Help with the second half question (a) and (b)

Xi~ Betal x, 1) fx ( * ) = 02 - the likelihood function of the sample values 24, x2,... an i's L= IT f (xilex ) = x". TT( aci ) a - 1 O - 1 = ofIT xil log- likelihood is given by, 1 (x ) = nlog + ( a-1) 2 Log xi differentiating w.r.t. a to obtain the value of a for which Log- likelihood is maximum. = + I log * i = 0 a = -n E Log Xi therefore, OMLE = Log 24i The likelihood function can be written as L= a $ Trail therefore, by factorization theorem the sufficient statistic for a is IT xi. since v(xi ) = d ( at ) (a+ 2 ) by invariance property of MLE introduced by zehna the MLE of V(xil = 2 where & = ( at) ( 2+ 2 ) I'MSI Log 24 E ( X i ) = xox dx = a [ x dx = Let 1+- = 1- X & = x 1- X L's 1- 27 method of moment estimator for of, where a= - I di6.2.7 If (x1, .. ., xn) is a sample from a Beta(a, 1) distribution where a > 0 is unknown, then determine the MLE of a. you can express the pdf in a form that does not have gamma functions. (What is the right constant so it integrates to 1?) . It may help to identify the distribution of Yi = -log(Xi) (and hence its mean and variance). (a) Show that the MLE for a is consistent. (b) Also, find the Fisher information and determine the asymptotical normal distribution for the MLE of a