Question

Suppose X , X ,..., X n 1 2 be a random sample drawn from a population with mean θand varianceσ² . Show that Ti=1/i∑X_j is unbiased estimator of θ (i = 1,2,…,n). Also show var(T_i ) is monotonically decreasing. Hence find the best unbiased estimator for θ.

Answer 1

$(i)~E(T_1)=\frac{1}{n}\sum_{i=1}^nE(X_iY_i)=\frac{1}{n}\sum_{i=1}^n E(X_i)E(Y_i)~~(since~X_i~and~Y_i's~are~independent)\\\\ =\frac{1}{n}\times n(p/q)q=p.\\\\ Hence~T_1~is~unbiased~estimator~of~p.\\\\ (ii)~Var(T_1)=\frac{1}{n^2}\sum_{i=1}^nVar(X_iY_i)~~(since~Z_i=X_iY_i's~are~independent)\\\\ =\frac{1}{n^2}\sum_{i=1}^n\left(E(X_i^2Y_i^2)-E^2(X_iY_i) \right )\\\\ =\frac{1}{n^2}\sum_{i=1}^n\left(E(X_i^2)E(Y_i^2)-E^2(X_i)E^2(Y_i) \right )~~(since~X_i~and~Y_i's~are~independent)\\\\ =\frac{1}{n^2}\sum_{i=1}^n((\sigma_1^2+(p/q)^2)(\sigma_2^2+q^2)-p^2)=\frac{\sigma_1^2\sigma_2^2+\frac{p^2\sigma_2^2}{q^2}+q^2\sigma_1^2}{n}\\\\ Vow,~since~E(T_1)=p~and~Var(T_1)\rightarrow 0~as~n\rightarrow \infty.\\\\ (iii)~E(T_2)=E\left(\frac{1}{n}\sum_{i=1}^nX_i \right )E\left(\frac{1}{n}\sum_{i=1}^nY_i \right )~(since~X_i~and~Y_i's~are~independent)\\\\ =\left(\frac{1}{n}\sum_{i=1}^nE(X_i) \right )\left(\frac{1}{n}\sum_{i=1}^nE(Y_i) \right )\\\\ =\frac{p}{q}\times q=p\\\\ Hence~T_2~is~unbiased~estimator~of~p.$