Consider estimators of the form a∑$(X_i − X)$$^2$ for $σ^2$ , the variance of a normal distribution with unknown mean µ and $σ^2$ , given a random sample of size n. Find the value for a that minimizes MSE[a∑$(X_i − X)$$^2$]
In order to answer this question is used the fact that ∑$(X_i − X)$$^2$ = (n-1)$S^2$ and
$MSE[α]$ = $Bias[α]^2$ + $Var[α]$
I got Bias[a∑$(X_i − X)$$^2$] = E[a∑$(X_i − X)$$^2$] - $σ^2$ = E[a(n-1)$S^2$] - $σ^2$
Therefore Bias[a∑$(X_i − X)$$^2$] =(an-n-1)$σ^2$, and so Bias[a∑(Xi−X)2]$^2$ = ($a^2$$n^2$-2$a^2$n-2an+$a^2$+2a+1)$σ^4$
Var[a∑$(X_i − X)$$^2$] = Var[a(n-1)$S^2$] = $a^2$$(n-1)^2$*2$σ^4$/(n-1) = (2$a^2$n - 2$a^2$)$σ^4$
Overall MSE[a∑$(X_i − X)$$^2$] = ($a^2$$n^2$-2an-$a^2$+2a+1)$σ^4$, and so the value of a which minimises this is a = 1/(n+1)
Is this correct?