Flexible modeling based on copulas in nonparametric median regression

Consider the model Y=m(X)+ε, where m(⋅)=med(Y|⋅) is unknown but smooth. It is often assumed that ε and X are independent. However, in practice this assumption is violated in many cases. In this paper we propose modeling the dependence between ε and X by means of a copula model, i.e. (ε,X)∼C_θ(F_ε(⋅),F_X(⋅)), where C_θ is a copula function depending on an unknown parameter θ, and F_ε and F_X are the marginals of ε and X. Since many parametric copula families contain the independent copula as a special case, the so-obtained regression model is more flexible than the ‘classical’ regression model.We estimate the parameter θ via a pseudo-likelihood method and prove the asymptotic normality of the estimator, based on delicate empirical process theory. We also study the estimation of the conditional distribution of Y given X. The procedure is illustrated by means of a simulation study, and the method is applied to data on food expenditures in households.     

Likelihood-based belief function: Justification and some extensions to low-quality data

Given a parametric statistical model, evidential methods of statistical inference aim at constructing a belief function on the parameter space from observations. The two main approaches are Dempster's method, which regards the observed variable as a function of the parameter and an auxiliary variable with known probability distribution, and the likelihood-based approach, which considers the relative likelihood as the contour function of a consonant belief function. In this paper, we revisit the latter approach and prove that it can be derived from three basic principles: the likelihood principle, compatibility with Bayes' rule and the minimal commitment principle. We then show how this method can be extended to handle low-quality data. Two cases are considered: observations that are only partially relevant to the population of interest, and data acquired through an imperfect observation process.     

Prediction of future observations using belief functions: A likelihood-based approach

We study a new approach to statistical prediction in the Dempster–Shafer framework. Given a parametric model, the random variable to be predicted is expressed as a function of the parameter and a pivotal random variable. A consonant belief function in the parameter space is constructed from the likelihood function, and combined with the pivotal distribution to yield a predictive belief function that quantifies the uncertainty about the future data. The method boils down to Bayesian prediction when a probabilistic prior is available. The asymptotic consistency of the method is established in the iid case, under some assumptions. The predictive belief function can be approximated to any desired accuracy using Monte Carlo simulation and nonlinear optimization. As an illustration, the method is applied to multiple linear regression.     

Multivalued backward stochastic differential equations with time delayed generators

Our aim is to study the following new type of multivalued backward stochastic differential equation: $$\left\{ \begin{gathered} - dY\left( t \right) + \partial \phi \left( {Y\left( t \right)} \right)dt \ni F\left( {t,Y\left( t \right),Z\left( t \right),Y_t ,Z_t } \right)dt + Z\left( t \right)dW\left( t \right), 0 \leqslant t \leqslant T, \hfill \\ Y\left( T \right) = \xi , \hfill \\ \end{gathered} \right.$$ where ? _φ is the subdifferential of a convex function and (Y _t , Z _t ):= (Y(t + θ), Z(t + θ)) _θ∈[?T,0] represent the past values of the solution over the interval [0, t]. Our results are based on the existence theorem from Delong & Imkeller, Ann. Appl. Probab., 2010, concerning backward stochastic differential equations with time delayed generators.     

Un survol de la theorie de l'integrale stochastique

The objective of this paper is to present the principal results of a large part of stochastic calculus in a manner that should be comprehensible to readers having only the general notions of stochastic processes. Not all the theorems are proved in detail, but all the fundamental theorems are explained with clarity and precision, and with special attention to the motivations behind them.Given two real valued stochastic processes X and Y, the basic problem is to give a meaning to Z = ∫ Y dX in such a way that the integral sign is not misused. If X is a process whose paths are of bounded variation, then Z should coincide with the ordinary Lebesgue-Stieltjes integral taken path by path. If Y is a left continuous step function, then Z should coincide with the obvious choice: if Y is constant on ]t, u], then Z_u-Z_t is that constant times X_u-X_t. And finally, the Lebesgue dominated convergence theorem should hold: if the processes Yⁿ converge to Y and all the Yⁿ are dominated by a process Y' for which ∝ Y' dX is well defined, then Z_n = ∫ Yⁿ dX should converge to Z = ∫ Y dX in some sense.Starting with these requirements, it is shown that, if ∫ Y dX is defined for all predictable Y, then X must be a semimartingale. Conversely, the integral is well defined for all predictable Y and all semimartingales X.With the integrals defined, a number of their important properties are discussed. In particular, the integral Z is a semimartingale, and a change of variable formula (Ito's formula) holds for $\text{?(Z)}$. Finally, stochastic integral equations are introduced, and a general theorem is given on the existence and uniqueness of solutions.A bibliography with commentaries supplements the text for the benefit of those who would like to go deeper into the subject.     

