Estimating a function of scale parameter of an exponential population with unknown location under general loss function
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Estimating a function of scale parameter of an exponential population with unknown location under general loss function Lakshmi Kanta Patra1 · Suchandan Kayal2 · Somesh Kumar3 Received: 1 March 2018 / Revised: 25 September 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract In the present study, we consider the problem of estimating a function of scale parameter ln σ under an arbitrary location invariant bowl-shaped loss function, when location parameter μ is unknown. Various improved estimators are proposed. Inadmissibility of the best affine equivariant estimator (BAEE) of ln σ is established by deriving a Stein-type estimator. This improved estimator is not smooth. We derive a smooth estimator improving upon the BAEE. Further, the integral expression of risk difference (IERD) approach of Kubokawa is used to derive a class of improved estimators. To illustrate these results, we consider two specific loss functions: squared error and linex loss functions, and derive various estimators improving upon the BAEE. Finally, a simulation study has been carried out to numerically compare the risk performance of the improved estimators. Keywords Best affine equivariant estimator · Location invariant loss function · Stein’s and Brewster–Zidek’s techniques · IERD approach · Inadmissible estimators
1 Introduction Inadmissibility of the best equivariant estimators for parameters of normal populations was first established by Stein (1956, 1964). These results were extended by Brown
B
Suchandan Kayal [email protected]; [email protected] Lakshmi Kanta Patra [email protected] Somesh Kumar [email protected]
1
Indian Institute of Petroleum and Energy, Visakhapatnam, India
2
Department of Mathematics, National Institute of Technology Rourkela, Rourkela 769008, India
3
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
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L. K. Patra et al.
(1966, 1968) to general class of location and scale families of probability distributions. This led to development of new methods for improving equivariant estimators. Two important techniques are by Brewster and Zidek (1974) and Kubokawa (1994). In this paper, we have applied various techniques for deriving improved estimators for the logarithmic function of the scale parameter of an exponential distribution when the location parameter is unknown. Note that this problem is equivalent to the study of estimating the Rényi entropy of an exponential distribution. Entropy can be considered as a measure of randomness of a probability model. One of the most fundamental measures of uncertainty is the Shannon entropy (see Shannon 1948). Several generalizations of the Shannon entropy have been proposed by various researchers. One of these is the Rényi entropy (see Rényi 1961). Let X be a random variable with probability density function f (x|θ ). Then, the Rényi entropy with parameter α ≥ 0 is given by Rα (θ ) =
1 ln 1−α
∞ −∞
f α (x|θ )d x =
1 ln E f α−1 (X |θ ) . 1−α
(1.1)
When the parameter α
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