Let X be a stochastic process obeying a stochastic differential equation of the form dXt = b(Xt, θ)dt + dYt, where Y is an adapted driving process possibly depending on X’s past history, and θ ∈ Θ ⊂ Rp is an unknown parameter. We consider estimation of θ when X is discretely observed at possibly non-equidistant time-points (tni)ni=0. We suppose hn := max1 ≤ i ≤ n(tni − tni − 1) → 0 and tnn → ∞ as n → ∞: the data becomes more high-frequency as its size increases. Under some regularity conditions including the ergodicity of X, we obtain √nhn-consistency of trajectory-fitting estimate as well as least-squares estimate, without identifying Y. Also shown is that some additional conditions, which requires Y's structure to some extent, lead to asymptotic normality. In particular, a Wiener-Poisson-driven setup is discussed as an important special case.
We shall propose a new computational scheme with the asymptotic method to achieve variance reduction of Monte Carlo simulation for numerical analysis particularly for finance. We not only provide general scheme of our method, but also show its effectiveness through numerical examples such as computing optimal portfolio and pricing an average option. Finally, we show mathematical validity of our method.
This article analyzes a Markov switching stochastic volatility (MSSV) model to accommodate the shift in the mean of log-volatility. Since it is difficult to estimate the parameters in this model based on the maximum likelihood method, a Bayesian Markov-chain Monte Carlo (MCMC) approach is adopted. A particle filter for the MSSV model, which is used for model comparison and diagnostics, is constructed. The estimation result, based on weekly returns of the TOPIX, confirms the finding by previous researchers that the estimate of the persistence parameter drops and the estimate of the error variance rises in the volatility equation of the MSSV model compared to those of the standard SV model. The model comparison provides evidence that the MSSV model is favored over the standard SV model. It is also found that the MSSV model passes the diagnostic tests based on the statistics obtained from the particle filter while the SV model does not.
This paper addresses the issue of deriving estimators improving on the best location equivariant (or Pitman) estimator under the squared error loss when a location parameter is restricted to a bounded interval. A class of improved estimators is constructed, and it is verified that the Bayes estimator for the uniform prior over the bounded interval and the truncated estimator belong to the class. This paper also obtains the sufficient conditions for the density under which the class includes the Bayes estimators with respect to the two-point boundary symmetric prior and general continuous prior distributions when a symmetric density is considered for the location family. It is demonstrated that the conditions on the symmetric density can be applied to logistic, double exponential and t-distributions as well as to a normal distribution. These conditions can be also applied to scale mixtures of normal distributions. Finally, some similar results are developed in the scale family.
In this paper, tests are developed for testing certain hypotheses on the covariance matrix Σ, when the sample size N = n + 1 is smaller than the dimension pof the data. Under the condition that (tr Σi⁄p) exists and > 0, as p → ∞, i =1,…,8, tests are developed for testing the hypotheses that the covariance matrix in a normally distributed data is an identity matrix, a constant time the identity matrix (spherecity), and is a diagonal matrix. The asymptotic null and non-null distributions of these test statistics are given.
For estimating the population variance S2y of study variable y, a class of chain estimators of S2y has been proposed in the presence of two auxiliary variables x and z by using known information on population mean and variance of the second auxiliary variable z. In this proposed class, the second auxiliary variable z is directly highly correlated with the first auxiliary variable x, whereas the variable z is correlated with the variable y due to only the high correlation between the variables y and x. Another generalized class of estimators of S2y has also been considered by using the same available information of auxiliary variable z when both the auxiliary variables x and z are directly highly correlated with the study variable y. The asymptotic expressions for the mean square errors and their optimum values have been obtained. A comparison between the two proposed classes of estimators of S2y has been made empirically.
Conventional spline procedures have proven to be effective and useful for estimating smooth functions. However, these procedures find piecewise and inhomogeneous smooth functions difficult to handle. Conventional spline procedures often yield an overly-smooth curve in regions where the true function or its derivative are discontinuous. To fully realize the potential of the spline methodology, this article proposes a function estimation procedure that uses adaptive free-knot splines and allows for multiple knots to be replaced at the same location and between design points. The article also proposes an associated algorithm for implementation in non-parametric regression. The proposed knot selection scheme uses a data-adaptive model selection criterion and an evolutionary algorithm that incorporates certain key features of simulated annealing. The evolutionary algorithm accurately locates the optimal knots, while the data-driven penalty guards against selection errors when searching through a large candidate knot space. The algorithm stochastically yields the globally optimal knots. The simulations suggest that the procedure performs competitively well against alternative methods and has a substantial advantage in relation to non-smooth and piecewise smooth functions.