Geodesic
An asymptotic expansion for the distribution of the maximum likelihood estimation of a~vektor parameter
Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 3, pp. 418-425

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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 $f(x,\theta)$. 
@article{TVP_1967_12_3_a1,
     author = {N. M. Mitrofanova},
     title = {An asymptotic expansion for the distribution of the maximum likelihood estimation of a~vektor parameter},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {418--425},
     publisher = {mathdoc},
     volume = {12},
     number = {3},
     year = {1967},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1967_12_3_a1/}
}
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N. M. Mitrofanova. An asymptotic expansion for the distribution of the maximum likelihood estimation of a~vektor parameter. Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 3, pp. 418-425. http://geodesic.mathdoc.fr/item/TVP_1967_12_3_a1/