A Generalized Stochastic Restricted Ridge Regression Estimator

In this article, we introduce a new stochastic restricted estimator for the unknown vector parameter in the linear regression model when stochastic linear restrictions on the parameters hold. We show that the new estimator is a generalization of the ordinary mixed estimator (OME), Liu estimator (LE), ordinary ridge estimator (ORR), (k-d) class estimator, stochastic restricted Liu estimator (SRLE), and stochastic restricted ridge estimator (SRRE). Performance of the new estimator in comparison to other estimators in terms of the mean squares error matrix (MMSE) is examined. Numerical example from literature have been given to illustrate the results.     

ON THE COMMON SCALE PARAMETER OF SEVERAL PARETO POPULATIONS IN CENSORED SAMPLES

The problem of estimation of an unknown common scale parameter of several Pareto distributions with unknown and possibly unequal shape parameters in censored samples is considered. A new class of estimators which includes both the maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (UMVUE) is proposed and examined under a squared error loss.     

Comparison of point estimators of normal percentiles

There are available several point estimators of the percentiles of a normal distribution with both mean and variance unknown. Consequently, it would seem appropriate to make a comparison among the estimators through some “closeness to the true value” criteria. Along these lines, the concept of Pitman-closeness efficiency is introduced. Essentially, when comparing two estimators, the Pit-man-closeness efficiency gives the “odds” in favor of one of the estimators being closer to the true value than is the other in a given situation. Through the use of Pitman-closeness efficiency, this paper compares (a) the maximum likelihood estimator, (b) the minimum variance unbiased estimator, (c) the best invariant estimator, and (d) the median unbiased estimator within a class of estimators which includes (a), (b), and (c). Mean squared efficiency is also discussed.     

An identity for exponential distributions with the common location parameter and its applications

An identity for exponential distributions with an unknown common location parameter and unknown and possibly unequal scale parameters is established.Through use of the identity the maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (UMVUE) of a quantile of an exponential population are compared under the squared error loss.A class of estimators dominating both MLE and UMVUE is obtained by using the identity.     

A new approach to use multivariate auxiliary information in sample surveys

A new approach to form multivariate difference estimator is suggested which does not require the knowledge of unknown population parameters as such. It gives minimum variance among the class of multivariate difference estimators. The performance of this estimator with respect to Des Raj's (J. Amer. Statist. Assoc. 60 (1965), 270–277) multivariate difference estimator is illustrated. Using the information on two auxiliary variates, the robustness of Des Raj's estimator y_d is studied empirically. Two new estimators to estimate population mean/total are developed on the same lines as that of y_d. The performance of these estimators is studied for a wide variety of populations.     

An improved class of estimators for the population mean

Starting from the Rao (Commun Stat Theory Methods 20:3325–3340, 1991) regression estimator, we propose a class of estimators for the unknown mean of a survey variable when auxiliary information is available. The bias and the mean square error of the estimators belonging to the class are obtained and the expressions for the optimum parameters minimizing the asymptotic mean square error are given in closed form. A simple condition allowing us to improve the classical regression estimator is worked out. Finally, in order to compare the performance of some estimators with the regression one, a simulation study is carried out when some population parameters are supposed to be unknown.     

Uniformly adaptive estimation for models with arma errors

A semiparametric estimator based on an unknown density isuniformly adaptive if the expected loss of the estimator converges to the asymptotic expected loss of the maximum liklihood estimator based on teh true density (MLE), and if convergence does not depend on either the parameter values or the form of the unknown density. Without uniform adaptivity, the asymptotic expected loss of the MLE need not approximate the expected loss of a semiparametric estimator for any finite sample I show that a two step semiparametric estimator is uniformly adaptive for the parameters of nonlinear regression models with autoregressive moving average errors.     

