Non Uniform Bounds on Geometric Approximation Via Stein's Method and w-Functions

In this article, we use Stein's method and w-functions to give uniform and non uniform bounds in the geometric approximation of a non negative integer-valued random variable. We give some applications of the results of this approximation concerning the beta-geometric, Pólya, and Poisson distributions.     

Maximizing Complex Likelihoods via Directed Stochastic Searching Algorithm

In this article, a directed stochastic searching algorithm is defined. It is a root or optimal parameter searching algorithm with stochastic searching directions. This algorithm is especially relevant when the objective function is complex or is observed with errors. We prove that the resulting roots or estimators have well-controlled biases under certain conditions. We examine the proposed method by finding the maximum likelihood estimates for which the corresponding likelihood function has or does not have a closed-form representation in both the simulations and the real cases. Finally, the limitations and the consequences when multiple solutions exist are addressed.     

Stochastic adaptation of importance sampler

Improving efficiency of the importance sampler is at the centre of research on Monte Carlo methods. While the adaptive approach is usually not so straightforward within the Markov chain Monte Carlo framework, the counterpart in importance sampling can be justified and validated easily. We propose an iterative adaptation method for learning the proposal distribution of an importance sampler based on stochastic approximation. The stochastic approximation method can recruit general iterative optimization techniques like the minorization–maximization algorithm. The effectiveness of the approach in optimizing the Kullback divergence between the proposal distribution and the target is demonstrated using several examples.     

Confidence Intervals from Stochastic Approximation

We propose a nonparametric method of constructing confidence interval for a scalar parameter from stochastic approximation through the efficient Robbins–Monro procedure proposed by Joseph (2004 Joseph, V.R. (2004). Efficient Robbins–Monro procedure for binary data. Biometrika 91:461–470.[Crossref], [Web of Science ®] , [Google Scholar]). Unlike the bootstrap method where the number of resampling is fixed in advance, the proposed procedure iteratively searches the endpoints in an optimal way such that the convergence is fast and the coverage is obtained accurately. Simulation and real data application illustrate its superiority over the usual Robbins–Monro procedure and common bootstrap methods.     

On Geometry and Scale of a Stochastic Chemostat

A chemostat is a fixed volume bioreactor in which micro–organisms are grown in a continuously renewed liquid medium. We propose a stochastic model for the evolution of the concentrations in the single species and single substrate case. It is obtained as a diffusion approximation of a pure jump Markov process, whose increments are comparable in mean with the deterministic model. A specific time scale, related to the noise intensity, is considered for each source of variation. The geometric structure of the problem, usable by identification procedures, is preserved both in the drift and diffusion term. We study the properties of this model by numerical experiments.     

Saddlepoint-Based Bootstrap Inference for Quadratic Estimating Equations

Abstract.  We propose an easy to implement method for making small sample parametric inference about the root of an estimating equation expressible as a quadratic form in normal random variables. It is based on saddlepoint approximations to the distribution of the estimating equation whose unique root is a parameter's maximum likelihood estimator (MLE), while substituting conditional MLEs for the remaining (nuisance) parameters. Monotoncity of the estimating equation in its parameter argument enables us to relate these approximations to those for the estimator of interest. The proposed method is equivalent to a parametric bootstrap percentile approach where Monte Carlo simulation is replaced by saddlepoint approximation. It finds applications in many areas of statistics including, nonlinear regression, time series analysis, inference on ratios of regression parameters in linear models and calibration. We demonstrate the method in the context of some classical examples from nonlinear regression models and ratios of regression parameter problems. Simulation results for these show that the proposed method, apart from being generally easier to implement, yields confidence intervals with lengths and coverage probabilities that compare favourably with those obtained from several competing methods proposed in the literature over the past half-century.     

Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract. We consider N independent stochastic processes (X _i (t), t ∈ [0,T _i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ _i . The distribution of the random effect φ _i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ _i and φ _i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T _i =T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.     

Bias Reduction using Stochastic Approximation

The paper studies stochastic approximation as a technique for bias reduction. The proposed method does not require approximating the bias explicitly, nor does it rely on having independent identically distributed (i.i.d.) data. The method always removes the leading bias term, under very mild conditions, as long as auxiliary samples from distributions with given parameters are available. Expectation and variance of the bias-corrected estimate are given. Examples in sequential clinical trials (non-i.i.d. case), curved exponential models (i.i.d. case) and length-biased sampling (where the estimates are inconsistent) are used to illustrate the applications of the proposed method and its small sample properties.     

Stochastic Bounds with a Low Rank Decomposition

We investigate how we can bound a discrete time Markov chain (DTMC) by a stochastic matrix with a low rank decomposition. In the first part of the article, we show the links with previous results for matrices with a decomposition of size 1 or 2. Then we show how the complexity of the analysis for steady-state and transient distributions can be simplified when we take into account the decomposition. Finally, we show how we can obtain a monotone stochastic upper bound with a low rank decomposition.     

