Multilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation

We address the approximation of functionals depending on a system of particles, described by stochastic differential equations (SDEs), in the mean-field limit when the number of particles approaches infinity. This problem is equivalent to estimating the weak solution of the limiting McKean–Vlasov SDE. To that end, our approach uses systems with finite numbers of particles and a time-stepping scheme. In this case, there are two discretization parameters: the number of time steps and the number of particles. Based on these two parameters, we consider different variants of the Monte Carlo and Multilevel Monte Carlo (MLMC) methods and show that, in the best case, the optimal work complexity of MLMC, to estimate the functional in one typical setting with an error tolerance of \(\mathrm {TOL}\), is Open image in new window when using the partitioning estimator and the Milstein time-stepping scheme. We also consider a method that uses the recent Multi-index Monte Carlo method and show an improved work complexity in the same typical setting of Open image in new window . Our numerical experiments are carried out on the so-called Kuramoto model, a system of coupled oscillators.     

Complete integral convergence for arrays of row-wise extended independent random variables under Sub-linear expectations

Abstract

In this paper, we establish the complete convergence and complete integral convergence for arrays of row-wise extended independent random variables under sub-linear expectation space with some conditions. At the same time we extend some complete convergence and complete integral convergence theorems from the classical probability space to the sub-linear expectation space. The results generalize corresponding results obtained by Wu et al. (2017 Zhang, L. X. 2016b. Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm. Science China Mathematics 59 (12):2503–26. doi: 10.1007/s11425-016-0079-1.[Crossref], [Web of Science ®] , [Google Scholar]).     

Large and moderate deviations for the total population in the nearly unstable INAR(1) model

This paper considers the non negative integer-valued autoregressive process with order one (INAR(1)), where the autoregression parameter is close to unity. Using the methods introduced by Yu, Wang, and Chen (2016 Yu, S. H., D. H. Wang, and X. Chen. 2016. Large and moderate deviations for the total population arising from a sub-critical Galton-Watson process with immigration. Journal of Theoretical Probabiltiy, doi:10.1007/s10959-016-0706-4.[Crossref] , [Google Scholar]), the large and moderate deviations with explicit rate functions for the total population of this process can be obtained.     

Multi-level Monte Carlo methods for the approximation of invariant measures of stochastic differential equations

We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology (Giles in Acta Numer. 24:259–328, 2015. https://doi.org/10.1017/S096249291500001X) to calculate expectations with respect to the invariant measure of an ergodic SDE. In that context, we study the (over-damped) Langevin equations with a strongly concave potential. We show that when appropriate contracting couplings for the numerical integrators are available, one can obtain a uniform-in-time estimate of the MLMC variance in contrast to the majority of the results in the MLMC literature. As a consequence, a root mean square error of $$\mathcal {O}(\varepsilon )$$ is achieved with $$\mathcal {O}(\varepsilon ^{-2})$$ complexity on par with Markov Chain Monte Carlo (MCMC) methods, which, however, can be computationally intensive when applied to large datasets. Finally, we present a multi-level version of the recently introduced stochastic gradient Langevin dynamics method (Welling and Teh, in: Proceedings of the 28th ICML, 2011) built for large datasets applications. We show that this is the first stochastic gradient MCMC method with complexity $$\mathcal {O}(\varepsilon ^{-2}|\log {\varepsilon }|^{3})$$, in contrast to the complexity $$\mathcal {O}(\varepsilon ^{-3})$$ of currently available methods. Numerical experiments confirm our theoretical findings.     

Learning-based EM algorithm for normal-inverse Gaussian mixture model with application to extrasolar planets

