Vtyjkyuk

A given statistic : $T_c = sum_j=1^n frac(X_j - bar X)^2c$, where $c$ is a constant, as an estimator of variance $sigma^2.$

$X_1,ldots, X_n$ denote a random sample from a population which has normal distribution with unknown mean $mu$ and unknown variance $sigma^2$.

The statistic is distributed as $x^2_n-1$ (a chi-squared variate with $n-1$ degrees of freedom).

I am tasked to find the bias of $T_c$.

I know the formula for bias is $mathbb E hat theta - theta$.

I found $theta$ as $mu = n - 1$ for a chi-squared distribution of $n-1$ degrees of freedom.

However, I am confused as to how to calculate $mathbb E hat theta$.

What i thought of doing is to calculate $mathbb E T_c$, but i got stuck halfway through, so I am not sure if I am doing the right thing. Any help please. thanks in advance.

Since $T_c$ is an estimator of $sigma^2$. $T_c$ will be unbiased for $theta = sigma^2$ if $mathbbE_mu,sigma^2(T_c) = sigma^2 quad forall mu, sigma^2 in mathbbR times mathbbR_0^+$

beginalign*
mathbbE(T_c)&= frac1cmathbbE Big( sum_j=1^n X_j^2 + barX^2 -2X_j barX Big) \
&= frac1c sum_j=1^n mathbbE(X_j^2) + mathbbE(barX^2) - 2 mathbbE (X_j barX)
endalign*

Now calculate $mathbbE(X_j^2),mathbbE(barX^2)$ & $ mathbbE (X_j barX)$.

1.
Using Var$(X) = mathbbE(X^2) - mathbbE(X)^2$ we find $mathbbE(X_j^2) = sigma^2 + mu^2$.

2.
beginalign*
mathbbE(barX^2) &= mathbbEBig( big(frac1n sum_i^n X_ibig) big(frac1n sum_i^n X_ibig)Big) \
&= frac1n^2 mathbbEBig( big(sum_i^n X_ibig) big(sum_i^n X_ibig)Big)\
&= frac1n^2 mathbbEBig( sum_i neq j X_iX_j + sum_i X_i^2Big) \
&=frac1n^2 Big(sum_i neq j mathbbE(X_i)mathbbE(X_j) + sum_i mathbbE(X_i^2)Big)\
&= frac1n^2 Big( n(n-1)mu^2 + n(mu^2 + sigma^2) Big)\
&= frac(n-1)mu^2 + mu^2 + sigma^2n =mu^2 + fracsigma^2n
endalign*

3.

beginalign*
mathbbE (X_j barX)&= frac1n sum_i^n mathbbE(X_jX_i) \
&=frac1n big( sum_ineq jmathbbE(X_j)mathbbE(X_i) big) + frac1n mathbbE(X_j^2)\
&= frac1n (n-1) mu^2 + frac1n(mu^2 + sigma^2)\
&= mu^2 + fracsigma^2n
endalign*
By using the values we have found we obtain

beginalign*
mathbbE(T_c)
&= frac1c sum_j=1^n mathbbE(X_j^2) + mathbbE(barX^2) - 2 mathbbE (X_j barX)\
&= frac1c sum_j=1^n (sigma^2 + mu^2) + ( mu^2 + fracsigma^2n ) - 2 (mu^2 + fracsigma^2n) \
&= frac(n-1)sigma^2c
endalign*

So $T_c$ is biased for $sigma^2$ for all values of $c neq n-1$

$$
frac 1 sigma^2 sum_i=1^n (X_i-overline X)^2 sim chi^2_n-1.
$$
Therefore
$$
operatorname Eleft( frac 1 sigma^2 sum_i=1^n (X_i - overline X)^2 right) = n-1.
$$
So
$$
operatorname Eleft( frac 1 c sum_i=1^n (X_i-overline X)^2 right) = fracsigma^2 c (n-1).
$$
Subtract $sigma^2$ from that to get the bias.