Let X1,...,Xn be i.i.d. random variables with density function Defin where Obtain the asymptotic distribution of the normalized form of Mn and, hence, show that as n → ∞, Mn does not converge to...

Let X₁,...,X_n
be i.i.d. random variables with density function

Defin

where

Obtain the asymptotic distribution of the normalized form of M_n
and, hence, show that as n → ∞, M_n
does not converge to 0, in probability. Obtain the asymptotic distribution of the normalized form of Y_n, and show that

for every

Compare the asymptotic variance of Y_n
for

p ∈ (0, 1.2) with that of p = 1/2 (i.e., the median), and comment on their asymptotic relative efficien y.