On Some Shrinkage Techniques for Estimating the Parameters of Exponential Distribution
نویسندگان
چکیده
A variety of shrinkage methods have been proposed for estimation of some unknown parameter by considering estimators based on a prior guess of the value of the parameter. We compare some of the shrunken estimators for the parameters)! and 6 of the exponential distribution through simulation. INTRODUCTION In the estimation of an unknown parameter there often exists some form of prior knowledge about the parameter which one would like to utilize in order to get a better estimate. Thompson ( 1968 ) described a shrinkage technique for estimating the mean of a population. Mehta and Srinivasan ( 1971 ) proposed another class of shrunken estimator for the mean of a population and have shown that this class has better performance than that of Thompson ( 1968 ) in terms of mean squared error. Pandey and Singh ( 1977 ) and Pandey ( 1979 ) described shrinkage techniques for estimating the variance of a normal population. Lemmer ( 1981 ) considered a shrunken estimator for the parameter of the binomial distribution. His estimator is similar to the Pandey ( 1979 ) estimator for the variance of a normal distribution. We consider a variety of shrinkage methods for estimating the parameters u and 0 of the exponential distribution. These estimators are compared through simulation. ESTIMATORS CONSIDERED Let the length oflife X of a certain system be distributed as 1 f( X, 6, Jl) =a exp [(X-)1) I 6 ], 0 ~ ,u ~x. On some shrinkage techniques A random sample of n such systems is subjected to test and the test terminated as soon as the first r ( ~ n ) items fail. Let x = { x ) < ... , < x ( ) } be the first r ~ ( 1 r ordered failure times. It is well known from Epstein and Sobel ( I954) that and r e = [i ~ x ( i ) + ( n r ) x ( r) nx ( 1 ) ] I ( rI ), r > I "=X -91n ..( 1) ' are the minimum variance unbiased estimators of e and u respectively. The variances of these estimators age given by (see Bain (1978 ), p-I63 ). var (e) = e2 I ( rI ) var(.~) = r92 1n2 (r-I) The first estimator considered is : v J.lT = ).l +C(J.l-).l ) 0 ~ c~ I (2.I) 0 0 where J.l is the guessed value of J.l. vJ.l is the actual Thompson-type estimator. o T Thompson suggested to determine C from v o MSE(uT) =0, ~c v 2 v with MSE ( ).lT ) = E ( ).lT-Jl) , the mean squared error of pT. It follows that C = (J.l-].1 ) 1[().1-).l )2 +var(~)] (2.2) 0 0 In practice C in ( 2.2 ) is estimated by replacing the unknown parameters by their sample estimates. Substituting the estimated value ofC in ( 2.I ) we have ( 2.3) Secondly, we consider the Mehta and Srinivasan-type estimator ( cf. Mehta and Srinivasan ( I97I )for Jl :
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