An Approximation of Subfractional Brownian Motion

In this article, we obtain an approximation theorem for subfractional Brownian motion with H > 1/2, using martingale differences. The proof involves the tightness and identification of finite dimensional distributions.     

The Asymptotic Behavior of a Brownian Motion with a Drift from a Random Domain

Consider a Brownian motion with drift starting at an interior point of a random domain D in R ^d+1, d ≥ 1, let τ _D denote the first time the Brownian motion exits from D. Estimates with exact constants for the asymptotics of log P(τ _D  > T) are given for T → ∞, depending on the shape of the domain D and the order of the drift. The problem is motivated by the model in insurance and early works of Lifshits and Shi. The methods of proof are based on the calculus of variations and early works of Li, Lifshits and Shi in the drift free case.     

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.     

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.     

Variations and estimators for self-similarity parameter of sub-fractional Brownian motion via Malliavin calculus

Using multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of the adjusted quadratic variation for a sub-fractional Brownian motion. We apply our results to construct strongly consistent statistical estimators for the self-similarity of sub-fractional Brownian motion.     

A Generalization of Geometric Brownian Motion with Applications

Although geometric Brownian motion has a great variety of applications, it can not cover all the random phenomena. The purpose of this article is to propose a model that generalizes geometric Brownian motion. We present some interesting applications of this model in financial engineering and statistical inferences for the unknown parameters.     

Repeated Confidence Intervals Under Fractional Brownian Motion in Long-Term Clinical Trials

Repeated confidence interval (RCI) is an important tool for design and monitoring of group sequential trials according to which we do not need to stop the trial with planned statistical stopping rules. In this article, we derive RCIs when data from each stage of the trial are not independent thus it is no longer a Brownian motion (BM) process. Under this assumption, a larger class of stochastic processes fractional Brownian motion (FBM) is considered. Comparisons of RCI width and sample size requirement are made to those under Brownian motion for different analysis times, Type I error rates and number of interim analysis. Power family spending functions including Pocock, O'Brien-Fleming design types are considered for these simulations. Interim data from BHAT and oncology trials is used to illustrate how to derive RCIs under FBM for efficacy and futility monitoring.     

Some Asymptotic Formulas for a Brownian Motion With a Regular Variation From a Parabolic Domain

Consider a Brownian motion with a regular variation starting at an interior point of a domain D in R^{d + 1}, d ? 1 and let τ_D denote the first time the Brownian motion exits from D. Estimates with exact constants for the asymptotics of log?P(τ_D > T) are given for T → ∞, depending on the shape of the domain D and the order of the regular variation. Furthermore, the asymptotically equivalence are obtained. The problem is motivated by the early results of Lifshits and Shi, Li in the first exit time, and Karamata in the regular variation. The methods of proof are based on their results and the calculus of variations.     

Maximum Likelihood Estimators in a Two Step Model for PLS

Univariate partial least squares regression (PLS1) is a method of modeling relationships between a response variable and explanatory variables, especially when the explanatory variables are almost collinear. The purpose is to predict a future response observation, although in many applications there is an interest to understand the contributions of each explanatory variable. It is an algorithmic approach. In this article, we are going to use the algorithm presented by Helland (1988 Helland , I. S. ( 1988 ). On the structure of partial least squares regression . Commun. Statist. Simul. Computat. 17 : 581 – 607 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The population PLS predictor is linked to a linear model including a Krylov design matrix and a two-step estimation procedure. For the first step, the maximum likelihood approach is applied to a specific multivariate linear model, generating tools for evaluating the information in the explanatory variables. It is shown that explicit maximum likelihood estimators of the dispersion matrix can be obtained where the dispersion matrix, besides representing the variation in the error, also includes the Krylov structured design matrix describing the mean.     

Estimation and testing stationarity for double-autoregressive models

Summary.  The paper considers the double-autoregressive model y _t  =  φ y _{t −1}+ ɛ _t with ɛ _t  =     . Consistency and asymptotic normality of the estimated parameters are proved under the condition E  ln | φ  +√ α η _t |<0, which includes the cases with | φ |=1 or | φ |>1 as well as     . It is well known that all kinds of estimators of φ in these cases are not normal when ɛ _t are independent and identically distributed. Our result is novel and surprising. Two tests are proposed for testing stationarity of the model and their asymptotic distributions are shown to be a function of bivariate Brownian motions. Critical values of the tests are tabulated and some simulation results are reported. An application to the US 90-day treasury bill rate series is given.     

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.     

A Note on the Comparison of the Stein Estimator and the James-Stein Estimator

The seminal work of Stein (1956 Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc. Third Berkeley Symp. Mathemat. Statist. Probab., University of California Press, 1:197–206. [Google Scholar]) showed that the maximum likelihood estimator (MLE) of the mean vector of a p-dimensional multivariate normal distribution is inadmissible under the squared error loss function when p ? 3 and proposed the Stein estimator that dominates the MLE. Later, James and Stein (1961 James, W., Stein, C. (1961). Estimation with quadratic loss. Proc. Fourth Berkeley Symp. Mathemat. Statist. Probab., University of California Press, 1:361–379. [Google Scholar]) proposed the James-Stein estimator for the same problem and received much more attention than the original Stein estimator. We re-examined the Stein estimator and conducted an analytic comparison with the James-Stein estimator. We found that the Stein estimator outperforms the James-Stein estimator under certain scenarios and derived the sufficient conditions.     

Model selection for stock prices data

The geometric Brownian motion (GBM) is very popular in modeling the dynamics of stock prices. However, the constant volatility assumption is questionable and many models with nonconstant volatility have been developed. In the papers [7 M.L. Esquível and P.P. Mota, On some auto-induced regime switching double-threshold glued diffusions, J. Stat. Theory Pract. 8 (2014), pp. 760–771. doi: 10.1080/15598608.2013.854184.[Taylor &; Francis Online] , [Google Scholar],12 P. P. Mota and M.L. Esquível, On a continuous time stock price model with regime switching, delay, and threshold, Quant. Financ. 14 (2014), pp. 1479–1488. doi: 10.1080/14697688.2013.879990.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] the authors introduce a regime switching process where in each regime the process is driven by GBM and the change in regime is defined by the crossing of a threshold. In this paper we used Akaike's and Bayesian information criteria to show that the GBM with regimes provides a better fit than the GBM. We also perform a forecasting comparison of the models for two selected companies.     

A strong convergence to the tempered fractional Brownian motion

In this paper, we give a strong approximation of the tempered fractional Brownian motion via transport processes and derive the rate of convergence.     

Comparing Nonparametric Estimators for Length-Biased Data

Length-biased data appear when sampling lifetimes by cross-section. Right-censoring may affect the sampled information due to time limitation in following-up, lost to follow-up cases, etc. In this article, we compare by simulations two alternative nonparametric estimators of the lifetime distribution function when the data are length-biased and right-censored. These estimates, recently introduced in the literature, are based on nonparametric maximum-likelihood and moment-based principles. It is shown that the relative benefits associated to each estimator depend on several factors, such as the shape of the underlying distribution, sample size, or censoring level.     

On sharper bounds for the risk of james-stein estimators

Two sampling strategies for estimation of population mean in overlapping clusters have been proposed, In the first strategy clusters are selected with egual probabilities, whereas in the second case selection probabilities are taken proportional to cluster size. The sampling efficiency of the latter is expected to be more in comparison to the former.     

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.