A solution to the multivariate behrens-fisher problem

Some new algebra on pattern and transition matrices is used to determine the degrees of freedom and the parameter matrix, if the distribution of a linear sum of Wishart matrices is approximated by a single Wishart distribution. This approximation is then used to find a solution to the multivariate Behrens-Fisher problem similar to the Welch (1947) solution in the univariate case.     

Consistency of the Tikhonov’s regularization in an ill-posed problem with α-mixing random data

In this article, we establish exponential inequalities for the probability of the distance between the approximated solution and the exact one for an operator equation with an exact right-hand side. In addition, the second member of the operator equation is observed with α-mixing random errors. These inequalities yield the almost complete convergence of the approximate solution.     

Computational Aspects Related to Martingale Estimating Functions for a Discretely Observed Diffusion

Martingale estimating functions for a discretely observed diffusion have turned out to provide estimators with nice asymptotic properties. However, their expression usually involves some conditional expectation that has to be evaluated through Monte Carlo simulations giving rise to an approximated estimator. In this work we study, for ergodic models, the asymptotic properties of the approximated estimator and describe the influence of the number of independent simulated trajectories involved in the Monte Carlo method as well as of the approximation scheme used. Our results are of practical relevance to assess the implementation of martingale estimating functions for discretely observed diffusions.     

Some reflections on almon estimators

In the first part of the paper we give a brief survey of methods to estimate distributed lag models where the coefficients can be approximated by a polynomial. In the second part we give a corrected version of Maddala's (1977) formula for the Almon estimator and some further results. To facilitate reference to Maddala's work, we use his notation, where possible.     

Testing for Equivalence of Means under Heteroskedasticity by an Approximate Solution of a Partial Differential Equation of Infinite Order

Abstract. A test for two‐sided equivalence of means has been developed under the assumption of normally distributed populations with heterogeneous variances. Its rejection region is limited by functions ± h that depend on the empirical variances. h is stated implicitly by a partial differential equation, an exact solution of which would provide a test that is exactly similar at the boundary of the null hypothesis of non‐equivalence. h is approximated by Taylor series up to third powers in the reciprocal number of degrees of freedom. This suffices to obtain error probabilities of the first kind that are very close to a nominal level of α = 0 . 05 at the boundary of the null hypothesis. For more than 10 data points in each group, they range between 0.04995 and 0.05005, and are thus much more precise than those obtained by other authors.     

A note about calibrated prediction regions and distributions

In this work we address the problem of the construction of prediction regions and distribution functions, with particular regard to the multidimensional setting. Firstly, we define a simple procedure for calculating the predictive distribution function which gives improved prediction limits. Secondly, with a multivariate generalization of a result presented in Ueki and Fueda (2007), we propose a method for correcting estimative prediction regions, to reduce their coverage error to the third-order accuracy. The improved prediction regions and the associated distribution functions are easy to calculate using a suitable bootstrap procedure. Examples of application are included, showing the good performance of the proposed method, even if we consider an approximated model for prediction purposes.     

Stochastic approximation results for variational inequality problem using random-type iterative schemes

Real world problems are embedded with uncertainties. Therefore, to tackle these problems, one must consider probabilistic nature of the problems both in modeling and solution. In this work, concepts of convergence of the solution of variational inequality in classical functional analysis are extended to a stochastic domain for a random Mann-type iterative and Ishikawa-type iterative schemes in a Banach space. A mean square convergence result is proved for this extension.     

Laplace based approximate posterior inference for differential equation models

Ordinary differential equations are arguably the most popular and useful mathematical tool for describing physical and biological processes in the real world. Often, these physical and biological processes are observed with errors, in which case the most natural way to model such data is via regression where the mean function is defined by an ordinary differential equation believed to provide an understanding of the underlying process. These regression based dynamical models are called differential equation models. Parameter inference from differential equation models poses computational challenges mainly due to the fact that analytic solutions to most differential equations are not available. In this paper, we propose an approximation method for obtaining the posterior distribution of parameters in differential equation models. The approximation is done in two steps. In the first step, the solution of a differential equation is approximated by the general one-step method which is a class of numerical numerical methods for ordinary differential equations including the Euler and the Runge-Kutta procedures; in the second step, nuisance parameters are marginalized using Laplace approximation. The proposed Laplace approximated posterior gives a computationally fast alternative to the full Bayesian computational scheme (such as Makov Chain Monte Carlo) and produces more accurate and stable estimators than the popular smoothing methods (called collocation methods) based on frequentist procedures. For a theoretical support of the proposed method, we prove that the Laplace approximated posterior converges to the actual posterior under certain conditions and analyze the relation between the order of numerical error and its Laplace approximation. The proposed method is tested on simulated data sets and compared with the other existing methods.     

A Monte Carlo method for filtering a marked doubly stochastic Poisson process

The author provides an approximated solution for the filtering of a state-space model, where the hidden state process is a continuous-time pure jump Markov process and the observations come from marked point processes. Each state k corresponds to a different marked point process, defined by its conditional intensity function λ _k (t). When a state is visited by the hidden process, the corresponding marked point process is observed. The filtering equations are obtained by applying the innovation method and the integral representation theorem of a point process martingale. Since the filtering equations belong to the family of Kushner–Stratonovich equations, an iterative solution is calculated. The theoretical solution is approximated and a Monte Carlo integration technique employed to implement it. The sequential method has been tested on a simulated data set based on marked point processes widely used in the statistical analysis of seismic sequences: the Poisson model, the stress release model and the Etas model.     

