Question

5. Suppose that X, X, ..., X, is a random sample from a distribution with the density function (@+1)x®, if 0 < x <1 1 0, otherwise (where @ > -1 is unknown). (a) Show that the moments estimator of e is à 28-1 1-X (b) (c) (where X denotes the sample mean, as usual). Show that is a consistent estimator of e. U = - h, In X, is a sufficient statistic for 8. Is a function of U? Explain why this means that the moments estimator is NOT a MVUE for e. Find the MLE of 8. Show that the MLE is a function of the sufficient statistic from (c). (d)

Answer 1

sample from o en l a) Let x , ..., an be a random & Collol. falol = @+1) 6 mi= 1 Exi = è Hi= E(x) = x fcelo) obc - 1x. 60 +1) xo che = 0+10 So ocet abc +60+) [ 2012 ] 012 J C + Co . + @ +2 o with There S for we can estimate ot - =X +8 s - e cott - Cotal 6 - 6 : x + 2 ) -1 . 1 = 8ł thi - 1 - Ô dx = 25-1 a> (1-x) = 25-14 => 0 = 2x-1 1x That is the moment estimator of ois x _ / 2x Y

b) we have Ecx) = x + 1 by WLLN, Dt 2 | 0 = 2x - -> 2(+) - | +2 ich cô -> 200+ 21 0 +2 072-0-1 0+2 20+2-Q-2 © oth in ô lo That is cê is a consistent estimater of @.