Robert L. Taylor

A consistent nonparametric density estimator for the deconvolution problem

The problem of nonparametric estimation of a probability density function when the sample observations are contaminated with random noise is studied. A particular estimator fn(x) is proposed which uses kernel-density and deconvolution techniques. The estimator fn(x) is shown to be uniformly consistent, and its appearance and properties are affected by constants Mn and hn which the user may choose. The optimal choices of Mn and hn depend on the sample size n, the noise distribution, and the true distribution which is being estimated. Particular selections for Mn and hn which minimize upper-bound functions of the mean squared error for fn(x) are recommended. On etudie le probleme de ľestimation ďune fonction de densite a partir ďun echantillon contamine par un bruit aleatoire. Un estimateur fn(x) de type “noyau” et faisant intervenir des techniques de deconvolution est propose. II est demontre que fn(x) est uniformement convergent. Sa forme et ses proprietes sont affectees par des constantes Mn et hn que ľutilisateur peut choisir. Les choix optimaux de Mn et hn dependent de la taille echantillonnale n, de la loi du bruit aleatoire et de la vraie densite que ľon estime. On recommande des choix de Mn et hn qui minimisent des bornes superieures de ľerreur quadratique moyenne de fn(x).

A strong law of large numbers for arrays of rowwise negatively dependent random variables

A strong law of large numbers for arrays of rowwise negatively dependent random variables is obtained which relaxes the usual assumption of rowwise independence. The moment conditions of the main result are similar to previous results, and the stochastic bounded condition also provides a relaxation of the usual distributional assumptions.

Bootstrap test of significance and sequential bootstrap estimation for unstable first order autoregressive processes

The asymptotic validity of the bootstrap for a test of criticality in a first order autoregressive, AR(1) process is established. To circumvent the asymptotic invalidity of the standard bootstrap least squares estimator for the unstable case, a sequential bootstrap procedure for the estimation of the parameter β in the AR(1) model , is studied. The asymptotic validity of the sequential bootstrap estimator is established for all |β|≤1.

Strong laws of large numbers for weighted sums of random elements in normed linear spaces.

Consider a sequence of independent random elements {Vn, n > in a real separable normed linear space (assumed to be a Banach space in most of the results), and sequences of con-

On the strong law for arrays and for the bootstrap mean and variance

Chung type strong laws of large numbers are obtained for arrays of rowwise independent random variables under various moment conditions. An interesting application of these results is the consistency of the bootstrap mean and variance.

Convergence of weighted sums of random sets

Stochastic convergence of weighted sums of the form is obtained for random sets {Xk} whose values are compact subsets of a Banach space and where {ank} is a Toeplitz sequence of real numbers. In particular , conditions imposed by some previous authors in comparable strong laws of large numbers that the compact subsets also be convex or that the random sets be identically distributed are removed

Strong laws of large numbers for arrays of rowwise independent random elements

Let {Xnk} be an array of rowwise independent random elements in a separable Banach space of type p

A weak law for normed weighted sums of random elements in Rademacher type p Banach spaces

For weighted sums [Sigma]j = 1najVj of independent random elements {Vn, n >= 1} in real separable, Rademacher type p (1 p 0 is established, where {vn, n >= 1} and bn --> [infinity] are suitable sequences. It is assumed that {Vn, n >= 1} is stochastically dominated by a random element V, and the hypotheses involve both the behavior of the tail of the distribution of V and the growth behaviors of the constants {an, n >= 1} and {bn, n >= 1}. No assumption is made concerning the existence of expected values or absolute moments of the {Vn, n >- 1}.

Laws of Large Numbers for Random Sets

The probabilistic study of geometrical objects has motivated the formulation of a general theory of random sets. Central to the general theory of random sets are questions concerning the convergence for averages of random sets which are known as laws of large numbers. General laws of large numbers for random sets are examined in this paper with emphasis on useful characterizations for possible applications.

Asymptotic bootstrap validity for finite markov chains

Two methods of bootstrap, viz., standard, and conditional, are presented for estimating the transition probabilities of a finite state Markov chain. Asymptotic validity of the bootstrap estimates are established for both methods. An applica- tion to a bootstrapped statistic for testing independence is briefly discussed together with some simulation results.