Karlis and Santourian [14 D. Karlis and A. Santourian, Model-based clustering with non-elliptically contoured distribution, Stat. Comput. 19 (2009), pp. 73–83. doi: 10.1007/s11222-008-9072-0[Crossref], [Web of Science ®] , [Google Scholar]] proposed a model-based clustering algorithm, the expectation–maximization (EM) algorithm, to fit the mixture of multivariate normal-inverse Gaussian (NIG) distribution. However, the EM algorithm for the mixture of multivariate NIG requires a set of initial values to begin the iterative process, and the number of components has to be given a priori. In this paper, we present a learning-based EM algorithm: its aim is to overcome the aforementioned weaknesses of Karlis and Santourian's EM algorithm [14 D. Karlis and A. Santourian, Model-based clustering with non-elliptically contoured distribution, Stat. Comput. 19 (2009), pp. 73–83. doi: 10.1007/s11222-008-9072-0[Crossref], [Web of Science ®] , [Google Scholar]]. The proposed learning-based EM algorithm was first inspired by Yang et al. [24 M.-S. Yang, C.-Y. Lai, and C.-Y. Lin, A robust EM clustering algorithm for Gaussian mixture models, Pattern Recognit. 45 (2012), pp. 3950–3961. doi: 10.1016/j.patcog.2012.04.031[Crossref], [Web of Science ®] , [Google Scholar]]: the process of how they perform self-clustering was then simulated. Numerical experiments showed promising results compared to Karlis and Santourian's EM algorithm. Moreover, the methodology is applicable to the analysis of extrasolar planets. Our analysis provides an understanding of the clustering results in the ln?P?ln?M and ln?P?e spaces, where M is the planetary mass, P is the orbital period and e is orbital eccentricity. Our identified groups interpret two phenomena: (1) the characteristics of two clusters in ln?P?ln?M space might be related to the tidal and disc interactions (see [9 I.G. Jiang, W.H. Ip, and L.C. Yeh, On the fate of close-in extrasolar planets, Astrophys. J. 582 (2003), pp. 449–454. doi: 10.1086/344590[Crossref], [Web of Science ®] , [Google Scholar]]); and (2) there are two clusters in ln?P?e space.     

Expert Knowledge Elicitation: Subjective but Scientific

ABSTRACT

Expert opinion and judgment enter into the practice of statistical inference and decision-making in numerous ways. Indeed, there is essentially no aspect of scientific investigation in which judgment is not required. Judgment is necessarily subjective, but should be made as carefully, as objectively, and as scientifically as possible.

Elicitation of expert knowledge concerning an uncertain quantity expresses that knowledge in the form of a (subjective) probability distribution for the quantity. Such distributions play an important role in statistical inference (for example as prior distributions in a Bayesian analysis) and in evidence-based decision-making (for example as expressions of uncertainty regarding inputs to a decision model). This article sets out a number of practices through which elicitation can be made as rigorous and scientific as possible.

One such practice is to follow a recognized protocol that is designed to address and minimize the cognitive biases that experts are prone to when making probabilistic judgments. We review the leading protocols in the field, and contrast their different approaches to dealing with these biases through the medium of a detailed case study employing the SHELF protocol.

The article ends with discussion of how to elicit a joint probability distribution for multiple uncertain quantities, which is a challenge for all the leading protocols. Supplementary materials for this article are available online.     

A Strong Limit Theorem for Weighted Sums of Negatively Dependent Random Variables

In this article, we establish a new complete convergence theorem for weighted sums of negatively dependent random variables. As corollaries, many results on the almost sure convergence and complete convergence for weighted sums of negatively dependent random variables are obtained. In particular, the results of Jing and Liang (2008 Jing, B.Y., Liang, H.Y. (2008). Strong limit theorems for weighted sums of negatively associated random variables. J. Theor. Probab. 21:890–909.[Crossref], [Web of Science ®] , [Google Scholar]), Sung (2012 Sung, S.H. (2012). Complete convergence for weighted sums of negatively dependent random variables. Stat. Pap. 53:73–82.[Crossref], [Web of Science ®] , [Google Scholar]), and Wu (2010) can be obtained.     

Stochastic proximal-gradient algorithms for penalized mixed models

Motivated by penalized likelihood maximization in complex models, we study optimization problems where neither the function to optimize nor its gradient has an explicit expression, but its gradient can be approximated by a Monte Carlo technique. We propose a new algorithm based on a stochastic approximation of the proximal-gradient (PG) algorithm. This new algorithm, named stochastic approximation PG (SAPG) is the combination of a stochastic gradient descent step which—roughly speaking—computes a smoothed approximation of the gradient along the iterations, and a proximal step. The choice of the step size and of the Monte Carlo batch size for the stochastic gradient descent step in SAPG is discussed. Our convergence results cover the cases of biased and unbiased Monte Carlo approximations. While the convergence analysis of some classical Monte Carlo approximation of the gradient is already addressed in the literature (see Atchadé et al. in J Mach Learn Res 18(10):1–33, 2017), the convergence analysis of SAPG is new. Practical implementation is discussed, and guidelines to tune the algorithm are given. The two algorithms are compared on a linear mixed effect model as a toy example. A more challenging application is proposed on nonlinear mixed effect models in high dimension with a pharmacokinetic data set including genomic covariates. To our best knowledge, our work provides the first convergence result of a numerical method designed to solve penalized maximum likelihood in a nonlinear mixed effect model.

