= Reciprocal moments of an estimated variance
\b[Reciprocal moments of an estimated variance] determine whether a precision estimate such as $1/S^2$ has a finite <expected value> or <variance>. Small positive values of a <sample variance> become large values after inversion. An <unbiased estimator> of a <variance> therefore need not give an unbiased or even integrable estimate of its reciprocal.
For $n\geq2$, let $X_1,\ldots,X_n$ be independent <normal random variables> with common mean $\mu$ and <variance> $\sigma^2>0$. For the unbiased <sample variance> $S^2=(n-1)^{-1}\sum_i(X_i-\overline X)^2$, put $\nu=n-1$. Then $Z=\nu S^2/\sigma^2$ has a <chi-squared distribution>. Its <probability density function> is $z^{\nu/2-1}e^{-z/2}/[2^{\nu/2}\Gamma(\nu/2)]$ for $z>0$.<SciPy 1.16.2 documentation>[scipy.stats.chi2, Notes, chi-squared density and gamma parameterization]{ref}
For any real $p>0$, direct integration and the <Gamma function> give
$$
\mathbb E[Z^{-p}]
=\frac1{2^{\nu/2}\Gamma(\nu/2)}\int_0^\infty z^{\nu/2-p-1}e^{-z/2}\,dz
=2^{-p}\frac{\Gamma(\nu/2-p)}{\Gamma(\nu/2)},\qquad \nu>2p.
$$
The convergence restriction is essential: near zero the integrand has exponent $\nu/2-p-1$, so the integral diverges when $\nu\leq2p$. Substitution of a negative <Gamma function> argument into the displayed quotient does not define a nonexistent moment.
Rescaling gives the useful general result
$$
\boxed{\mathbb E[(S^2)^{-p}]
=\left(\frac\nu{2\sigma^2}\right)^p
\frac{\Gamma(\nu/2-p)}{\Gamma(\nu/2)},\qquad \nu>2p.}
$$
In particular, the mean reciprocal exists for $n>3$, while its <variance> requires $n>5$:
$$
\mathbb E[1/S^2]=\frac{\nu}{(\nu-2)\sigma^2},\qquad
\operatorname{Var}(1/S^2)=\frac{2\nu^2}{\sigma^4(\nu-2)^2(\nu-4)}.
$$
For $n=6$ the mean is $5/(3\sigma^2)$, the <variance> is $50/(9\sigma^4)$ and the <coefficient of variation> is $\sqrt2$. For $n=4$ the mean is finite but the <variance> is infinite. For $n=3$ even the mean diverges. Increasing the number of groups does not repair a nonexistent moment of each raw reciprocal estimate.
<Inverse-variance weighting> commonly normalizes these reciprocals. Those normalized weights lie between zero and one, so their moments can exist even when the raw reciprocal moments fail. This distinction matters in <uncertainty in estimated inverse-variance weights>: neither the moment calculation nor a plug-in precision formula alone establishes the accuracy of the final weighted <estimator>. The results here use the exact normal-sample law; a nonnormal sample requires a different analysis.
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