Estimation of a Gamma Scale Parameter Under Asymmetric Squared Log Error Loss

Abstract

Estimation of scale parameter under the squared log error loss function is considered with restriction to the principle of invariance and risk unbiasedness. An explicit form of minimum risk scale-equivariant estimator under this loss is obtained. The admissibility and inadmissibility of a class of linear estimators of the form (cT + d) are considered, where T follows a gamma distribution with an unknown scale parameter η and a known shape parameter ν. This includes the admissibility of the minimum risk equivariant estimator on η (MRE).     

Strawderman-type estimators for a scale parameter with application to the exponential distribution

It is shown that Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique for estimating the variance of a normal distribution can be extended to estimating a general scale parameter in the presence of a nuisance parameter. Employing standard monotone likelihood ratio-type conditions, a new class of improved estimators for this scale parameter is derived under quadratic loss. By imposing an additional condition, a broader class of improved estimators is obtained. The dominating procedures are in form analogous to those in Strawderman [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198]. Application of the general results to the exponential distribution yields new sufficient conditions, other than those of Brewster and Zidek [1974. Improving on equivariant estimators. Ann. Statist. 2, 21–38] and Kubokawa [1994. A unified approach to improving equivariant estimators. Ann. Statist. 22, 290–299], for improving the best affine equivariant estimator of the scale parameter. A class of estimators satisfying the new conditions is constructed. The results shed new light on Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique.     

A note on comparison and improvement of estimators based on likelihood

The problem of estimation of a parameter of interest in the presence of a nuisance parameter, which is either location or scale, is considered. Three estimators are taken into account: usual maximum likelihood (ML) estimator, maximum integrated likelihood estimator and the bias-corrected ML estimator. General results on comparison of these estimators w.r.t. the second-order risk based on the mean-squared error are obtained. Possible improvements of basic estimators via the notion of admissibility and methodology given in Ghosh and Sinha [A necessary and sufficient condition for second order admissibility with applications to Berkson's bioassay problem. Ann Stat. 1981;9(6):1334–1338] are considered. In the recent paper by Tanaka et al. [On improved estimation of a gamma shape parameter. Statistics. 2014; doi:10.1080/02331888.2014.915842], this problem was considered for estimating the shape parameter of gamma distribution. Here, we perform more accurate comparison of estimators for this case as well as for some other cases.     

Simultaneous equivariant estimation of the parameters of matrix scale and matrix location-scale models

Eaton and Olkin (1987) discussed the problem of best equivariant estimator of the matrix scale parameter with respect to different scalar loss functions. Edwin Prabakaran and Chandrasekar (1994) developed simultaneous equivariant estimation approach and illustrated the method with examples. The problems considered in this paper are simultaneous equivariant estimation of the parameters of (i) a matrix scale model and (ii) a multivariate location-scale model. By considering matrix loss function (Klebanov, Linnik and Ruhin, 1971) a characterization of matrix minimum risk equivariant (MMRE) estimator of the matrix parameter is obtained in each case. Illustrative examples are provided in which MMRE estimators are obtained with respect to two matrix loss functions.     

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.     

Admissible estimation in an one parameter nonregular family of absolutely continuous distributions

Consider the problem of estimating under entropy loss an arbitrarily positive, strictly increasing or decreasing parametric function based on a sample of size n in an one parameter noregular family of absolutly continuous distributions with both endpoints of the support depending on a single parameter. We first provide sufficient conditions for the admissibility of generalized Bayes estimator with respect to some specific priors and then treat several examples which illustrate the admissibility of best invariant estimators is some location or scale parameter problems.     

Estimation of the parameter in the problem of the nile

Estimation of the parameter in the problem of the Nile is treated as a decision problem with squared error loss, It is shown that the minimum risk scale equivariant estimator dominates the incomplete sufficient unbiased estimators considered by Iwase and Seto, Sharper bounds for the equivariant estimator are derived which may be used to obtain the values of the same from the sample with sufficient accuracy.     

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.     

Estimation of scale parameter under entropy loss function

Estimation of scale parameter under the entropy loss function is considered with restrictions to the principles of invariance and risk unbiasedness. An explicit form of minimum risk scale-equivariant estimator under entropy loss is obtained. The admissibility and inadmissibility of a class of linear estimators of the form cT + d, where T ≈ Γ(v,η), which includes the admissibility of the MRE estimator of parameter of interest are studied.     

Comparison of normal variance estimators under multiple criteria and towards a compromise estimator

For estimating a normal variance under the squared error loss function it is well known that the best affine (location and scale) equivariant estimator, which is better than the maximum likelihood estimator as well as the unbiased estimator, is also inadmissible. The improved estimators, e.g., stein type, brown type and Brewster–Zidek type, are all scale equivariant but not location invariant. Lately, a good amount of research has been done to compare the improved estimators in terms of risk, but comparatively less attention had been paid to compare these estimators in terms of the Pitman nearness criterion (PNC) as well as the stochastic domination criterion (SDC). In this paper, we have undertaken a comprehensive study to compare various variance estimators in terms of the PNC and the SDC, which has been long overdue. Finally, using the results for risk, the PNC and the SDC, we propose a compromise estimator (sort of a robust estimator) which appears to work ‘well’ under all the criteria discussed above.     

Estimating the exponential reliability function for a system in series with type II censored data

The problem of estimating the one parameter exponential reliability function for a system composed of l componentes in series is considered. Under the type II censoring scheme, the Bayes nature of the minimum variance unbiased estimator is demonstrated and the admissibility of related generalized Bayes estimators is established. For the one component case, the best unbiased estimator is admissible.     