     

A Bottom-Up Dynamic Model of Portfolio Credit Risk with Stochastic Intensities and Random Recoveries

In Bielecki et al. (2014a Bielecki , T. R. , Cousin , A. , Crépey , S. , Herbertsson , A. ( 2014a ). Dynamic hedging of portfolio credit risk in a markov copula model . J. Optimiz. Theor. Applic . doi: DOI 10.1007/s10957-013-0318-4 (forthcoming) .[Crossref] , [Google Scholar]), the authors introduced a Markov copula model of portfolio credit risk where pricing and hedging can be done in a sound theoretical and practical way. Further theoretical backgrounds and practical details are developed in Bielecki et al. (2014b Bielecki , T. R. , Cousin , A. , Crépey , S. , Herbertsson , A. ( 2014b ). A bottom-up dynamic model of portfolio credit risk - Part I: Markov copula perspective . In: Recent Adv. Fin. Eng. 2012 , World Scientific (preprint version available at http://dx.doi.org/10.2139/ssrn.1844574) . [Google Scholar],c) where numerical illustrations assumed deterministic intensities and constant recoveries. In the present paper, we show how to incorporate stochastic default intensities and random recoveries in the bottom-up modeling framework of Bielecki et al. (2014a Bielecki , T. R. , Cousin , A. , Crépey , S. , Herbertsson , A. ( 2014a ). Dynamic hedging of portfolio credit risk in a markov copula model . J. Optimiz. Theor. Applic . doi: DOI 10.1007/s10957-013-0318-4 (forthcoming) .[Crossref] , [Google Scholar]) while preserving numerical tractability. These two features are of primary importance for applications like CVA computations on credit derivatives (Assefa et al., 2011 Assefa , S. , Bielecki , T. R. , Crépey , S. , Jeanblanc , M. ( 2011 ). CVA computation for counterparty risk assessment in credit portfolios . In: Bielecki , T.R. , Brigo , D. , Patras , F. , Eds., Credit Risk Frontiers . Hoboken : Wiley/Bloomberg-Press . [Google Scholar]; Bielecki et al., 2012 Bielecki , T. R. , Crépey , S. , Jeanblanc , M. , Zargari , B. ( 2012 ). Valuation and Hedging of CDS counterparty exposure in a markov copula model . Int. J. Theoret. Appl. Fin. 15 ( 1 ): 1250004 .[Crossref] , [Google Scholar]), as CVA is sensitive to the stochastic nature of credit spreads and random recoveries allow to achieve satisfactory calibration even for “badly behaved” data sets. This article is thus a complement to Bielecki et al. (2014a Bielecki , T. R. , Cousin , A. , Crépey , S. , Herbertsson , A. ( 2014a ). Dynamic hedging of portfolio credit risk in a markov copula model . J. Optimiz. Theor. Applic . doi: DOI 10.1007/s10957-013-0318-4 (forthcoming) .[Crossref] , [Google Scholar]), Bielecki et al. (2014b Bielecki , T. R. , Cousin , A. , Crépey , S. , Herbertsson , A. ( 2014b ). A bottom-up dynamic model of portfolio credit risk - Part I: Markov copula perspective . In: Recent Adv. Fin. Eng. 2012 , World Scientific (preprint version available at http://dx.doi.org/10.2139/ssrn.1844574) . [Google Scholar]) and Bielecki et al. (2014c Bielecki , T. R. , Cousin , A. , Crépey , S. , Herbertsson , A. ( 2014c ). A bottom-up dynamic model of portfolio credit risk - Part II: Common-shock interpretation, calibration and hedging issues . Recent Adv. Fin. Eng. 2012 , World Scientific (preprint version available at http://dx.doi.org/10.2139/ssrn.2245130) . [Google Scholar]).     

ß-expectation tolerance region for the heteroscedastic multiple regression model with multivariate Student-t error

A ß-expectation tolerance region has been constructed for the multivariate regression model with heteroscedastic errors which follow a multivariate Student-t distribution with an unknown number of degrees of freedom. The ß-expectaion tolerance region obtained in this paper is optimal in the sense of having minimum enclosure among all such tolerance regions that guarantees that it would cover any preassigned proportions, namely, ß×100 percent of the future responses from the model.     