Alternative estimators of the common regression matrix in two GMANOVA models under weighted quadratic losses

We consider the problem of estimating the common regression matrix of two GMANOVA models with different unknown covariance matrices under certain type of loss functions which include a weighted quadratic loss function as a special case. We consider a class of estimators, which contains the Graybill–Deal-type estimator proposed by Sugiura and Kubokawa (Ann. Inst. Statist. Math. 40 (1988) 119), and we give its risk representation via Kubokawa and Srivastava's (Ann. Statist. 27 (1999) 600; J. Multivariate Anal. 76 (2001) 138) identities when the error matrices follow the elliptically contoured distributions. Using the method similar to an approximate minimization of the unbiased risk estimate due to Stein (Studies in the Statistical Theory of Estimation, vol. 74, Nauka, Leningrad, 1977, p. 4), we obtain an alternative estimator to the Graybill–Deal-type estimator which was given under the normality assumption. However, it seems difficult to evaluate the risk of our proposed estimator analytically because of complex nature of its risk function. Instead, we conduct a Monte-Carlo simulation to evaluate the performance of our proposed estimator. The results indicate that our proposed estimator compares favorably with the Graybill–Deal-type estimator.     

Uniformly adaptive estimation for models with arma errors

A semiparametric estimator based on an unknown density isuniformly adaptive if the expected loss of the estimator converges to the asymptotic expected loss of the maximum liklihood estimator based on teh true density (MLE), and if convergence does not depend on either the parameter values or the form of the unknown density. Without uniform adaptivity, the asymptotic expected loss of the MLE need not approximate the expected loss of a semiparametric estimator for any finite sample I show that a two step semiparametric estimator is uniformly adaptive for the parameters of nonlinear regression models with autoregressive moving average errors.     

Estimating the common hazard rate of two exponential distributions with ordered location parameters

This paper is concerned with estimating the common hazard rate of two exponential distributions with unknown and ordered location parameters under a general class of bowl-shaped scale invariant loss functions. The inadmissibility of the best affine equivariant estimator is established by deriving an improved estimator. Another estimator is obtained which improves upon the best affine equivariant estimator. A class of improving estimators is derived using the integral expression of risk difference approach of Kubokawa [A unified approach to improving equivariant estimators. Ann Statist. 1994;22(1):290–299]. These results are applied to specific loss functions. It is further shown that these estimators can be derived for four important sampling schemes: (i) complete and i.i.d. sample, (ii) record values, (iii) type-II censoring, and (iv) progressive Type-II censoring. A simulation study is carried out for numerically comparing the risk performance of these proposed estimators.     

A target distribution model for nonparametric density estimation

In this paper a model is proposed which represents a wide class of continuous distributions. It is shown how the parameters of this model can be estimated leading to a distribution estimator and a corresponding density estimator. An important property of this estimator is that it can be structured to reflect a priori knowledge of the unknown distribution.

Finally, some examples are shown and some comparisons made with kernel and orthogonal series estimators.     

Minimax estimation of a lower-bounded scale parameter of a gamma distribution for scale-invariant squared-error loss

Let X have a gamma distribution with known shape parameter θr;aL and unknown scale parameter θ. Suppose it is known that θ ≥ a for some known a > 0. An admissible minimax estimator for scale-invariant squared-error loss is presented. This estimator is the pointwise limit of a sequence of Bayes estimators. Further, the class of truncated linear estimators C = {θ_ρ|θ_ρ(x) = max(a, ρ), ρ > 0} is studied. It is shown that each θ_ρ is inadmissible and that exactly one of them is minimax. Finally, it is shown that Katz's [Ann. Math. Statist., 32, 136–142 (1961)] estimator of θ is not minimax for our loss function. Some further properties of and comparisons among these estimators are also presented.     

Bounds for the variance of the graybill-deal estimator of the common mean of two normal distributions

In this note we derive sharp lower and upper bounds for the variance of the Graybill-Deal estimator of the common mean of two normal distributions with unknown variances when the sample sizes are not necessarily equal. We also derive similar bounds for the variance of the Brown-Cohen (1974) T _a(1) class of unbiased es-timators to which the Graybill-Deal estimator belongs. Further, we illustrate the sharpness of the bounds by numerical computations in the case of the Graybill-Deal estimator.     

Estimating the common hazard rate of several exponential distributions

Abstract

In the present communication, we consider the estimation of the common hazard rate of several exponential distributions with unknown and unequal location parameters with a common scale parameter under a general class of bowl-shaped scale invariant loss functions. We have shown that the best affine equivariant estimator (BAEE) is inadmissible by deriving a non smooth improved estimator. Further, we have obtained a smooth estimator which improves upon the BAEE. As an application, we have obtained explicit expressions of improved estimators for special loss functions. Finally, a simulation study is carried out for numerically comparing the risk performance of various estimators.     

