Bias, variance, MSE, and consistency

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Problem

For i.i.d. $X_i\sim\operatorname{Bernoulli}(p)$ , let $S_n=\sum_iX_i$ and $T_n=(S_n+1)/(n+2)$ . Derive the bias, variance, and MSE of $T_n$ as an estimator of $p$ . Is $T_n$ consistent? Evaluate the bias and MSE at $n=10,p=1/2$ .

Reveal answer or reference solution

Since $E[S_n]=np$ and $\operatorname{Var}(S_n)=np(1-p)$ ,

\operatorname{Bias}(T_n)=\frac{np+1}{n+2}-p=\frac{1-2p}{n+2},

\operatorname{Var}(T_n)=\frac{np(1-p)}{(n+2)^2},

and

\operatorname{MSE}(T_n)=\frac{np(1-p)+(1-2p)^2}{(n+2)^2}

Both bias and variance tend to zero, so $T_n$ is consistent. At $n=10,p=1/2$ , the bias is $0$ and the MSE is $10(1/4)/12^2=5/288$ .

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Bias, variance, MSE, and consistency

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