On drift parameter estimation in models with fractional Brownian motion

We consider a stochastic differential equation involving standard and fractional Brownian motion with unknown drift parameter to be estimated. We investigate the standard maximum likelihood estimate of the drift parameter, two non-standard estimates and three estimates for the sequential estimation. Model strong consistency and some other properties are proved. The linear model and Ornstein–Uhlenbeck model are studied in detail. As an auxiliary result, an asymptotic behaviour of the fractional derivative of the fractional Brownian motion is established.     

On Estimation of Hurst Parameter Under Noisy Observations

It is widely accepted that some financial data exhibit long memory or long dependence, and that the observed data usually possess noise. In the continuous time situation, the factional Brownian motion B^H and its extension are an important class of models to characterize the long memory or short memory of data, and Hurst parameter H is an index to describe the degree of dependence. In this article, we estimate the Hurst parameter of a discretely sampled fractional integral process corrupted by noise. We use the preaverage method to diminish the impact of noise, employ the filter method to exclude the strong dependence, and obtain the smoothed data, and estimate the Hurst parameter by the smoothed data. The asymptotic properties such as consistency and asymptotic normality of the estimator are established. Simulations for evaluating the performance of the estimator are conducted. Supplementary materials for this article are available online.     

Statistical inference on the drift parameter in fractional Brownian motion with a deterministic drift

In statistical inference on the drift parameter a in the fractional Brownian motion W^H_t with the Hurst parameter H ∈ (0, 1) with a constant drift Y^H_t = at + W^H_t, there is a large number of options how to do it. We may, for example, base this inference on the properties of the standard normal distribution applied to the differences between the observed values of the process at discrete times. Although such methods are very simple, it turns out that more appropriate is to use inverse methods. Such methods can be generalized to non constant drift. For the hypotheses testing about the drift parameter a, it is more proper to standardize the observed process, and to use inverse methods based on the first exit time of the observed process of a pre-specified interval until some given time. These procedures are illustrated, and their times of decision are compared against the direct approach. Other generalizations are possible when the random part is a symmetric stochastic integral of a known, deterministic function with respect to fractional Brownian motion.     

Estimation of the Hurst parameter in some fractional processes

We propose to estimate the Hurst parameter involved in fractional processes via a method based on the Karhunen–Loève expansion of a Gaussian process. We specifically investigate the cases of the fractional Brownian motion, the fractional Ornstein–Uhlenbeck family and the fractional Brownian bridge. We numerically compare our results with the ones obtained by the maximum-likelihood method, which show the validity of our proposal.     

Fisher Information for Fractional Brownian Motion Under High-Frequency Discrete Sampling

We investigate the issue of the validation of the local asymptotic normality property of three characterizing parameters of the fractional Brownian motion under high-frequency discrete sampling. We prove that the local asymptotic normality property holds true for the likelihood only when at least one of the volatility parameter and the Hurst exponent is known. We provide optimal rates of convergence of the three parameters and Fisher information matrix in closed form.     

Minimum Contrast Estimation for Fractional Diffusions

Abstract.  When the Hurst coefficient of a fractional Brownian motion     is greater than 1/2 it is possible to define a stochastic integral with respect to     , as the pathwise limit of Riemann sums, and thus to consider pathwise solutions to fractional diffusion equations. In this paper, we consider the vanishing drift case and assume that the solution X _t is parameterized by θ in a compact parameter space Θ . Our main interest is the estimation of θ based on discrete time, but with very frequent observations. It is shown that the estimation problem in this context is locally asymptotically mixed normal. The asymptotic behaviour of a certain class of minimum contrast estimators is then studied and asymptotic efficiency is discussed.     

On approximate likelihood inference in a poisson mixed model

A two-step estimation approach is proposed for the fixed-effect parameters, random effects and their variance σ² of a Poisson mixed model. In the first step, it is proposed to construct a small σ²-based approximate likelihood function of the data and utilize this function to estimate the fixed-effect parameters and σ². In the second step, the random effects are estimated by minimizing their posterior mean squared error. Methods of Waclawiw and Liang (1993) based on so-called Stein-type estimating functions and of Breslow and Clayton (1993) based on penalized quasilikelihood are compared with the proposed likelihood method. The results of a simulation study on the performance of all three approaches are reported.     

