Mathamatical Statistics?

LET X1,X2,鈥?.Xn denote a random sample from a population Bernoulli distribution with the probability distrubution function,


P(Xi=x)=p耍 (1-p)鹿 炉 耍 for x=0,1 and parameter p.


a)Find the maximum likelihood estimator(MLE) of (1-p)虏


b)Is the estimator in (a) unbiased? If it is biased,Find an unbiased estimator of (1-p)虏.|||As you know MLE of p is Xbar (sample mean)


using Zehna rule (invariance of MLE) the MLE of (1-p)虏 = (1-Xbar)虏





But its biased because


E((1-Xbar)虏)=Var(1-Xbar)+E虏(1-Xbar)=p(鈥β? which is not equal to (1-p)虏





E(1-Xbar)=1-E(Xbar)=1-p


Var(1-Xbar)=Var(Xbar)=p(1-p)/n





------------


T(X) = (1-Xbar)虏 - S虏/n


in which S虏=Sample variance of X


T(X) is an unbiased estimator


because


E(T(X)) = E((1-Xbar)虏 - S/n) = E((1-Xbar)虏) - E(S虏/n)=p(1-p)/n+(1-p)虏 - p(1-p)/n=(1-p)虏|||ask an asian