[ Pobierz caÅ‚ość w formacie PDF ] µ0 2 T ùø T - µ0 ëø± ± - " 2 ~ ìø ÷ø ~ ~ CT " -± A± - 2bT ± ìø ÷ø ìø ÷ø p× p [C0 + C(2 - 2bT ± + ± A ±] ~ ~ ~ íø øø o) ïø úø p× p ~ ~ ~ ìø ÷ø 2n ðø ûø ~ ~ ~ íø øø n íø øø 2µ0 ± ± -± µ0 -± Æ µ0 T µg (14) ëø öø ëø öø µ0 2 T ~ ~ 2 ~ ~ [± A± -CT - bT ±] ìø ÷ø ìø ÷ø [C0 + C(2 - 2bT ± + ± A ±] o) ~ ~ ~ ìø ÷ø n ~ ~ ~ íø øø n íø øø 56 Table 3.1 Biases and mean squared errors of various estimators of µ0 ESTIMATOR BIAS MSE g(1)(µ0,eT) g(2)(µ0,eT) g(12)(µ0,eT) T öø, µ0 ¸ , µ0 ëø¸ ¸ - ˜ ¸ ìø ÷ø ~ p× p ~ ~ ~ ~ íø øø v 0 2 Æ µ0 îø¸ T µg (15) ëø öø ëø öø µ0 2 ùø T A¸ - CT ˜ + 2bT ¸ ìø ÷ø ìø ÷ø [C0 + C(2 + 2bT ¸ +¸ A ¸] ~ 0) ïø úø ìø ÷ø ~ ~ p× p ~ ~ ~ ~ 2n ðø ûø n where ˜ =diag{¸1,¸2,& ¸p} íø øø íø øø ~ p× p T 2 µ0 ¸ ¸ Æ µ0 ¸ ¸ µ0 T µg (16) ëø öø ëø öø µ0 2 T ~ ~ ~ ~ [¸ A¸ + 2bT ¸] ìø ÷ø ìø ÷ø [C0 + C(2 + 2bT ¸ +¸ A ¸] 0) ìø ÷ø ~ ~ ~ ~ ~ ~ 2n n íø øø íø øø µ0 ¸ ˜* µ0 , ¸ p× p Æ µ0 µg (17) ëø öø ëø öø ~ µ0 2 T 2 ~ ~ [CT ˜*p× p +2bT ¸] ìø ÷ø ìø ÷ø [C0 + C(2 + 2bT ¸ +¸ A¸] ~ o) ~ ìø ÷ø 2n ~ ~ ~ íø øø n íø øø ëø öø ¸ ëø öø ¸1 where ˜*p× p =diag{¸1 ìø -1÷ø & ,¸p ìø p -1÷ø } ìøÉ ÷ø ìøÉ ÷ø ~ íø 1 øø p íø øø Unbiased ± * O O 1 ëø öø îøC + C(2 + 2µ0bT ± *+ ± *T A ± *ùø 2 ~ ~ p× p p× p ~ ìø ÷ø 0 o) ïø úø ~ ~ n ðø ~ ûø íø øø Æ µg (18) T where ± * =(±1*,±2*,...,±p*) with ± *i =(±i,µi, i=1,2,...,p) ~ ~ 57 4. ESTIMATORS BASED ON ESTIMATED OPTIMUM It may be noted that the minimum MSE (2.6) is obtained only when the optimum values of constants involved in the estimator, which are functions of the unknown population parameters µ0, b and A, are known quite accurately. To use such estimators in practice, one has to use some guessed values of the parameters µ0, b and A, either through past experience or through a pilot sample survey. Das and Tripathi (1978, sec.3) have illustrated that even if the values of the parameters used in the estimator are not exactly equal to their optimum values as given by (2.5) but are close enough, the resulting estimator will be better than the conventional unbiased estimator y . For further discussion on this issue, the reader is referred to Murthy (1967), Reddy (1973), Srivenkataramana and Tracy (1984) and Sahai and Sahai (1985). On the other hand if the experimenter is unable to guess the values of population parameters due to lack of experience, it is advisable to replace the unknown population parameters by their consistent estimators. Let ÆÆ be a consistent estimator of Æ=A-1b. We then replace Æ by ÆÆ and also µ0 by y if necessary, in the Æ Æ optimum µg resulting in the estimator µg(est) , say, which will now be a function of y , u and Æ. Thus we define a family of estimators (based on estimated optimum values) of µ0 as Æ µg(est) = g **(y,uT ,ÆÆT ) (4.1) where g**(Å") is a function of (y,uT ,ÆÆT ) such that T g **(µ0 ,eT ,Æ )= µ0 for all µ0 , "g **(Å") Ò! = 1 "y (µ0 ,eT ,ÆT ) "g **(Å") "g(Å") = = -µ0 A-1b = -µ0Æ "u "u (µ0 ,eTÆT ) (µ0 ,e)T (4.2) and "g **(Å") = 0 "ÆÆ ,eT ,ÆT (µ0 ) With these conditions and following Srivastava and Jhajj (1983), it can be shown to the first degree of approximation that Æ Æ MSE(µg(est))= min.MSE(µg) ëø öø µ0 2 2 ìø ÷ø = [C0 + C(2 - bT A-1b] 0) ìø ÷ø n íø øø Thus if the optimum values of constants involved in the estimator are replaced by their consistent Æ estimators and conditions (4.2) hold true, the resulting estimator µg(est) will have the same asymptotic Æ mean square error, as that of optimum µg . Our work needs to be extended and future research will explore the computational aspects of the proposed algorithm. 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Sankhya, A, 32, 99-106. 61 CONTENTS Forward & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & 4 Estimation of Weibull Shape Parameter by Shrinkage Towards An Interval Under Failure Censored Sampling, [ Pobierz caÅ‚ość w formacie PDF ] zanotowane.pl doc.pisz.pl pdf.pisz.pl blacksoulman.xlx.pl