A modified sequential Monte Carlo procedure for the efficient recursive estimation of extreme quantiles

• Published date: 01 August 2019
• DOI: http://doi.org/10.1002/for.2568
• Date: 01 August 2019

Received: 6 June 2018 Revised: 3 September 2018 Accepted: 3 January 2019

DOI: 10.1002/for.2568

RESEARCH ARTICLE

A modified sequential Monte Carlo procedure for the

efficient recursive estimation of extreme quantiles

Serdar Neslihanoglu1Paresh Date2

1Department of Statistics, Faculty of

Science and Letters, Eskisehir Osmangazi

University, Eskisehir,Turkey

2Department of Mathematics, College of

Engineering, Design and Physical

Sciences, Brunel University, London, UK

Correspondence

Serdar Neslihanoglu, Department of

Statistics, Faculty of Science and Letters,

Eskisehir Osmangazi University,Meselik

Yerleskesi,26480 Eskisehir, Turkey.

Email: sneslihanoglu@ogu.edu.tr

Funding information

Eskisehir Osmangazi University Scientific

Research Fund (ESOGU BAP),

Grant/AwardNumber: 201519046

Abstract

Many applications in science involve finding estimates of unobserved variables

from observed data, by combining model predictions with observations. The

sequential Monte Carlo (SMC) is a well-established technique for estimating

the distribution of unobserved variables that are conditional on current observa-

tions. While the SMC is very successful at estimating the first central moments,

estimating the extreme quantiles of a distribution via the current SMC methods

is computationally very expensive. The purpose of this paper is to develop a new

framework using probability distortion. We use an SMC with distorted weights

in order to make computationally efficient inferences about tail probabilities of

future interest rates using the Cox–Ingersoll–Ross (CIR) model, as well as with

an observed yield curve. We show that the proposed method yields acceptable

estimates about tail quantiles at a fraction of the computational cost of the full

Monte Carlo.

KEYWORDS

extreme event simulation, risk analysis, sequential Monte Carlo, simulation

1INTRODUCTION

This study concerns the problem of estimating the latent

states of a dynamic system from observed time series data.

Such problems arise in many branches of the physical and

social sciences, including the geosciences, navigation, and

econometrics. Filtering is an indefinitely repetitive proce-

dure of merging model predictions and noisy observations

for the purpose of making estimates. The estimate of a state

at a particular point in time is a probability distribution

(conditional upon the observations that have been made

up to that time) which is, in some cases, characterized by

its first few central moments. The Kalman filter (Kalman,

1960) is a commonly used estimator for linear systems.

Provided that the conditional densities are Gaussian and

that the Kalman filter is used as a closed-form solution, the

Kalman filter is most satisfactory.

When a system is nonlinear,a stochastic partial differen-

tial equation solution must typically be solved in order to

generate a conditional distribution. A numerical solution

for such an equation is often intractable. This is espe-

cially the case when a real-time solution is being sought

(e.g., in navigation and tracking) or when the state or the

observation dimension is high (e.g., in the geosciences

or econometrics). Nonlinear filtering in different disci-

plines requires various Bayesian approximation methods,

each of which require concessions to the accurate estima-

tion and computational loads that are particular to those

implementations. To illustrate, the extended Kalman fil-

ter (EKF) (Anderson & Moore, 1979) and the unscented

Kalman filter (UKF) (Julier & Uhlmann, 2004), as well as

their variants as cited in Date, Jalen, and Mamon (2008)

and Date, Mamon, and Jalen (2010), can all be listed

as different approximation methods. A sequential Monte

Carlo (SMC), also called a particle filter, extends a dis-

crete approximation of conditional density. The SMC can

also be used as a recursive procedure for approximation

of the conditional density at random together with novel

390 © 2019 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/for Journal of Forecasting. 2019;38:390–399.