On maximum likelihood estimation of the long-memory parameter in fractional Gaussian noise

Approximate normality and unbiasedness of the maximum likelihood estimate (MLE) of the long-memory parameter H of a fractional Brownian motion hold reasonably well for sample sizes as small as 20 if the mean and scale parameter are known. We show in a Monte Carlo study that if the latter two parameters are unknown the bias and variance of the MLE of H both increase substantially. We also show that the bias can be reduced by using a parametric bootstrap procedure. In very large samples, maximum likelihood estimation becomes problematic because of the large dimension of the covariance matrix that must be inverted. To overcome this difficulty, we propose a maximum likelihood method based upon first differences of the data. These first differences form a short-memory process. We split the data into a number of contiguous blocks consisting of a relatively small number of observations. Computation of the likelihood function in a block then presents no computational problem. We form a pseudo-likelihood function consisting of the product of the likelihood functions in each of the blocks and provide a formula for the standard error of the resulting estimator of H. This formula is shown in a Monte Carlo study to provide a good approximation to the true standard error. The computation time required to obtain the estimate and its standard error from large data sets is an order of magnitude less than that required to obtain the widely used Whittle estimator. Application of the methodology is illustrated on two data sets.     

Innovative methods for modeling of scale invariant processes

This paper presents some innovative methods for modeling discrete scale invariant (DSI) processes and evaluation of corresponding parameters. For the case where the absolute values of the increments of DSI processes are in general increasing, we consider some moving sample variance of the increments and present some heuristic algorithm to characterize successive scale intervals. This enables us to estimate scale parameter of such DSI processes. To present some superior structure for the modeling of DSI processes, we consider the possibility that the variations inside the prescribed scale intervals show some further self-similar behavior. Such consideration enables us to provide more efficient estimators for Hurst parameters. We also present two competitive estimation methods for the Hurst parameters of self-similar processes with stationary increments and prove their efficiency. Using simulated samples of some simple fractional Brownian motion, we show that our estimators of Hurst parameter are more efficient as compared with the celebrated methods of convex rearrangement and quadratic variation. Finally we apply the proposed methods to evaluate DSI behavior of the S&P500 indices in some period.     

Stochastically Curtailed Tests Under Fractional Brownian Motion

Stochastic curtailment has been considered for the interim monitoring of group sequential trials (Davis and Hardy, 1994). Statistical boundaries in Davis and Hardy (1994) were derived using theory of Brownian motion. In some clinical trials, the conditions of forming a Brownian motion may not be satisfied. In this paper, we extend the computations of Brownian motion based boundaries, expected stopping times, and type I and type II error rates to fractional Brownian motion (FBM). FBM includes Brownian motion as a special case. Designs under FBM are compared to those under Brownian motion and to those of O’Brien–Fleming type tests. One- and two-sided boundaries for efficacy and futility monitoring are also discussed. Results show that boundary values decrease and error rates deviate from design levels when the Hurst parameter increases from 0.1 to 0.9, these changes should be considered when designing a study under FBM.     

Estimators for the Drift of Subfractional Brownian Motion

In this paper, we consider, using technique based on Girsanov theorem, the problem of efficient estimation for the drift of subfractional Brownian motion S^H ? (S^H_t)_{t ∈ [0, T]}. We also construct a class of biased estimators of James-Stein type which dominate, under the usual quadratic risk, the natural maximum likelihood estimator.     

Operating Characteristics for Group Sequential Trials Monitored Under Fractional Brownian Motion

Brownian motion has been used to derive stopping boundaries for group sequential trials, however, when we observe dependent increment in the data, fractional Brownian motion is an alternative to be considered to model such data. In this article we compared expected sample sizes and stopping times for different stopping boundaries based on the power family alpha spending function under various values of Hurst coefficient. Results showed that the expected sample sizes and stopping times will decrease and power increases when the Hurst coefficient increases. With same Hurst coefficient, the closer the boundaries are to that of O'Brien-Fleming, the higher the expected sample sizes and stopping times are; however, power has a decreasing trend for values start from H = 0.6 (early analysis), 0.7 (equal space), 0.8 (late analysis). We also illustrate study design changes using results from the BHAT study.     

