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A Course in Mathematical Statistics phần 7

Tham khảo tài liệu 'a course in mathematical statistics phần 7', ngoại ngữ, ngữ pháp tiếng anh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 316 12 Point Estimation REMARK 7 We know see Remark 4 in Chapter 3 that if a p 1 then the Beta distribution becomes U 0 1 . In this case the corresponding Bayes estimate is . YL xi 1 ổ x1 . xn 1------- n 2 as follows from 21 . EXAMPLE 15 Let X1 . Xn be i.i.d. r.v. s from N e 1 . Take Ả to be N d 1 where d is known. Then I1 f x e --- f Xn e Ả e dd r_exp -1Y x - e 21 xp d Ide 1 12 Y JI 2 J . _1. exp -4Yx2 d Ì n 2 j h 2n L 2V1 1 J xf expj-1 n 1 e2 - 2 nx d e de. But n 1 e2 -2 nx d e n 1 e2 -2 n d el n 1 - 2 e - 2 nx -d e in- id n 1 V n 1 J 2 C d V n 1 Therefore n 1 2 e - n V n 1J 2 TT V n 1 J I 2 expl 1 2 n Y x2 d2 j 1 n 1 1 X2í n ĩ - exp 2 1 n nx d 2 1 vhĩ 2 V n 1 1 Jde 1 1 2 - 42n n expl 1 2 n Y x2 d2 j 1 nx d n 1 22 Exercises 317 Next 12 ef X1 e . f xn e Ả e de 1 n 1 n Oexp-1I x-e exP L 2 j 1 J e-p 2 de de 2 1 1 pny exp n 1 XT 2 - 2 Ixj p j 1 2 nx p 2 n 1 1 4ĩn 1 n 1 i ớexP - -- 1 2 1 4 1 2 e V n 1 y V de 1 1 n exPi 1 n - V Ĩ Vũ - ị 1 xỉ p 2 j 1 2 nx p 2 I nx p n 1 23 By means of 22 and 23 one has on account of 15 s x.r nx p x1 . . . xn v n 1 24 Exercises 12.7.1 Refer to Example 14 and i Determine the posterior p.d.f. h e x ii Construct a 100 1 - a Bayes confidence interval for e that is determine a set e e 0 1 fi e x c x where c x is determined by the requirement that the Prprobability of this set is equal to 1 - a iii Derive the Bayes estimate in 21 as the mean of the posterior p.d.f. h e x . Hint For simplicity assign equal probabilities to the two tails. 12.7.2 Refer to Example 15 and i Determine the posterior p.d.f. h e x ii Construct the equal-tail 100 1 - a Bayes confidence interval for e iii Derive the Bayes estimate in 24 as the mean of the posterior p.d.f. h e x . 318 12 Point Estimation 12.7.3 Let X be an r.v. distributed as P ớ and let the prior p.d.f. Ả of ớ be Negative Exponential with parameter T. Then on the basis of X i Determine the posterior p.d.f. h ớ x ii Construct the equal-tail 100 1 - a Bayes confidence interval for ớ iii Derive the Bayes .