Simplified Estimating Functions for Diffusion Models with a High-dimensional Parameter

We consider estimating functions for discretely observed diffusion processes of the following type: for one part of the parameter of interest we propose to use a simple and explicit estimating function of the type studied by Kessler (2000); for the remaining part of the parameter we use a martingale estimating function. Such an approach is particularly useful in practical applications when the parameter is high-dimensional. It is also often necessary to supplement a simple estimating function by another type of estimating function because only the part of the parameter on which the invariant measure depends can be estimated by a simple estimating function. Under regularity conditions the resulting estimators are consistent and asymptotically normal. Several examples are considered in order to demonstrate the idea of the estimating procedure. The method is applied to two data sets comprising wind velocities and stock prices. In one example we also propose a general method for constructing diffusion models with a prescribed marginal distribution which have a flexible dependence structure.     

对计量经济模型中随机误差项的认识

计量经济模型与数理经济模型的主要区别之一就是计量经济模型是随机模型。文中对计量经济模型中的随机误差项u谈了三点认识:如何理解随机误差项u的不可观测性;为了澄清概念上的模糊性,定义了序列残差和随机残差,并在此基础上讨论如何理解序列残差的方差和随机残差的方差;如何从数学和经济学两个方面去理解用序列残差的方差去估计随机误差项的方差。     

A Note on Ergodic Processes Prediction via Estimation of the Conditional Mode Function

We establish the uniform almost-sure convergence of a kernel estimate of the conditional density for an ergodic process. A useful application to the prediction of the ergodic process via the conditional mode function is also given.     

Takacs–Fiksel Method for Stationary Marked Gibbs Point Processes

Abstract. This article studies a method to estimate the parameters governing the distribution of a stationary marked Gibbs point process. This procedure, known as the Takacs–Fiksel method, is based on the estimation of the left and right hand sides of the Georgii–Nguyen–Zessin formula and leads to a family of estimators due to the possible choices of test functions. We propose several examples illustrating the interest and flexibility of this procedure. We also provide sufficient conditions based on the model and the test functions to derive asymptotic properties (consistency and asymptotic normality) of the resulting estimator. The different assumptions are discussed for exponential family models and for a large class of test functions. A short simulation study is proposed to assess the correctness of the methodology and the asymptotic results.     

Transform Analysis and Asset Pricing for Affine Jump‐diffusions

In the setting of ‘affine’ jump‐diffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems. Example applications include fixed‐income pricing models, with a role for intensity‐based models of default, as well as a wide range of option‐pricing applications. An illustrative example examines the implications of stochastic volatility and jumps for option valuation. This example highlights the impact on option ‘smirks’ of the joint distribution of jumps in volatility and jumps in the underlying asset price, through both jump amplitude as well as jump timing.     

Large and moderate deviations in testing time inhomogeneous diffusions

This paper studies hypothesis testing in time inhomogeneous diffusion processes. With the help of large and moderate deviations for the log-likelihood ratio process, we give the negative regions and obtain the decay rates of the error probabilities. Moreover, we apply our results to hypothesis testing in α‐Wiener bridge.     

Non-parametric Estimation of the Death Rate in Branching Diffusions

We consider finite systems of diffusing particles in with branching and immigration. Branching of particles occurs at position dependent rate. Under ergodicity assumptions, we estimate the position-dependent branching rate based on the observation of the particle process over a time interval [0, t ]. Asymptotics are taken as t  → ∞. We introduce a kernel-type procedure and discuss its asymptotic properties with the help of the local time for the particle configuration. We compute the minimax rate of convergence in squared-error loss over a range of Hölder classes and show that our estimator is asymptotically optimal.     

Non-Parametric Testing of Conditional Variance Functions in Time Series

This paper proposes a non‐parametric test for examining hypotheses about variance functions under stationarity and ergodicity conditions. Special cases of nonlinear time series models are studied, and it is found that under mild conditions the test is consistent. Its power is examined in a simulation study.     

The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes

Abstract.  The Pearson diffusions form a flexible class of diffusions defined by having linear drift and quadratic squared diffusion coefficient. It is demonstrated that for this class explicit statistical inference is feasible. A complete model classification is presented for the ergodic Pearson diffusions. The class of stationary distributions equals the full Pearson system of distributions. Well-known instances are the Ornstein–Uhlenbeck processes and the square root (CIR) processes. Also diffusions with heavy-tailed and skew marginals are included. Explicit formulae for the conditional moments and the polynomial eigenfunctions are derived. Explicit optimal martingale estimating functions are found. The discussion covers GMM, quasi-likelihood, non-linear weighted least squares estimation and likelihood inference too. The analytical tractability is inherited by transformed Pearson diffusions, integrated Pearson diffusions, sums of Pearson diffusions and Pearson stochastic volatility models. For the non-Markov models, explicit optimal prediction-based estimating functions are found. The estimators are shown to be consistent and asymptotically normal.     

Stochastic approximation methods in nonlinear regression models 1

The brief review of theory of stochastic approximation methods in nonlinear regression models is given. Particularly the situation when regression function has several extrema is discussed. The special attention is given to recent on the weakening of conditions on distrubances and on regression function, which guarantee the convergrence of the provedures. The problems of asymptotically optimal choice of procedures, parameters and applications to the parametric estimation are touched too.