Strong Decomposition of Random Variables

A random variable X is called strongly decomposable into (strong) components Y,Z, if X=Y+Z where Y=φ(X), Z=X−φ(X) are independent nondegenerate random variables and φ is a Borel function. Examples of decomposable and indecomposable random variables are given. It is proved that at least one of the strong components Y and Z of any random variable X is singular. A necessary and sufficient condition is given for a discrete random variable X to be strongly decomposable. Phenomena arising when φ is not Borel are discussed. The Fisher information (on a location parameter) in a strongly decomposable X is necessarily infinite.     

Malliavin Calculus Approach to Statistical Inference for Lévy Driven SDE’s

By means of the Malliavin calculus, integral representations for the likelihood function and for the derivative of the log-likelihood function are given for a model based on discrete time observations of the solution to equation dX _t = a _θ (X _t )dt + dZ _t with a Lévy process Z. Using these representations, regularity of the statistical experiment and the Cramer-Rao inequality are proved.     

Estimation for the change point of volatility in a stochastic differential equation

We consider a multidimensional Itô process Y=(Y_t)_t∈[0,T] with some unknown drift coefficient process b_t and volatility coefficient σ(X_t,θ) with covariate process X=(X_t)_t∈[0,T], the function σ(x,θ) being known up to θ∈Θ. For this model, we consider a change point problem for the parameter θ in the volatility component. The change is supposed to occur at some point t^∗∈(0,T). Given discrete time observations from the process (X,Y), we propose quasi-maximum likelihood estimation of the change point. We present the rate of convergence of the change point estimator and the limit theorems of the asymptotically mixed type.     

On the greatest splitting topology

It is well known that (see, for example, [H. Render, Nonstandard topology on function spaces with applications to hyperspaces, Trans. Amer. Math. Soc. 336 (1) (1993) 101-119; M. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105-145; D.N. Georgiou, S.D. Iliadis, F. Mynard, in: Elliott Pearl (Ed.), Function Space Topologies, Open Problems in Topology, vol. 2, Elsevier, 2007, pp. 15-22]) the intersection of all admissible topologies on the set C(Y,Z) of all continuous maps of an arbitrary space Y into an arbitrary space Z, is always the greatest splitting topology. However, this intersection maybe not admissible. In the case, where Y is a locally compact Hausdorff space the compact-open topology on the set C(Y,Z) is splitting and admissible (see [R.H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945) 429-432; R. Arens, A topology for spaces of transformations, Ann. of Math. 47 (1946) 480-495; R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31]), which means that the intersection of all admissible topologies on C(Y,Z) is admissible. In [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31] an example of a non-locally compact Hausdorff space Y is given having the same property for the case, where Z=[0,1], that is on the set C(Y,[0,1]) the compact-open topology is splitting and admissible. This space Y is the set [0,1] with a topology τ, whose semi-regular reduction coincides with the usual topology on [0,1]. Also, in [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31, Theorem 5.3] another example of a non-locally compact space Y is given such that the compact-open topology on the set C(Y,[0,1]) is distinct from the greatest splitting topology.In this paper first we construct non-locally compact Hausdorff spaces Y such that the intersection of all admissible topologies on the set C(Y,Z), where Z is an arbitrary regular space, is admissible. Furthermore, for a Hausdorff splitting topology t on C(Y,Z) we find sufficient conditions in order that t to be distinct from the greatest splitting topology. Using this result, we construct some concrete non-locally compact spaces Y such that the compact-open topology on C(Y,Z), where Z is a Hausdorff space, is distinct from the greatest splitting topology. Finally, we give some open problems.     