Wavelet-based estimation of regression function with strong mixing errors under fixed design

We consider wavelet-based non linear estimators, which are constructed by using the thresholding of the empirical wavelet coefficients, for the mean regression functions with strong mixing errors and investigate their asymptotic rates of convergence. We show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes B^s_{p, q}. The theory is illustrated with some numerical examples.

A new ingredient in our development is a Bernstein-type exponential inequality, for a sequence of random variables with certain mixing structure and are not necessarily bounded or sub-Gaussian. This moderate deviation inequality may be of independent interest.     

Nonparametric Estimation of the Conditional Distribution at Regression Boundary Points

Abstract

Nonparametric regression is a standard statistical tool with increased importance in the Big Data era. Boundary points pose additional difficulties but local polynomial regression can be used to alleviate them. Local linear regression, for example, is easy to implement and performs quite well both at interior and boundary points. Estimating the conditional distribution function and/or the quantile function at a given regressor point is immediate via standard kernel methods but problems ensue if local linear methods are to be used. In particular, the distribution function estimator is not guaranteed to be monotone increasing, and the quantile curves can “cross.” In the article at hand, a simple method of correcting the local linear distribution estimator for monotonicity is proposed, and its good performance is demonstrated via simulations and real data examples. Supplementary materials for this article are available online.     

On characterizations and variance bounds of discrete α-unimodality

Characterizations of α-unimodality for integer-valued random variables about a specific mode are established in terms of their probability mass functions, distribution functions and characteristic functions. Using these characterizations variance lower bounds in terms of α and the mode are derived. For α=1 all these results are reduced to ordinary unimodality. The new variance lower bounds for discrete unimodality is sharper than its continuous counterpart. An upper bound for the variance of discrete unimodal distribution defined on a finite support is discussed.     

On the Evolution and Asymptotic Analysis of Open Markov Populations: Application to Consumption Credit

In this paper, we study, by means of randomized sampling, the long-run stability of some open Markov population fed with time-dependent Poisson inputs. We show that state probabilities within transient states converge—even when the overall expected population dimension increases without bound—under general conditions on the transition matrix and input intensities.

Following the convergence results, we obtain ML estimators for a particular sequence of input intensities, where the sequence of new arrivals is modeled by a sigmoidal function. These estimators allow for the forecast, by confidence intervals, of the evolution of the relative population structure in the transient states.

Applying these results to the study of a consumption credit portfolio, we estimate the implicit default rate.     

Correcting Corrupt Research: Recommendations for the Profession to Stop Misuse of p-Values

ABSTRACT

p-Values and Null Hypothesis Significance Testing (NHST), combined with a large number of institutional factors, jointly define the Generally Accepted Soft Social Science Publishing Process (GASSSPP) that is now dominant in the social sciences and is increasingly used elsewhere. The case against NHST and the GASSSPP has been abundantly articulated over past decades, and yet it continues to spread, supported by a large number of self-reinforcing institutional processes. In this article, the author presents a number of steps that may be taken to counter the spread of this corruption that directly address the institutional forces, both as individuals and through collaborative efforts. While individual efforts are indispensable to this undertaking, the author argues that these alone cannot succeed unless the institutional forces are also addressed. Supplementary materials for this article are available online.     

Large sample properties of nonparametric copula estimators under bivariate censoring

A new nonparametric estimator is proposed for the copula function of a bivariate survival function for data subject to random right-censoring. We consider two censoring models: univariate and copula censoring. We show strong consistency and we obtain an i.i.d. representation for the copula estimator. In a simulation study we compare the new estimator to the one of Gribkova and Lopez [Nonparametric copula estimation under bivariate censoring; doi:10.1111/sjos.12144].     

Bayesian inference and data cloning in the calibration of population projection matrices

Discrete time models are used in Ecology for describing the dynamics of an age-structured population. They can be introduced from a deterministic or from a stochastic viewpoint. We analyze a stochastic model for the case in which the dynamics of the population is described by means of a projection matrix. In this statistical model, fertility rates and survival rates are unknown parameters which are estimated by using a Bayesian approach and also data cloning, which is a simulation-based method especially useful with complex hierarchical models.

Both methodologies are applied to real data from the population of Steller sea lions located in the Alaska coast since 1978–2004. The estimates obtained from these methods show a good behavior when they are compared to the nonmissing actual values.