Geodesic
Asymptotical behavior of some statistical estimators in the smooth case. I.~Study of the likelihood ratio
Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 3, pp. 469-486

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The paper considers properties of likelihood ratio determined by (1.1). We prove that the distributions in functional space $\mathbf C_0$, generated by the processes $Z_n(\theta)$ $(-\infty\theta\infty)$ tend to the distribution in $\mathbf C_0$, generated by the process $Z(\theta)$ defined by (2.1), provided conditions I–IV of section 1 are satisfied. As a consequence, we have asymptotical normality of the maximum likelihood estimator without assumptions of continuity of $\log f(x,\theta)$ and existence of $f''_{\theta\theta}(x,\theta)$. We deduce several other consequences of this result useful in the second part of the paper. 
@article{TVP_1972_17_3_a5,
     author = {I. A. Ibragimov and R. Z. Khas'minskii},
     title = {Asymptotical behavior of some statistical estimators in the smooth case. {I.~Study} of the likelihood ratio},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {469--486},
     publisher = {mathdoc},
     volume = {17},
     number = {3},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1972_17_3_a5/}
}
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I. A. Ibragimov; R. Z. Khas'minskii. Asymptotical behavior of some statistical estimators in the smooth case. I.~Study of the likelihood ratio. Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 3, pp. 469-486. http://geodesic.mathdoc.fr/item/TVP_1972_17_3_a5/