A Bayesian nonparametric estimator based on left censored data

Summary This paper introduces a Bayesian nonparametric estimator for an unknown distribution function based on left censored observations. Hjort (1990)/Lo (1993) introduced Bayesian nonparametric estimators derived from beta/beta-neutral processes which allow for right censoring. These processes are taken as priors from the class ofneutral to the right processes (Doksum, 1974). The Kaplan-Meier nonparametric product limit estimator can be obtained from these Bayesian nonparametric estimators in the limiting case of a vague prior. The present paper introduces what can be seen as the correspondingleft beta/beta-neutral process prior which allow for left censoring. The Bayesian nonparametyric estimator is obtained as in the corresponding product limit estimator based on left censored data.     

A class of high-breakdown scale estimators based on subranges

We consider a new class of scale estimators with 50% breakdown point. The estimators are defined as order statistics of certain subranges. They all have a finite-sample breakdown point of [n/2]/n, which is the best possible value. (Here, [...] denotes the integer part.) One estimator in this class has the same influence function as the median absolute deviation and the least median of squares (LMS) scale estimator (i.e., the length of the shortest half), but its finite-sample efficiency is higher. If we consider the standard deviation of a subsample instead of its range, we obtain a different class of 50% breakdown estimators. This class contains the least trimmed squares (LTS) scale estimator. Simulation shows that the LTS scale estimator is nearly unbiased, so it does not need a small-sample correction factor. Surprisingly, the efficiency of the LTS scale estimator is less than that of the LMS scale estimator.     

A stochastic restricted k–d class estimator

In this paper, we introduce a stochastic restricted k–d class estimator for the vector of parameters in a linear model when additional linear restrictions on the parameter vector are assumed to hold. The stochastic restricted k–d class estimator is a generalization of the ordinary mixed estimator and the k–d class estimator. We show that our new biased estimator is superior in the mean squared error matrix sense to the k–d class estimator [S. Sakall?o?lu and S. Kaçiranlar, A new biased estimator based on ridge estimation, Statist. Papers 49 (2008), pp. 669–689] and the stochastic restricted Liu estimator [H. Yang and J.W. Xu, An alternative stochastic restricted Liu estimator in linear regression, Statist. Papers 50 (2009), pp. 639–647]. Finally, a numerical example is given to show the theoretical results.     

A family of shrinkage estimators for Weibull shape parameter in censored sampling

In this paper a new class of shrinkage estimators has been introduced for the shape parameter in an independently identically distributed two-parameterWeibull model under censored sampling. The main idea is to incorporate the prior guessed value by correcting the standard estimator, which is essentially an unbiased estimator, with optimally weighted ratios of the guessed value and the standard estimator, instead of considering a convex combination of the standard estimator and the difference of the guessed value and the standard estimator. The resulting estimator dominates the standard estimator in a surprisingly large neighborhood of the guessed value. The suggested estimator has also been compared with the minimum mean squared error estimator and a class of estimators suggested by Singh and Shukla in IAPQR Trans 25(2), 107–118, 2000. It is found that the suggested class of estimators has lesser bias as well as lesser mean squared error than its competitors subject to certain conditions.      

Estimation of the common scale of several shifted exponential distributions with unknown locations

We consider the estimation of the common scale parameter of two or more independent shifted exponential distributions with unknown locations. Under a large class of bowl-shaped loss functions, the best location-scale in-variant estimator is shown to be inadmissible. A class of improved estimators is derived. Some numerical results are presented to show the magnitude of risk reduction.     

Nonparametric local polynomial approximation of the time-varying frequency and amplitude

The problem of estimating the time-varying frequency, phase and amplitude of a real-valued harmonic signal is considered. It is assumed that the frequency and amplitude are unspecified rapidly time-varying functions of time. The technique is based on fitting a local polynomial approximation of the phase and amplitude which implements a high-order nonlinear nonparametric estimator. The estimator is shown to be strongly consistent and Gaussian. In particular, the convergence ratesO(h^-3/2 )and O(h^-5/2 ), where $i;h$ei; is the number of observations, are obtained for the frequency estimator when the amplitude is unknown constant or linear in time respectively. The orders of the bias and Gaussian distribution are obtained for a class of the time-varying frequency and amplitude with bounded second derivatives. The a priori amplitude information about the unknown time-varying frequency and amplitude and their derivatives can be incorporated to improve the accuracy of the estimation. Simulation results are given.