Confidence intervals for marginal parameters under fractional linear regression imputation for missing data

Item nonresponse occurs frequently in sample surveys and other applications. Imputation is commonly used to fill in the missing item values in a random sample {Y_i;i=1,…,n}. Fractional linear regression imputation, based on the model with independent zero mean errors ?_i, is used to create one or more imputed values in the data file for each missing item Y_i, where {X_i,i=1,…,n}, is observed completely. Asymptotic normality of the imputed estimators of the mean μ=E(Y), distribution function θ=F(y) for a given y, and qth quantile θ_q=F^-1(q),0<q<1 is established, assuming that Y is missing at random (MAR) given X. This result is used to obtain normal approximation (NA)-based confidence intervals on μ,θ and θ_q. In the case of θ_q, a Bahadur-type representation and Woodruff-type confidence intervals are also obtained. Empirical likelihood (EL) ratios are also obtained and shown to be asymptotically scaled variables. This result is used to obtain asymptotically correct EL-based confidence intervals on μ,θ and θ_q. Results of a simulation study on the finite sample performance of NA-based and EL-based confidence intervals are reported.     

Simple regression in view of elliptical models

For the simple linear model Y = θ1 + βx + ? where the error vector follows the elliptically contoured distribution, we consider the unrestricted, restricted, preliminary test and shrinkage estimators for the intercept parameter, θ when it is suspected that the slope parameter β may be β_o. The exact bias and MSE expressions are derived and the mean-square relative efficiency is taken to determine the relative dominance properties of the proposed estimators in comparison. In the continuation, the optimal level of significance of the preliminary test estimator is tabulated and some graphical result are also displayed.     

Inner functions as improving multipliers

If X⊂Y are two classes of analytic functions in the unit disk D and θ is an inner function, θ is said to be (X,Y)-improving, if every function f∈X satisfying fθ∈Y must actually satisfy fθ∈X. This notion has been recently introduced by K.M. Dyakonov. In this paper we study the (X,Y)-improving inner functions for several pairs of spaces (X,Y). In particular, we prove that for any p∈(0,1) the (Qp,BMOA)-improving inner functions and the (Qp,B)-improving inner functions are precisely the inner functions which belong to the space Qp. Here, B is the Bloch space. We also improve some results of Dyakonov on the subject regarding Lipschitz spaces and Besov spaces.     

A semiparametric Bernstein–von Mises theorem for Gaussian process priors

This paper is a contribution to the Bayesian theory of semiparametric estimation. We are interested in the so-called Bernstein–von Mises theorem, in a semiparametric framework where the unknown quantity is (θ, f), with θ the parameter of interest and f an infinite-dimensional nuisance parameter. Two theorems are established, one in the case with no loss of information and one in the information loss case with Gaussian process priors. The general theory is applied to three specific models: the estimation of the center of symmetry of a symmetric function in Gaussian white noise, a time-discrete functional data analysis model and Cox’s proportional hazards model. In all cases, the range of application of the theorems is investigated by using a family of Gaussian priors parametrized by a continuous parameter.     

The bitwisted Cartesian model for the free loop fibration

Using the notion of truncating twisting function from a simplicial set to a cubical set a special, bitwisted, Cartesian product of these sets is defined. For the universal truncating twisting function, the (co)chain complex of the corresponding bitwisted Cartesian product agrees with the standard Cartier (Hochschild) chain complex of the simplicial (co)chains. The modelling polytopes Fn are constructed. An explicit diagonal on Fn is defined and a multiplicative model for the free loop fibration ΩY→ΛY→Y is obtained. As an application we establish an algebra isomorphism H^∗(ΛY;Z)≈S(U)⊗Λ(s−1U) for the polynomial cohomology algebra H^∗(Y;Z)=S(U).     

Methods for generating random variates with Polya characteristic functions

Polya has shown that real even continuous functions that are convex on (0,∞), for 1 t = 0, and decreasing to 0 as t → ∞ are characteristic functions. Dugué and Girault (1955) have shown that the corresponding random variables are distributed as $\text{Y}\text{Z}$ where Y is a random variable with density (2π)^?1$\text{(}\text{sin}\text{(}\text{x}\text{2}\text{)/(}\text{x}\text{2}\text{))}{}^2$, and Z is independent of Y and has distribution function 1 ? φ + tφ′, t > 0. This property allows us to develop fast algorithms for this class of distributions. This is illustrated for the symmetric stable distribution, Linnik's distribution and a few other distributions. We pay special attention to the generation of Y.     

