A simple and conservative empirical likelihood function

Abstract

Multiple measurements with an unknown Gaussian likelihood function are treated probabilistically. The likelihood function

$L(\mu)\propto \left( (n-1)\sigma_x^2 +n(\mu-\overline{x})^2 \right)^{-n/4}$ is derived

where $\mu$ is the true value, $\overline{x}$ is the mean,

and $\sigma_x^2$ is the variance obtained from the $n$ measurements.