Estimation of common location parameter of two exponential populations based on records

Consider the problem of estimating the common location parameter of two exponential populations using record data when the scale parameters are unknown. We derive the maximum likelihood estimator (MLE), the modified maximum likelihood estimator (MMLE) and the uniformly minimum variance unbiased estimator (UMVUE) of the common location parameter. Further, we derive a general result for inadmissibility of an equivariant estimator under the scaled-squared error loss function. Using this result, we conclude that the MLE and the UMVUE are inadmissible and better estimators are provided. A simulation study is conducted for comparing the performances of various competing estimators.     

On the improved estimation of a function of the scale parameter of an exponential distribution based on doubly censored sample

In the present article, we have studied the estimation of entropy, that is, a function of scale parameter ln⁡σ of an exponential distribution based on doubly censored sample when the location parameter is restricted to positive real line. The estimation problem is studied under a general class of bowl-shaped non monotone location invariant loss functions. It is established that the best affine equivariant estimator (BAEE) is inadmissible by deriving an improved estimator. This estimator is non-smooth. Further, we have obtained a smooth improved estimator. A class of estimators is considered and sufficient conditions are derived under which these estimators improve upon the BAEE. In particular, using these results we have obtained the improved estimators for the squared error and the linex loss functions. Finally, we have compared the risk performance of the proposed estimators numerically. One data analysis has been performed for illustrative purposes.     

Admissibility of linear estimators with respect to restricted parameter sets

In this paper the admissibility of a linear estimator for a linear regression parameter is characterized for such cases, where the considered parameter varies in an ellipsoid. We obtain a certain subset of the set of all linear estimators which are admissible with respect to the unrestricted parameter set. Furthermore, various linear estimators which have been proposed for improving the least squares estimator in cases of a restricted parameter set are investigated for admissibility. It turns out that only some shrunken estimators and some estimators of ridge type are admissible, whereas the KUKS-OLMAN estimator and all estimators of MARQUARDT type can be improved.     

Estimation of a multivariate mean with constraints on the norm

Estimation of the mean θ of a spherical distribution with prior knowledge concerning the norm ||θ|| is considered. The best equivariant estimator is obtained for the local problem ||θ|| = λ₀, and its risk is evaluated. This yields a sharp lower bound for the risk functions of a large class of estimators. The risk functions of the best equivariant estimator and the best linear estimator are compared under departures from the assumption ||θ|| = λ₀.     

Simultaneous equivariant estimation of the parameters of linear models

Consider a family of distributions which is invariant under a group of transformations. In this paper, we define an optimality criterion with respect to an arbitrary convex loss function and we prove a characterization theorem for an equivariant estimator to be optimal. Then we consider a linear model Y=Xβ+ε, in which ε has a multivariate distribution with mean vector zero and has a density belonging to a scale family with scale parameter σ. Also we assume that the underlying family of distributions is invariant with respect to a certain group of transformations. First, we find the class of all equivariant estimators of regression parameters and the powers of σ. By using the characterization theorem we discuss the simultaneous equivariant estimation of the parameters of the linear model.     

Estimation of the shape parameter of a Pareto distribution

We consider the problem of estimating the shape parameter of a Pareto distribution with unknown scale under an arbitrary strictly bowl-shaped loss function. Classes of estimators improving upon minimum risk equivariant estimator are derived by adopting Stein, Brown, and Kubokawa techniques. The classes of estimators are shown to include some known procedures such as Stein-type and Brewster and Zidek-type estimators from literature. We also provide risk plots of proposed estimators for illustration purpose.     

Shrinkage estimation of location parameters in a multivariate skew-normal distribution

Abstract

This paper studies decision theoretic properties of Stein type shrinkage estimators in simultaneous estimation of location parameters in a multivariate skew-normal distribution with known skewness parameters under a quadratic loss. The benchmark estimator is the best location equivariant estimator which is minimax. A class of shrinkage estimators improving on the best location equivariant estimator is constructed when the dimension of the location parameters is larger than or equal to four. An empirical Bayes estimator is also derived, and motivated from the Bayesian procedure, we suggest a simple skew-adjusted shrinkage estimator and show its dominance property. The performances of these estimators are investigated by simulation.     

Some remarks on an admissibility condition

Summary In this note we deal with some admissibility conditions proved by G. B. Tranquilli to be sufficient in the class of unbiased estimators of finite population parameters and with respect to (w.r.t.) a quadratic loss function. We show that the same conditions:i) are sufficient for the admissibility of an unbiased estimator with any loss function;ii) imply hyperadmissibility with reference to a particular (critical) population of the. From this fact we deduce that, for a fixed critical population, there is at most one estimator, in the class of all unbiased estimator of a finite population parameter, which satisfies Tranquilli condition. This research was partially supported by a M.U.R.S.T. grant ?Metodi inferenziali basati sul ricampionamento?.     

Universal admissibility in second order asymptotics

Some sufficient conditions for an estimator to be universally second order admissible are derived. Those sufficient conditions consist of the elementary integrals with respect to the Fisher information and the limits of some functions characterized by the dealt statistical model, and thus can be checked with comparative ease. In location model and scale model, the sufficient condition for the linear estimator with respect to the maximum likelihood estimator (MLE) to be universally second order admissible is given. Furthermore, a guide for classifying any estimator into either the universal admissibility or the non-universal admissibility is proposed.