Statistical analysis of the number of self-overlapping leftmost repeats in an homogeneous stationary Markov chain on finite states

ABSTRACT

This article addresses the problem of repeats detection used in the comparison of significant repeats in sequences. The case of self-overlapping leftmost repeats for large sequences generated by an homogeneous stationary Markov chain has not been treated in the literature. In this work, we are interested by the approximation of the number of self-overlapping leftmost long enough repeats distribution in an homogeneous stationary Markov chain. Using the Chen–Stein method, we show that the number of self-overlapping leftmost long enough repeats distribution is approximated by the Poisson distribution. Moreover, we show that this approximation can be extended to the case where the sequences are generated by a m-order Markov chain.     

Accelerating backpropagation using effective parameters at each step and an experimental evaluation

An acceleration of backpropagation algorithm with momentum (BPM) is introduced. At every stage of the learning process, local quadratic approximation of the error function is performed and the Hessian matrix of the quadratic function is approximated. Effective learning rate and momentum factor are determined by means of maximum and minimum eigenvalues of the approximated Hessian matrix at each step. BPM algorithm is modified so as to work automatically with these effective parameters. Performance of this new approach is demonstrated in comparison with well-known training algorithms on conventional problems by an experimental evaluation.     

Gaussian proposal density using moment matching in SMC methods

In this article we introduce a new Gaussian proposal distribution to be used in conjunction with the sequential Monte Carlo (SMC) method for solving non-linear filtering problems. The proposal, in line with the recent trend, incorporates the current observation. The introduced proposal is characterized by the exact moments obtained from the dynamical system. This is in contrast with recent works where the moments are approximated either numerically or by linearizing the observation model. We show further that the newly introduced proposal performs better than other similar proposal functions which also incorporate both state and observations. This work was supported by a research grant from THALES Nederland BV.     

Polynomial approximation of nonlinear regression functions

In the present paper a nonlinear regression function is approximated by a polynomial estimator according to the expectation of the quadratic L ²-distance as risk is given. For special experimental designs with repeating experimental points this estimator coincides with the estimator by the method of the reproducing kernel.

Considerations about the relation for the sample size and the degree of the approximation polynomial and about the quadratic mean are given.     

Approximating the equilibrium quantity traded and welfare in large markets

We consider the efficient outcome of a canonical economic market model involving buyers and sellers with independent and identically distributed random valuations and costs, respectively. When the number of buyers and sellers is large, we show that the joint distribution of the equilibrium quantity traded and welfare is asymptotically normal. Moreover, we bound the approximation rate. The proof proceeds by constructing, on a common probability space, a representation consisting of two independent empirical quantile processes, which in large markets can be approximated by independent Brownian bridges. The distribution of interest can then be approximated by that of a functional of a Gaussian process. This methodology applies to a variety of mechanism design problems.     

α-stable laws for noncoding regions in DNA sequences

In this work, we analyze the long-range dependence parameter for a nucleotide sequence in several different transformations. The long-range dependence parameter is estimated by the approximated maximum likelihood method, by a novel estimator based on the spectral envelope theory, by a regression method based on the periodogram function, and also by the detrended fluctuation analysis method. We study the length distribution of coding and noncoding regions for all Homo sapiens chromosomes available from the European Bioinformatics Institute. The parameter of the tail rate decay is estimated by the Hill estimator ?α. We show that the tail rate decay is greater than 2 for coding regions, while for almost all noncoding regions it is less than 2.     

Exponential inequalities in linear calibration problem

ABSTRACT

Calibration, also called inverse regression, is a classical problem which appears often in a regression setup under fixed design. The aim of this article is to propose a stochastic method which gives an estimated solution for a linear calibration problem. We establish exponential inequalities of Bernstein–Frechet type for the probability of the distance between the approximate solutions and the exact one. Furthermore, we build a confidence domain for the so-mentioned exact solution. To check the validity of our results, a numerical example is proposed.     

On limiting likelihood ratio processes of some change-point type statistical models

Different change-point type models encountered in parametric statistical inference give rise to different limiting likelihood ratio processes. In this paper we consider two such likelihood ratios. The first one is an exponential functional of a two-sided Poisson process driven by some parameter, while the second one is an exponential functional of a two-sided Brownian motion. We establish that for sufficiently small values of the parameter, the Poisson type likelihood ratio can be approximated by the Brownian type one. As a consequence, several statistically interesting quantities (such as limiting variances of different estimators) related to the first likelihood ratio can also be approximated by those related to the second one. Finally, we discuss the asymptotics for large values of the parameter and illustrate the results by numerical simulations.     

Approximating discrete multivariate distributions prom known moments

To approximate the joint distribution of the two-colony stepping-stone model, a finite mixture approach is proposed for constructing discrete multi-variate distributions. This approach generahzes the classic method of linear combinations of independent variables. The stepping-stone model is approximated through matching known moments. Numerical examples from entomology are given. Comparisons are made with the work by Wehrly et al (1993).     

Opportunities of the minimum Anderson–Darling estimator as a variant of the maximum likelihood method

We reveal that the minimum Anderson–Darling (MAD) estimator is a variant of the maximum likelihood method. Furthermore, it is shown that the MAD estimator offers excellent opportunities for parameter estimation if there is no explicit formulation for the distribution model. The computation time for the MAD estimator with approximated cumulative distribution function is much shorter than that of the classical maximum likelihood method with approximated probability density function. Additionally, we research the performance of the MAD estimator for the generalized Pareto distribution and demonstrate a further advantage of the MAD estimator with an issue of seismic hazard analysis.     

Exponential Inequalities in Stochastic Inverse Problems Using an Iterative Method

We consider an iterative method in order to solve linear inverse problems. We establish exponential inequalities for the probability of the distance between the approximated solution and the exact one for a calibration problem. The approximate is given by an iterative method with Gaussian errors. We treat an operator equation of the form Ax = u, where A is a compact operator.