Radhey S. Singh

Estimation of error variance in linear regression models with errors having multivariate student-t distribution with unknown degrees of freedom

Abstract Linear regression model y = Xβ + μ , where the components of the disturbance vector u have jointly multivariate Student - t distribution with unknown degrees of freedom, is considered. An estimator of the degrees of freedom parameter is provided. This estimator is used to provide estimates, computable from the data, of the usual unbiased and minimum mean square error estimators of the error variance and of their variances, which are, otherwise, not computable from the data.

Note: Optimal allocation of resources to nodes of series systems with respect to failure‐rate ordering

Allocation of spare components in a system in order to optimize the lifetime of the system with respect to a suitable criterion is of considerable interest in reliability, engineering, industry, and defense. We consider the problem of allocation of K active spares to a series system of independent and identical components in order to optimize the failure-rate function of the system.

Empirical Bayes estimation in a multiple linear regression model

SummaryEstimation of the vector β of the regression coefficients in a multiple linear regressionY=Xβ+ε is considered when β has a completely unknown and unspecified distribution and the error-vector ε has a multivariate standard normal distribution. The optimal estimator for β, which minimizes the overall mean squared error, cannot be constructed for use in practice. UsingX, Y and the information contained in the observation-vectors obtained fromn independent past experiences of the problem, (empirical Bayes) estimators for β are exhibited. These estimators are compared with the optimal estimator and are shown to be asymptotically optimal. Estimators asymptotically optimal with rates nearO(n−1) are constructed.

Speed of convergence in nonparametric estimation of a multivariate μ-density and its mixed partial derivatives☆

Let f be a m-variate unknown density with respect to a σ-finite measure on Em, the m- dimensional Euclidean space, and let x=(x1,…,xm) and p=(p1,…,pm) be in Em, where p j≥0 are arbitrary integers. This paper exhibits kernel estimators of mixed partial derivatives f(p)(x)=(∂p1+…+pmf∂(x))/(∂xP11…∂xPmm) of f based on a random sample from f and investigates the speed of convergence for various asymptotic properties (of the estimators) including asymptotic unbiasedness, mean square consistency, strong consistency and asymptotic normality. Local bounds for the bias and mean square error as well as uniform convergence rate results for asymptotic unbiasedness, mean square consistency and strong consistency of the estimators are obtained. Rate result for covariance of the estimators evaluated at two distinct points in Em is also given.

Nonparametric Time Series Estimation of Joint DGP, Conditional DGP and Vector Autoregression

In this paper we develop nonparametric estimators of the joint time series data generating process (DGP) of ( x , y ) at different t -values, of conditional DGP, of the conditional mean of x given the past values of x and y , and, more generally, the conditional mean of ( x , y ) given their past values (vector autoregression). We establish, among other results, the central limit theorems for these estimators under far weaker mixing conditions than those used in Robinson [23], where only the x series is considered. Uniform consistency and rate results for the consistencies of various estimators are also obtained. The results of the paper are useful in light of the fact that often the functional form of the dynamic regression is not known and also the assumption of the Gaussian process is not true.

Mise of kernel estimates of a density and its derivatives

For an integer p [greater-or-equal, slanted] 0, Singh has considered a class of kernel estimators [latin small letter f with hook]~(p) of the pth order derivative [latin small letter f with hook](p) of a density [latin small letter f with hook] and showed how specializations of some of the results there improve the corresponding existing results. In this paper these improved estimators are examined on a global measure of quality of an estimator, namely, the mean integrated square error (MISE) behavior. An upper bound, which can not be tightened any further for a wide class of kernels, is obtained for MISE ([latin small letter f with hook]~(p)). The exact asymptotic value for the same is also obtained. Under two alternative conditions, weaker than those assumed for the two results mentioned above, convergence of MISE ([latin small letter f with hook]~(p)) to zero is proved. Specializations of some of the results here improve the corresponding existing results by weakening the conditions, sharpening the rates of convergence or both.

Estimation of prior distribution and empirical Bayes estimation in a nonexponential family

Abstract Empirical Bayes squared error loss estimation for a nonexponential family with densities ⨍(x¦θ) = exp( - (x − θ))I(x > θ) for θ ϵ Θ, a subset of the real line, is considered. An almost sure consistent estimator of the prior distribution G, whatever it may be, on Θ, is proposed. Based on this estimator an empirical Bayes estimator consistent for the minimum risk optimal estimator is exhibited. Asymptotic optimality of this estimator is further established.

Nonparametric empirical bayes procedures, asymptotic optimality And rates Of convergence For two‐tail tests In exponential family *

This paper provides nonparametric empirical Bayes (EB) solutions to two-tail test in the exponential family , under the standard product loss function which is proportional to (θ-θ1) (θ-θ2) for incorrectly accepting H1. Based on empirical data X1,…X n , and the present data X from nonparametric (in the sense that G is completely unknown and unspecified) EB test procedures are proposed. These procedures are asymptotically optimal (a.o.) whenever Further, for every integer s > 0 a class of non-parametric EB test procedures is proposed. These procedures are shown to be a.o. with rates for 0 < λ ≤ 2 satisfying certain conditions. Examples of exponential families and gamma densities are given where these conditions reduce to some simple moment conditions on G. No assumption on the smoothness of the function u(.), (and hence of the density function of X), is made at all for any of the results of this paper. By an example of a family of distributions, it is demonstrated that the rates arbitrarily close to o(n -1) can be attained by these procedures in some situations. It is noted, however, that the actual rates of convergence really depends on the nature of the unknown prior distribution G.

Empirical Bayes linear loss hypothesis testing in a non-regular exponential family☆

Abstract Asymptotically optimal empirical Bayes (EB) two-action linear loss hypothesis test procedures are developed for the nonregular translated exponential family with densities f(x¦θ) = k exp (− k(x − θ))I (x > θ) , k known positive constant. Rates of convergence for the procedures to asymptotic optimality are investigated. The prior distribution G of the parameter θ is completely unknown and unspecified with support in ( a 1 , a 2 ), − ∞ ⩽ a 1 a 2 ⩽ ∞. The approach taken is to use the past data X 1 , …, X n , i.i.d. according to the marginal p.d.f. f(x) = ∫f(x¦θ) d G (θ) first to provide a sequence of uniformally strongly consistent estimators G n of G and then use this to develop asymptotically optimal EB test procedures δ n ( X ). X = X n + 1 ∼ f ( x ), for the present ( n + 1)st testing problem. It is shown that the difference between the actual ( n + 1) stage Bayes risk of δ n and the optimal risk (the minimum Bayes risk) R ( G ) is of the order o(1) as n → ∞ for every prior distribution G for which E | θ | δ n for the ( n + 1)st stage problem is developed and is shown that the difference between the Bayes risk of { δ n } and the optimal risk is of the order O (n − r (r + 1) log log n) γ 2 for every G with E|θ| 2 (1 − γ) for some 0 γ

Non-parametric recursive estimates of a probability density function and its derivatives

Recursive estimates of a probability density function (pdf) are known. This paper presents recursive estimates of a derivative of any desired order of a pdf. Let f be a pdf on the real line and p⩾0 be any desired integer. Based on a random sample of size n from f, estimators f(p)n of f(p), the pth order derivatives of f, are exhibited. These estimators are of the form n−1∑nj=1δjp, where δjp depends only on p and the jth observation in the sample, and hence can be computed recursively as the sample size increases. These estimators are shown to be asymptotically unbiased, mean square consistent and strongly consistent, both at a point and uniformly on the real line. For pointwise properties, the conditions on f(p) have been weakened with a little stronger assumption on the kernel function.