# Consider a random sample X1,…,x, from the population distribution with density f(;4) = 0x8–1, 1 € (0,1), and parameter 0 € (0,00). (1) Verify that the Assumptions

Consider a random sample X1,…,x, from the population distribution with density f(;4) = 0x8–1, 1 € (0,1), and parameter 0 € (0,00). (1) Verify that the Assumptions (1) and (2) of the Cramér-Rao Theorem (Theorem 2.22) are met. (2) Verify that the Assumption (3) of the same theorem is fulfilled as well. To do this, you can use the following result about a derivative of a (Lebesgue) integral whose integrand depends on a parameter (tailored to this particular problem): Suppose that T(X1,…,xn) is an unbiased estimator of (@) with finite variance; let f(11 …, In;) denote the density of (X1,…,x.). Fix >0 arbitrary. If 8 >0 is such that 0-8 >0 and that (i) and (ü) hold, where () f(\$1…,In;) is continously differentiable in 6 on (6-8, 69+8) for almost all (w.r.t. the Lebesgue measure) (11,…,In) € (0,1)” ; (ii) T (T1,…, n)f(11 … , In: 0) = g(11,…, In) for all ((1, …, In) € (0,1)” and all 0 € (06-8, 69+8) for some function 9 (which may depend on 0, and 5) which is integrable on (0,1)” (i.e. 8… So 9(21.In)dxı… dxn < 0);=”” then=”” a=”” t(91=”” ,=”” ..,=”” in)f(21…,=”” in;€)drz–=”” -=”” dinle=”” ae=”” oo=”” ols=”” |=”” l-lot(31…in)f=”” (21…,19;0)=”” dx.dan=”” ae=”” le=”0″ (3)=”” suppose=”” we=”” wish=”” to=”” estimate=”” 1/0.=”” use=”” the=”” cramer-rao=”” theorem=”” to=”” find=”” the=”” umvue.=”” theert=”” 222=”” cramkae=”” rats=”” suppose=”” that=”” xs,…x=”” is=”” a=”” random=”” sample=”” from=”” sr=”” let=”” tx,..,x)=”” be=”” an=”” unbiased=”” estimator=”” of=”” 8=”” (o),=”” ie.=”” woc=”” e,=”” t=”” -=”” let=”” xp=”” assume=”” that=”” the=”” condititious=”” cd,=”” (2),=”” (3)=”” below=”” hold=”” (1)=”” for=”” all=”” oc=”” p.=”” has=”” a=”” pdf=”” i=”” puf=”” #(*;o)=”” and=”” sf=”” 26=”” exists=”” for=”” all=”” and=”” all=”” x=”” en.=”” such=”” that=”” (=”” x+=”” n.)=”” -=”” for=”” alle=”” (2)=”” 46=”” e=”” (obot=”” (x,0)=”” -=”” )=”” and=”” el=”” choose=”” (x;=”” 0)))=”” -=”” i=”” @)=”” €=”” 6,-)=”” 20=”” for=”” all=”” ac=”” here,=”” ilois=”” called=”” the=”” fisher=”” information=”” (3)=”” т=”” var.=”” t.,=”” .=”” ,=”” xx)=””>< 0=”” for=”” all=”” sco=”” and=”” hagf=”” è=”” etc.-,=”” x).=”” (x8,0}-4=”” for=”” all=”” then=”” {to=”” rosno=”” tcxx)=”” in=”” i=”” (6)=”” gramer=”” -=”” ras=””>   