ﻻ يوجد ملخص باللغة العربية
An approximate maximum likelihood method of estimation of diffusion parameters $(vartheta,sigma)$ based on discrete observations of a diffusion $X$ along fixed time-interval $[0,T]$ and Euler approximation of integrals is analyzed. We assume that $X$ satisfies a SDE of form $dX_t =mu (X_t ,vartheta ), dt+sqrt{sigma} b(X_t ), dW_t$, with non-random initial condition. SDE is nonlinear in $vartheta$ generally. Based on assumption that maximum likelihood estimator $hat{vartheta}_T$ of the drift parameter based on continuous observation of a path over $[0,T]$ exists we prove that measurable estimator $(hat{vartheta}_{n,T},hat{sigma}_{n,T})$ of the parameters obtained from discrete observations of $X$ along $[0,T]$ by maximization of the approximate log-likelihood function exists, $hat{sigma}_{n,T}$ being consistent and asymptotically normal, and $hat{vartheta}_{n,T}-hat{vartheta}_T$ tends to zero with rate $sqrt{delta}_{n,T}$ in probability when $delta_{n,T} =max_{0leq i<n}(t_{i+1}-t_i )$ tends to zero with $T$ fixed. The same holds in case of an ergodic diffusion when $T$ goes to infinity in a way that $Tdelta_n$ goes to zero with equidistant sampling, and we applied these to show consistency and asymptotical normality of $hat{vartheta}_{n,T}$, $hat{sigma}_{n,T}$ and asymptotic efficiency of $hat{vartheta}_{n,T}$ in this case.
Estimating the matrix of connections probabilities is one of the key questions when studying sparse networks. In this work, we consider networks generated under the sparse graphon model and the in-homogeneous random graph model with missing observati
With a view to statistical inference for discretely observed diffusion models, we propose simple methods of simulating diffusion bridges, approximately and exactly. Diffusion bridge simulation plays a fundamental role in likelihood and Bayesian infer
We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=expvarphi_0$ where $varphi_0$ is a concave function on $mathbb{R}$. The pointwise limiting distributi
We present theoretical properties of the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in $mathbb{R}^d$. Our study covers both the case where the true underlying density is log-concav
The saddlepoint approximation gives an approximation to the density of a random variable in terms of its moment generating function. When the underlying random variable is itself the sum of $n$ unobserved i.i.d. terms, the basic classical result is t