Robust parameter estimation for the Ornstein–Uhlenbeck process

In this paper, we derive elementary M- and optimally robust asymptotic linear (AL)-estimates for the parameters of an Ornstein–Uhlenbeck process. Simulation and estimation of the process are already well-studied, see Iacus (Simulation and inference for stochastic differential equations. Springer, New York, 2008). However, in order to protect against outliers and deviations from the ideal law the formulation of suitable neighborhood models and a corresponding robustification of the estimators are necessary. As a measure of robustness, we consider the maximum asymptotic mean square error (maxasyMSE), which is determined by the influence curve (IC) of AL estimates. The IC represents the standardized influence of an individual observation on the estimator given the past. In a first step, we extend the method of M-estimation from Huber (Robust statistics. Wiley, New York, 1981). In a second step, we apply the general theory based on local asymptotic normality, AL estimates, and shrinking neighborhoods due to Kohl et?al. (Stat Methods Appl 19:333–354, 2010), Rieder (Robust asymptotic statistics. Springer, New York, 1994), Rieder (2003), and Staab (1984). This leads to optimally robust ICs whose graph exhibits surprising behavior. In the end, we discuss the estimator construction, i.e. the problem of constructing an estimator from the family of optimal ICs. Therefore we carry out in our context the One-Step construction dating back to LeCam (Asymptotic methods in statistical decision theory. Springer, New York, 1969) and compare it by means of simulations with MLE and M-estimator.     

Characteristic function estimation of non-Gaussian Ornstein–Uhlenbeck processes

Continuous non-Gaussian stationary processes of the OU-type are becoming increasingly popular given their flexibility in modelling stylized features of financial series such as asymmetry, heavy tails and jumps. The use of non-Gaussian marginal distributions makes likelihood analysis of these processes unfeasible for virtually all cases of interest. This paper exploits the self-decomposability of the marginal laws of OU processes to provide explicit expressions of the characteristic function which can be applied to several models as well as to develop efficient estimation techniques based on the empirical characteristic function. Extensions to OU-based stochastic volatility models are provided.     

LAN property for an ergodic Ornstein–Uhlenbeck process with Poisson jumps

In this article, we consider an ergodic Ornstein–Uhlenbeck process with jumps driven by a Brownian motion and a compensated Poisson process, whose drift and diffusion coefficients as well as its jump intensity depend on unknown parameters. Considering the process discretely observed at high frequency, we derive the local asymptotic normality property. To obtain this result, Malliavin calculus and Girsanov’s theorem are applied to write the log-likelihood ratio in terms of sums of conditional expectations, for which a central limit theorem for triangular arrays can be applied.     

Semi-parametric estimation of the autoregressive parameter in non-Gaussian Ornstein–Uhlenbeck processes

ABSTRACT

This paper considers the problem of estimating the autoregressive parameter in discretely observed Ornstein–Uhlenbeck processes. Two consistent estimators are proposed: one obtained by maximizing a kernel-based likelihood function, and another by minimizing a Kolmogorov-type distance from independence. After establishing the consistency of these estimators, their finite-sample performance and possible normality in large samples, is investigated by means of extensive simulations. An illustrative example to credit rating is discussed.     

Distribution of the mean reversion estimator in the Ornstein–Uhlenbeck process

ABSTRACT

We derive the exact distribution of the maximum likelihood estimator of the mean reversion parameter (κ) in the Ornstein–Uhlenbeck process using numerical integration through analytical evaluation of a joint characteristic function. Different scenarios are considered: known or unknown drift term, fixed or random start-up value, and zero or positive κ. Monte Carlo results demonstrate the remarkably reliable performance of our exact approach across all the scenarios. In comparison, misleading results may arise under the asymptotic distributions, including the advocated infill asymptotic distribution, which performs poorly in the tails when there is no intercept in the regression and the starting value of the process is nonzero.     

Parameter estimation for Ornstein–Uhlenbeck processes of the second kind driven by α-stable Lévy motions

In this article, we study the problem of parameter estimation for Ornstein–Uhlenbeck processes of the second kind driven by α-stable Lévy motions, based on continuous and discrete observations, respectively. Using the trajectory fitting method combined with the weighted least-squares technique, we discuss the consistency and the asymptotic distributions of the estimators for general weights in both the ergodic and the non ergodic cases.     

Exact simulation of tempered stable Ornstein–Uhlenbeck processes

Based on a representation of a stochastic integral of Ornstein–Uhlenbeck (O–U) type, the exact simulation algorithm of the tempered stable O–U process is given in this paper. The algorithm employs the double rejection method and the general acceptance–rejection technique. The time complexity of the double rejection method is uniformly bounded over all values of the parameter. And the acceptance probability of the acceptance–rejection technique can be improved to as close to 1 as possible. Thus, the implementation of the algorithm is efficient. The performance of the simulation method is evidenced by some empirical results.     

Unrestricted consumption under a deterministic wealth and an Ornstein–Uhlenbeck process as a discount rate

We consider an individual or household endowed with an initial capital and an income, modeled as a linear function of time. Assuming that the discount rate evolves as an Ornstein–Uhlenbeck process, we target to find an unrestricted consumption strategy such that the value of the expected discounted consumption is maximized. Differently than in the case with restricted consumption rates, we can determine the optimal strategy and the value function.     

