-like inequality for the fuzzy integral. The possibility of using this theory in Bayesian estimation with fuzzy prior information is explored.     

Central limit theorem for perturbed empirical distribution functions evaluated at a random point

Let be an estimator obtained by integrating a kernel type density estimator based on a random sample of size n from a (smooth) distribution function F. Sufficient conditions are given for the central limit theorem to hold for the target statistic where {U_n} is a sequence of U-statistics.     

On the rate of convergence in the central limit theorem for signed rank statistics

The problem of the rate of convergence of the cumulative distribution function of the one-sample rank order statistics S_N⁺ to the limiting normal one is studied for the case when the underlying observations are independent but not necessarily identically distributed. The results obtained are then used to derive a strong law and functional form of a central limit theorem for S_N⁺     

The covariance of uncertain variables: definition and calculation formulae

Uncertainty theory as a branch of axiomatic mathematics has been widely used to deal with human uncertainty. The two commonly used numerical characteristics of uncertain variables, the expected value and the variance together with their mathematical properties have been discussed and applied to real optimization problems in an uncertain environment. As a further study, in this paper, we focus on the covariance and correlation coefficient of uncertain variables. The definitions and calculation formulae of covariance and correlation coefficient of two uncertain variables are suggested by means of their inverse distributions. Then we show that the correlation coefficient of uncertain variables is essentially a measure of the relevance of distributions of uncertain variables. Finally, the relation between variance and covariance is analysed and represented with some equalities and inequalities.     

Uncertain stock model with periodic dividends

This paper presents an uncertain stock model with periodic dividends based on uncertainty theory. Some option pricing formulae related to the proposed model are investigated and several numerical examples are discussed to illustrate the related formula.     

Order statistics of uncertain random variables with application to <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system

Uncertain random variables are tools to deal with a mixture of uncertainty and randomness. A new concept of order statistics associated with uncertain random variables is proposed, and is applied to analyze k-out-of-n systems with uncertain random lifetimes. The chance distributions of order statistics of uncertain random variables are derived from the operational law of uncertain random variables. Finally, the reliability of k-out-of-n systems with uncertain random lifetimes is discussed.