An Asymptotic Expansion for a Class of Estimators Containing Maximum Likelihood Estimators

Let $X_1,\dots,X_n$ be a sample from a distribution dependent on a parameter $\theta=(\theta_1,\dots,\theta_s)^T$ and $\vartheta_n=\vartheta_n(X_1,\dots,X_n)$ a minimum contrast estimator for $\theta$ corresponding to a contrast function $f(x,\theta)$ (see, e.g., [7], [8], [9]). When the $X_i$'s have a density $p(x,\theta)$ and $f(x,\theta)=-\log p(x,\theta), \vartheta$ is the maximum likelihood estimator. Among the regularity conditions, it is assumed that the continous derivatives $f^{\alpha}(x,\theta)=(d^{\alpha_1+\dots+\alpha_s}/d\theta_1^{\alpha_1}\dots d\theta_s^{\alpha_s})f(x,\theta)$ exist in a neighbourhood of the true value $\theta_0$ for all $\alpha$ with $\alpha_1+\dots+\alpha_s\le k+1$ and $E_{\theta_0}|f^{(\alpha)}(X,\theta_0)|^r<\infty$ for some $r>2$. We obtain an expansion of the form
$$ \sqrt{n}(\vartheta_n-\theta_0)=h_1+n^{-1/2}h_2+\dots+n^{-(k-1)/2}h_k+\zeta_n $$
where the components of $h_j, j=1,\ldots,k$, are polynomials dependent on some random variables of the form $n^{-1/2}\sum\limits_{i=1}^n f^{(\alpha)}(X_i,\theta_0)$ and $\zeta_n$ is a random variable converging to zero at a certain rate.