Let $X_1, X_2 \cdots X_N$ be random variables, which follow a Gaussian distribution.
\begin{equation} X \sim N(\mu, \sigma^2) \end{equation} Let the parameters $\mu$ and $\sigma^2$ be unknown. I know that the unbiased estimator of $\mu$ is \begin{equation} \bar{X}=\frac{1}{N}(X_1+X_2 \cdots X_N) \end{equation} and that of $\sigma^2$ is \begin{equation} U = \frac{1}{N-1}\left((X_1 -\bar{X})^2 \cdots (X_N -\bar{X})^2\right) \end{equation}
Here, a new parameter $f$ is defined as \begin{equation} f = \frac{1}{\sigma^2 + \sigma_t^2} \exp\left(-\frac{1}{2}(\mu -\mu_t)^2(\sigma^2 + \sigma_t^2)^{-1}\right) \end{equation} , where $\mu_t$ and $\sigma_t^2$ are known parameters. I'm not sure how to obtain the unbiased estimator of $f$.
The simplest case is when $\mu$ is a known parameter and $\mu = \mu_t$. In this case,
\begin{equation} f = \frac{1}{\sigma^2 + \sigma_t^2} \end{equation}
and I'd like to calculate the mean value of $\tilde{F}=\frac{1}{U + \sigma_t^2}$
\begin{equation} \begin{split} E\left[\tilde{F} \right]&= E\left[\frac{1}{U + \sigma_t^2}\right] \\ &= E\left[\frac{N-1}{\sigma^2 Y + (N-1)\sigma_t^2}\right] \quad (Y \sim \chi_{N-1}^2) \\ &= (N-1) \int_0^{\infty} \frac{1}{\sigma^2 y + (N-1)\sigma_t^2} \frac{y^{\frac{N-1}{2}-1}e^{-\frac{y}{2}}} {2^{\frac{N-1}{2}} \Gamma(\frac{N-1}{2})} dy \end{split} \end{equation} How can I calculate this value? So, my question is
- how to obtain the unbiased estimator of $f$
- how to calculate $E[\tilde{F}]$
Either one or the other answer is fine.
I faced this problem when I was studying quantum mechanics. Gaussian quantum states can be represented by Gaussian distributions of operators of $\hat{x}$ and $\hat{p}$. A vector of the mean values of these operators can be defined as $\boldsymbol{m} = (\left<\hat{x}\right>, \left< \hat{p}\right>)^{\top}$ and the covariance matrix can be defined as \begin{equation} V = \begin{pmatrix} (\Delta \hat{x})^2 & \Delta \hat{x} \Delta \hat{p} \\ \Delta \hat{p} \Delta \hat{x} & (\Delta \hat{p})^2 \end{pmatrix} \end{equation} Fidelity $f$, which is a measure of closeness between two quantum states, can be written as \begin{equation} \begin{split} f&= \frac{\hbar}{\sqrt{\operatorname{det}\left(V_1+V_2\right)}} \exp \left[-\frac{1}{2}\left(\boldsymbol{m}_1-\boldsymbol{m}_2\right)^\top\left(V_1+V_2\right)^{-1}\left(\boldsymbol{m}_1-\boldsymbol{m}_2\right)\right] \\ &0 \le f \le 1 \end{split} \end{equation} If the two states are the same ($\boldsymbol{m}_1 = \boldsymbol{m}_2$, $V_1 = V_2$), the fidelity is 1 since $\operatorname{det} V=(\frac{\hbar}{2})^2$. The Gaussian states can be defined using two parameters $\hat{x}$ and $\hat{p}$, but the question is changed to one parameter in order to make the problem simpler.