An asymptotic expansion for the distribution of the maximum likelihood estimation of a vektor parameter

Let $X$ he a random variable with a distribution function $f(x,\theta)$ depending on a vector parameter $\theta=(\theta,\dots,\theta_r)$. Let $\widehat\theta_n$ be the maximum likelihood estimate of $\theta$ corresponding to a sample of size $n$. It is proved that under certain conditions on $f(x,\theta)$ the distribution function of $\widehat\theta_n$ has an asymptotic expansion on $n^{1/2}$ with the number of terms depending on properties of $f(x,\theta)$.