Estimation of the linear mixed integrated Ornstein–Uhlenbeck model

The linear mixed model with an added integrated Ornstein–Uhlenbeck (IOU) process (linear mixed IOU model) allows for serial correlation and estimation of the degree of derivative tracking. It is rarely used, partly due to the lack of available software. We implemented the linear mixed IOU model in Stata and using simulations we assessed the feasibility of fitting the model by restricted maximum likelihood when applied to balanced and unbalanced data. We compared different (1) optimization algorithms, (2) parameterizations of the IOU process, (3) data structures and (4) random-effects structures. Fitting the model was practical and feasible when applied to large and moderately sized balanced datasets (20,000 and 500 observations), and large unbalanced datasets with (non-informative) dropout and intermittent missingness. Analysis of a real dataset showed that the linear mixed IOU model was a better fit to the data than the standard linear mixed model (i.e. independent within-subject errors with constant variance).     

Infinite Variation Tempered Stable Ornstein–Uhlenbeck Processes with Discrete Observations

We investigate transition law between consecutive observations of Ornstein–Uhlenbeck processes of infinite variation with tempered stable stationary distribution. Thanks to the Markov autoregressive structure, the transition law can be written in the exact sense as a convolution of three random components; a compound Poisson distribution and two independent tempered stable distributions, one with stability index in (0, 1) and the other with index in (1, 2). We discuss simulation techniques for those three random elements. With the exact transition law and proposed simulation techniques, sample paths simulation proves significantly more efficient, relative to the known approximative technique based on infinite shot noise series representation of tempered stable Lévy processes.     

On Transition and First Hitting Time Densities and Moments of the Ornstein–Uhlenbeck Process

This paper derives transition and first hitting time densities and moments for the Ornstein–Uhlenbeck Process (OUP) between exponential thresholds. The densities are obtained by simplifying the process via Doob’s representation into Brownian motion between affine thresholds. The densities in this paper also offer easy-to-use and fast small-time approximations for the densities of OUP between constant thresholds given that exponential thresholds are virtually constant for a small time. This is of interest for estimation with high-frequency data given that extant approaches for constant thresholds impose a large demand on computing power. The moments of the transition distribution up to order n are derived within a closed-form recursive formula that offers valuable information for management. Expressions for the moments of the first hitting time distribution are also obtained in closed form by simplifying integrals via series expansions.     

Tempered stable Ornstein– Uhlenbeck processes: A practical view

We study the one-dimensional Ornstein–Uhlenbeck (OU) processes with marginal law given by tempered stable and tempered infinitely divisible distributions. We investigate the transition law between consecutive observations of these processes and evaluate the characteristic function of integrated tempered OU processes with a view toward practical applications. We then analyze how to draw a random sample from this class of processes by considering both the classical inverse transform algorithm and an acceptance–rejection method based on simulating a stable random sample. Using a maximum likelihood estimation method based on the fast Fourier transform, we empirically assess the simulation algorithm performance.     

Parametric bootstrap goodness-of-fit testing for Wehrly–Johnson bivariate circular distributions

The Wehrly–Johnson family of bivariate circular distributions is by far the most general one currently available for modelling data on the torus. It allows complete freedom in the specification of the marginal circular densities as well as the binding circular density which regulates any dependence that might exist between them. We propose a parametric bootstrap approach for testing the goodness-of-fit of Wehrly–Johnson distributions when the forms of their marginal and binding densities are assumed known. The approach admits the use of any test for toroidal uniformity, and we consider versions of it incorporating three such tests. Simulation is used to illustrate the operating characteristics of the approach when the underlying distribution is assumed to be bivariate wrapped Cauchy. An analysis of wind direction data recorded at a Texan weather station illustrates the use of the proposed goodness-of-fit testing procedure.     

Location-shifts for improved estimation under Midzuno–Sen sampling scheme

Simple location-shifts for the study or auxiliary character are proposed under Midzuno–Sen sampling from a finite population. These aim at improving the efficiency of the classical Horvitz–Thompson estimator or the unbiased ratio estimator of a population total. It is demonstrated that the choice of the translation parameters is flexible. A few methods for assessing these parameters are outlined. The gain in efficiency of estimation is illustrated.     

Reliability estimation in multicomponent stress–strength model for Topp-Leone distribution

In this paper, we consider the estimation reliability in multicomponent stress-strength (MSS) model when both the stress and strengths are drawn from Topp-Leone (TL) distribution. The maximum likelihood (ML) and Bayesian methods are used in the estimation procedure. Bayesian estimates are obtained by using Lindley’s approximation and Gibbs sampling methods, since they cannot be obtained in explicit form in the context of TL. The asymptotic confidence intervals are constructed based on the ML estimators. The Bayesian credible intervals are also constructed using Gibbs sampling. The reliability estimates are compared via an extensive Monte-Carlo simulation study. Finally, a real data set is analysed for illustrative purposes.