sp2 is an unbiased estimator for σ2We have, (n1−1)s12σ2∼X(n1−1)2(n2−1)s22σ2∼x(n2−1)2⇒(n1−1)s12σ2+(n2−1)s22σ2∼x(n1+n2−2)2⇒Var(sp2)=2σ4n1+n2−2

The variance of the&nbsp;Xn1+n2−22&nbsp;distribution is&nbsp;2(n1+n2−2).Since your estimator is obtained by multiplying that&nbsp;χ{n1+n2−2}2&nbsp;random variable by&nbsp;σ2n1+n2−2, the variance is2(n1+n2−2)⋅(σ2n1+n2−2)2=2σ4n1+n2−2.[This is using the fact that&nbsp;Var(aX)=a2Var(X).

Solved: "sp2 is an unbiased estimator for σ2We have,..."

alparcero97oy

Answered question

2022-04-01

s_{p}^{2}

is an unbiased estimator for

σ^{2}

We have,

\frac{(n_{1} - 1) s_{1}^{2}}{σ^{2}} \sim X_{(n_{1} - 1)}^{2}

\frac{(n_{2} - 1) s_{2}^{2}}{σ^{2}} \sim x_{(n_{2} - 1)}^{2}

\Rightarrow \frac{(n_{1} - 1) s_{1}^{2}}{σ^{2}} + \frac{(n_{2} - 1) s_{2}^{2}}{σ^{2}} \sim x_{(n_{1} + n_{2} - 2)}^{2}

\Rightarrow V a r (s_{p}^{2}) = \frac{2 σ^{4}}{n_{1} + n_{2} - 2}

Answer & Explanation

Carter Lin

Beginner2022-04-02Added 13 answers

The variance of the $X_{n_{1} + n_{2} - 2}^{2}$ distribution is $2 (n_{1} + n_{2} - 2)$ .
Since your estimator is obtained by multiplying that $χ_{{n_{1} + n_{2} - 2}}^{2}$ random variable by $\frac{σ^{2}}{n_{1} + n_{2} - 2}$ , the variance is
$2 (n_{1} + n_{2} - 2) \cdot {(\frac{σ^{2}}{n_{1} + n_{2} - 2})}^{2} = \frac{2 σ^{4}}{n_{1} + n_{2} - 2} .$
[This is using the fact that $Var (a X) = a^{2} Var (X)$ .

Do you have a similar question?

Recalculate according to your conditions!

New Questions in College Statistics

Didn't find what you were looking for?