Question

Let X₁, X₂, ..., X_n be a random sample from X which has pdf
f(20)
depending on a parameter $\theta$ and

(i)   (ii)  

where $-\infty$ < x < $\infty$ . In both these two cases

a) write down the log-likelihood function and find a 1-dimensional sufficient statistic for $\theta$

b) find the score function and the maximum likelihood estimator of $\theta$

c) find the observed information and evaluate the Fisher information at $\theta$ = 1.

Answer 1

Solution: Given that, X; , X2, X.3.,X4,-Xn be a semnale random sample from x n and f(xb) = (203\2 exp{-} (x-1)} (-00<x<«). a) we have to find al-dimensional sup rocient statistic for o, The kelihood function can be given as 1(210) - . f(xyl) = Ñ connexp $ ==} ( 27-03"} = (875" expé -4 (up =)2} = (273") exp{-} [3x;"- 20nī + n0>]} =(27)" lla 2x3?. enz = n(AX2,43.-- xn) 90(1). Неле чр(т) - епъө sufficient statistic of o There for e, T=x is the for 1 - dimension ESTE POST RUOTOTAPE

(6) we have We have to find the score function and the maximum likelihood estimator of O. NOW, log L(410) = un log 2 7 - nounie. a) o log 2(x10) 10 -2not nă=0 2 20 > no - Eno=0 =) no =sxi = X There fore log L(110) , score function = do - Î xq - no. (C) we have to find the observed in formation and the Fished in forma hºon at 0 = 1. Therefore, by using again, de log2 (x10) E-n. Fisher information - - EC log L(zlo)/07 =nu.

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