A framework for adaptive Monte-Carlo procedures

Adaptive Monte Carlo methods are recent variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented by Vazquez-Abad and Dufresne, Fu and Su, and Arouna. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to some examples of valuation of financial derivatives

Simulation in Statistics

Simulation has become a standard tool in statistics because it may be the only tool available for analysing some classes of probabilistic models. We review in this paper simulation tools that have been specifically derived to address statistical challenges and, in particular, recent advances in the areas of adaptive Markov chain Monte Carlo (MCMC) algorithms, and approximate Bayesian calculation (ABC) algorithms.Comment: Draft of an advanced tutorial paper for the Proceedings of the 2011 Winter Simulation Conferenc

Integral approximation by kernel smoothing

Let $(X_1,\ldots,X_n)$ be an i.i.d. sequence of random variables in $\mathbb{R}^d$, $d\geq 1$. We show that, for any function $\varphi :\mathbb{R}^d\rightarrow\mathbb{R}$, under regularity conditions, $$n^ {1/2}\Biggl(n^{-1}\sum_{i=1}^n\frac{\varphi(X_i)}{\widehat{f}^(X_i)}- \int \varphi(x)\,dx\Biggr)\stackrel{\mathbb{P}}{\longrightarrow}0,$$ where $\widehat{f}$ is the classical kernel estimator of the density of $X_1$. This result is striking because it speeds up traditional rates, in root $n$, derived from the central limit theorem when $\widehat{f}=f$. Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions $\varphi$ and $f$, the dimension $d$ and the bandwidth of the kernel estimator $\widehat{f}$. Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of $\varphi$. As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations.Comment: Published at http://dx.doi.org/10.3150/15-BEJ725 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin note: text overlap with arXiv:1312.449