Almost Periodic Solutions for Stochastic Differential Equations Driven By G-Brownian Motion

In this paper, we introduce the concept of the p-mean almost periodicity for stochastic processes in non linear expectation spaces. The existence and uniqueness of square-mean almost periodic solutions to some non linear stochastic differential equations driven by G-Brownian motion are established under some assumptions for the coefficients. The asymptotic stability of the unique square-mean almost periodic solution in the square-mean sense is also discussed.     

A Bayesian Meta-Modeling Approach for Gaussian Stochastic Process Models Using a Non Informative Prior

In this article, an efficient Bayesian meta-modeling approach is proposed for Gaussian stochastic process models in computer experiments. Different prior densities and particularly, a non informative hyper prior have been employed on the parameters involved in the correlation matrix. And the estimation of related parameters is obtained by the expectation-maximization algorithm. Compared with the recent work of Li and Sudjianto (2005 Li , R. , Sudjianto , A. ( 2005 ). Analysis of computer experiments using penalized likelihood in Kriging models . Technometrics 47 : 111 – 120 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), the proposed approach is not only of higher prediction accuracy but also of lower computational cost, due to the utilization of the non informative prior and the absence of tuning parameters. Experimental results demonstrate that our approach yields state-of-the-art performance.     

Stochastic approximation Monte Carlo EM for change-point analysis

In the expectation–maximization (EM) algorithm for maximum likelihood estimation from incomplete data, Markov chain Monte Carlo (MCMC) methods have been used in change-point inference for a long time when the expectation step is intractable. However, the conventional MCMC algorithms tend to get trapped in local mode in simulating from the posterior distribution of change points. To overcome this problem, in this paper we propose a stochastic approximation Monte Carlo version of EM (SAMCEM), which is a combination of adaptive Markov chain Monte Carlo and EM utilizing a maximum likelihood method. SAMCEM is compared with the stochastic approximation version of EM and reversible jump Markov chain Monte Carlo version of EM on simulated and real datasets. The numerical results indicate that SAMCEM can outperform among the three methods by producing much more accurate parameter estimates and the ability to achieve change-point positions and estimates simultaneously.     

Stochastic Algorithms for the Estimation of an Optimal Solution of a LP Problem. Convergence and Central Limit Theorem

In this article, we consider a Linear Programming (LP) problem with unknown objective function. We introduce a class of stochastic algorithms to estimate an optimal solution of the LP problem. The almost sure convergence and the speed of convergence of these algorithms are analyzed. We also prove a central limit theorem for the estimation errors of the algorithms.     

A Stochastic Approach to General Nonlinear Programming Problems

Nonlinear programming problem is the general case of mathematical programming problem such that both the objective and constraint functions are nonlinear and is the most difficult case of smooth optimization problem to solve. In this article, we suggest a stochastic search method to general nonlinear programming problems which is not an iterative algorithm but it is an interior point method. The proposed method finds the near-optimal solution to the problem. The results of a few numerical studies are reported. The efficiency of the new method is compared and is found to be reasonable.     

Evaluation of the parameters of su by rational fractions

A rational fraction approximation is given for a function of one of the parameters defining Johnson's S_UError assessment for a segment of the domain of validity shows remarkable accuracy.     

Non uniform bound on a combinatorial central limit theorem

ABSTRACT

In this paper, we give a non uniform bound on combinatorial central limit theorem by using Stein’s method. This improves the result of Neammanee and Rattanawong (2009 Neammanee, K., Rattanawong, P. (2009). Non-uniform bound on normal approximation of Latin hypercube sampling. JMR 1:28–42. [Google Scholar]) from the finiteness of the sixth moment to the finiteness of the third moment.     

Stochastic compartmental models with birth,death and immigration of particles

A stochastic multi-compartmental system with initial and immigrant particles performing birth and death processes is considered. Moments and approximate solutions for cumulant generating function for the number of particles in various compartments are obtained. Explicit expression have been obtained for the covariance function between the number of particles in any compartment at two different time points. Applications are discussed.     

Berry-Esseen Bounds for Random Index Non Linear Statistics via Stein's Method

Under the second moment condition, we obtain Berry-Esseen bounds for random index non linear statistics by using a technique discussed in Chen and Shao (2007 Chen, L. H.Y., Shao, Q.-M. (2007). Normal approximation for nonlinear statistics using a concentration inequality approach. Bernoulli 13(2):581–599.[Crossref], [Web of Science ®] , [Google Scholar]). A concept in this article is to approximate any random index non-linear statistic by a random index linear statistic. The bounds for random sums of independent random variables are also provided. Applications are the bounds for random U-statistics and random sums of the present values in investment analysis.     

Stochastic approximation Monte Carlo Gibbs sampling for structural change inference in a Bayesian heteroscedastic time series model

We consider a Bayesian deterministically trending dynamic time series model with heteroscedastic error variance, in which there exist multiple structural changes in level, trend and error variance, but the number of change-points and the timings are unknown. For a Bayesian analysis, a truncated Poisson prior and conjugate priors are used for the number of change-points and the distributional parameters, respectively. To identify the best model and estimate the model parameters simultaneously, we propose a new method by sequentially making use of the Gibbs sampler in conjunction with stochastic approximation Monte Carlo simulations, as an adaptive Monte Carlo algorithm. The numerical results are in favor of our method in terms of the quality of estimates.