Temporal Aggregation of Stationary and Non-stationary Continuous-Time Processes

Abstract.  We study the autocorrelation structure of aggregates from a continuous-time process. The underlying continuous-time process or some of its higher derivative is assumed to be a stationary continuous-time auto-regressive fractionally integrated moving-average (CARFIMA) process with Hurst parameter H . We derive closed-form expressions for the limiting autocorrelation function and the normalized spectral density of the aggregates, as the extent of aggregation increases to infinity. The limiting model of the aggregates, after appropriate number of differencing, is shown to be some functional of the standard fractional Brownian motion with the same Hurst parameter of the continuous-time process from which the aggregates are measured. These results are then used to assess the loss of forecasting efficiency due to aggregation.     

Long-range dependence and approximate Bayesian computation

In this work, we propose a method for estimating the Hurst index, or memory parameter, of a stationary process with long memory in a Bayesian fashion. Such approach provides an approximation for the posterior distribution for the memory parameter and it is based on a simple application of the so-called approximate Bayesian computation (ABC), also known as likelihood-free method. Some popular existing estimators are reviewed and compared to this method for the fractional Brownian motion, for a long-range binary process and for the Rosenblatt process. The performance of our proposal is remarkably efficient.     

Is a Brownian Motion Skew?

We study the asymptotic behaviour of the maximum likelihood estimator corresponding to the observation of a trajectory of a skew Brownian motion, through a uniform time discretization. We characterize the speed of convergence and the limiting distribution when the step size goes to zero, which in this case are non‐classical, under the null hypothesis of the skew Brownian motion being an usual Brownian motion. This allows to design a test on the skewness parameter. We show that numerical simulations can be easily performed to estimate the skewness parameter and provide an application in Biology.     

Parameter identification for the discretely observed geometric fractional Brownian motion

This paper deals with the problem of estimating all the unknown parameters of geometric fractional Brownian processes from discrete observations. The estimation procedure is built upon the marriage of the quadratic variation and the maximum likelihood approach. The asymptotic properties of the estimators are provided. Moveover, we compare our derived method with the approach proposed by Misiran et al. [Fractional Black-Scholes models: complete MLE with application to fractional option pricing. In International conference on optimization and control; Guiyang, China; 2010. p. 573–586.], namely the complete maximum likelihood estimation. Simulation studies confirm theoretical findings and illustrate that our methodology is efficient and reliable. To show how to apply our approach in realistic contexts, an empirical study of Chinese financial market is also presented.     

Parametric estimation for ARFIMA models via spectral methods

Given a fractional integrated, autoregressive, moving average,ARFIMA (p, d, q) process, the simultaneous estimation of the short and long memory parameters can be achieved by maximum likelihood estimators. In this paper, following a two-step algorithm, the coefficients are estimated combining the maximum likelihood estimators with the general orthogonal decomposition of stochastic processes. In particular, the principal component analysis of stochastic processes is exploited to estimate the short memory parameters, which are plugged into the maximum likelihood function to obtain the fractional differencingd.     

Numerical maximum likelihood estimation for the g-and-k and generalized g-and-h distributions

Continuing increases in computing power and availability mean that many maximum likelihood estimation (MLE) problems previously thought intractable or too computationally difficult can now be tackled numerically. However, ML parameter estimation for distributions whose only analytical expression is as quantile functions has received little attention. Numerical MLE procedures for parameters of new families of distributions, the g-and-k and the generalized g-and-h distributions, are presented and investigated here. Simulation studies are included, and the appropriateness of using asymptotic methods examined. Because of the generality of these distributions, the investigations are not only into numerical MLE for these distributions, but are also an initial investigation into the performance and problems for numerical MLE applied to quantile-defined distributions in general. Datasets are also fitted using the procedures here. Results indicate that sample sizes significantly larger than 100 should be used to obtain reliable estimates through maximum likelihood.