Robust kernel-based regression with bounded influence for outliers

The kernel-based regression (KBR) method, such as support vector machine for regression (SVR) is a well-established methodology for estimating the nonlinear functional relationship between the response variable and predictor variables. KBR methods can be very sensitive to influential observations that in turn have a noticeable impact on the model coefficients. The robustness of KBR methods has recently been the subject of wide-scale investigations with the aim of obtaining a regression estimator insensitive to outlying observations. However, existing robust KBR (RKBR) methods only consider Y-space outliers and, consequently, are sensitive to X-space outliers. As a result, even a single anomalous outlying observation in X-space may greatly affect the estimator. In order to resolve this issue, we propose a new RKBR method that gives reliable result even if a training data set is contaminated with both Y-space and X-space outliers. The proposed method utilizes a weighting scheme based on the hat matrix that resembles the generalized M-estimator (GM-estimator) of conventional robust linear analysis. The diagonal elements of hat matrix in kernel-induced feature space are used as leverage measures to downweight the effects of potential X-space outliers. We show that the kernelized hat diagonal elements can be obtained via eigen decomposition of the kernel matrix. The regularized version of kernelized hat diagonal elements is also proposed to deal with the case of the kernel matrix having full rank where the kernelized hat diagonal elements are not suitable for leverage. We have shown that two kernelized leverage measures, namely, the kernel hat diagonal element and the regularized one, are related to statistical distance measures in the feature space. We also develop an efficiently kernelized training algorithm for the parameter estimation based on iteratively reweighted least squares (IRLS) method. The experimental results from simulated examples and real data sets demonstrate the robustness of our proposed method compared with conventional approaches.     

On the orthogonal component of BSDEs in a Markovian setting

In this note we consider a quadratic growth backward stochastic differential equation (BSDE) driven by a continuous martingale M. We prove (in Theorem 3.2) that if M is a strong Markov process and if the BSDE has the form (2.2) with regular data then the unique solution (Y,Z,N) of the BSDE is reduced to (Y,Z), i.e. the orthogonal martingale N is equal to zero, showing that in a Markovian setting the “usual” solution (Y,Z) (of a BSDE with regular data) has not to be completed by a strongly orthogonal component even if M does not enjoy the martingale representation property.     

Laguerre deconvolution with unknown matrix operator

In this paper we consider the convolutionmodel Z = X + Y withX of unknown density f, independent of Y, when both random variables are nonnegative. Our goal is to estimate the unknown density f of X from n independent identically distributed observations of Z, when the law of the additive process Y is unknown. When the density of Y is known, a solution to the problem has been proposed in [17]. To make the problem identifiable for unknown density of Y, we assume that we have access to a preliminary sample of the nuisance process Y. The question is to propose a solution to an inverse problem with unknown operator. To that aim, we build a family of projection estimators of f on the Laguerre basis, well-suited for nonnegative random variables. The dimension of the projection space is chosen thanks to a model selection procedure by penalization. At last we prove that the final estimator satisfies an oracle inequality. It can be noted that the study of the mean integrated square risk is based on Bernstein’s type concentration inequalities developed for random matrices in [23].     

On attainable Cramér - Rao type lower bounds for weighted loss functions

Let Y = (Y₁,…,Y_n) be any random vector, with density φ γ (γ, θ) where θ ϵ Φ ⊂ R¹. Suppose that φ is regular. Let g(Y) = − ϖ²logφ/ϖθ². An attainable lower bound for Eg(Y)(θ−θ)² is developed and an application to the first order autoregressive process is cited.     

Estimation of rating classes and default probabilities in credit risk models with dependencies

Let Y = m(X) + ε be a regression model with a dichotomous output Y and a one‐step regression function m . In the literature, estimators for the three parameters of m , that is, the breakpoint θ and the levels a and b , are proposed for independent and identically distributed (i.i.d.) observations. We show that these standard estimators also work in a non‐i.i.d. framework, that is, that they are strongly consistent under mild conditions. For that purpose, we use a linear one‐factor model for the input X and a Bernoulli mixture model for the output Y . The estimators for the split point and the risk levels are applied to a problem arising in credit rating systems. In particular, we divide the range of individuals' creditworthiness into two groups. The first group has a higher probability of default and the second group has a lower one. We also stress connections between the standard estimator for the cutoff θ and concepts prevalent in credit risk modeling, for example, receiver operating characteristic. Copyright © 2014 John Wiley & Sons, Ltd.