Results 1  10
of
35
Estimating Equations Based on Eigenfunctions for a Discretely Observed Diffusion Process
, 1995
"... : A new type of martingale estimating function is proposed for inference about classes of diffusion processes based on discretetime observations. These estimating functions can be tailored to a particular class of diffusion processes by utilizing a martingale property of the eigenfunctions of the g ..."
Abstract

Cited by 58 (13 self)
 Add to MetaCart
: A new type of martingale estimating function is proposed for inference about classes of diffusion processes based on discretetime observations. These estimating functions can be tailored to a particular class of diffusion processes by utilizing a martingale property of the eigenfunctions of the generators of the diffusions. Optimal estimating functions in the sense of Godambe and Heyde are found. Inference based on these is invariant under transformations of data. A result on consistency and asymptotic normality of the estimators is given for ergodic diffusions. The theory is illustrated by several examples and by a simulation study. Keywords: generator, optimal estimating function, stochastic differential equation, quasilikelihood. 1 Introduction Martingale estimating functions have turned out to give good and relatively simple estimators for discretely observed diffusion models, for which the likelihood function is only explicitly known in special cases. These estimators have th...
Local polynomial kernel regression for generalized linear models and quasilikelihood functions
 Journal of the American Statistical Association,90
, 1995
"... were introduced as a means of extending the techniques of ordinary parametric regression to several commonlyused regression models arising from nonnormal likelihoods. Typically these models have a variance that depends on the mean function. However, in many cases the likelihood is unknown, but the ..."
Abstract

Cited by 57 (7 self)
 Add to MetaCart
were introduced as a means of extending the techniques of ordinary parametric regression to several commonlyused regression models arising from nonnormal likelihoods. Typically these models have a variance that depends on the mean function. However, in many cases the likelihood is unknown, but the relationship between mean and variance can be specified. This has led to the consideration of quasilikelihood methods, where the conditionalloglikelihood is replaced by a quasilikelihood function. In this article we investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to these more general contexts. In the ordinary regression case local polynomial fitting has been seen to possess several appealing features in terms of intuitive and mathematical simplicity. One noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators. These properties are shown to carryover to the generalized linear model and quasilikelihood model. The end result is a class of kernel type estimators for smoothing in quasilikelihood models. These estimators can be viewed as a straightforward generalization of the usual parametric estimators. In addition, their simple asymptotic distributions allow for simple interpretation
Estimating Functions for Discretely Sampled DiffusionType Models. Chapter of the Handbook of financial econometrics, AitSahalia and Hansen eds. http://home.uchicago.edu/ lhansen/handbook.htm Birgé
 in Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics
, 2004
"... Estimating functions provide a general framework for finding estimators and studying their properties in many different kinds of statistical models, including stochastic process models. An estimating function is a function of the data as well as of the parameter to be estimated. An estimator is obta ..."
Abstract

Cited by 26 (9 self)
 Add to MetaCart
Estimating functions provide a general framework for finding estimators and studying their properties in many different kinds of statistical models, including stochastic process models. An estimating function is a function of the data as well as of the parameter to be estimated. An estimator is obtained by equating the estimating function to zero and solving the resulting
A diffusion model for exchange rates in a target zone
 In preparation
, 2003
"... We present two relatively simple and analytically tractable diffusion models for an exchange rate in a target zone. One model generalize a model proposed by De Jong, Drost & Werker (2001) to allow asymmetry between the currencies and thus obtain a better fit to data, in which asymmetries are often a ..."
Abstract

Cited by 14 (9 self)
 Add to MetaCart
We present two relatively simple and analytically tractable diffusion models for an exchange rate in a target zone. One model generalize a model proposed by De Jong, Drost & Werker (2001) to allow asymmetry between the currencies and thus obtain a better fit to data, in which asymmetries are often an important feature. Optimal estimation of the model parameter using eigenfunctions of the generator is investigated in detail and shown to give wellbehaved estimators that are easy to calculate. The model is demonstrated to fit data on exchange rates in the European Monetary System well. Also an alternative diffusion model is presented, which has similar properties in the centre of the target zone, but with a more realistic dynamics near the boundaries of the target zone. Estimators based on eigenfunctions work well in this case too. For both models noarbitrage pricing of derivative assets is discussed. Finally, problems concerning adjustments of the central parity are discussed. Key words: currency options, eigenfunctions, estimating functions, exchange rate target zones, Jacobi diffusion, option pricing, realignments. 1 1
Inference for Observations of Integrated Diffusion Processes
"... Estimation of parameters in diusion models is usually based on observations of the process at discrete time points. Here we investigate estimation when a sample of discrete observations is not available, but, instead, observations of a running integral of the process with respect to some weight ..."
Abstract

Cited by 12 (7 self)
 Add to MetaCart
Estimation of parameters in diusion models is usually based on observations of the process at discrete time points. Here we investigate estimation when a sample of discrete observations is not available, but, instead, observations of a running integral of the process with respect to some weight function. This type of observations is, for example, obtained when a realization of the process is observed after passage through an electronic lter. Another example is provided by the icecore data on oxygen isotopes used to investigate paleotemperatures. Finally, such data play a role in connection with the stochastic volatility models of nance. The integrated process is no longer a Markov process which render the use of martingale estimating functions dicult. Therefore, a generalization of the martingale estimating functions, namely the predictionbased estimating functions, is applied to estimate parameters in the underlying diusion process. The estimators are shown to be consistent and asymptotically normal. The method is applied to inference based on integrated data from OrnsteinUhlenbeck processes and from the CIRmodel for both of which an explicit estimating function can be found.
QuasiLikelihood Models and Optimal Inference
 Ann. Statist
"... Consider an ergodic Markov chain on the real line, with parametric models for the conditional mean and variance of the transition distribution. Such a setting is an instance of a quasilikelihood model. The customary estimator for the parameter is the maximum quasilikelihood estimator. It is not ef ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
Consider an ergodic Markov chain on the real line, with parametric models for the conditional mean and variance of the transition distribution. Such a setting is an instance of a quasilikelihood model. The customary estimator for the parameter is the maximum quasilikelihood estimator. It is not efficient, but as good as the best estimator that ignores the parametric model for the conditional variance. We construct two efficient estimators. One is a convex combination of solutions of two estimating equations, the other a weighted nonlinear onestep least squares estimator, with weights involving predictors for the third and fourth centered conditional moments of the transition distribution. Additional restrictions on the model can lead to further improvement. We illustrate this with an autoregressive model whose error variance is related to the autoregression parameter. 1 Introduction According to Wedderburn (1974), a quasilikelihood model is defined by a relation between mean and v...
Estimation for Discretely Observed Diffusions using Transform Functions
 J. APPL. PROB
, 2003
"... This paper introduces a new estimation technique for discretely observed diffusion processes. Transform functions are applied to transform the data to obtain good and easily calculated estimators of both the drift and diffusion coefficients. Consistency and asymptotic normality of the resulting ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
This paper introduces a new estimation technique for discretely observed diffusion processes. Transform functions are applied to transform the data to obtain good and easily calculated estimators of both the drift and diffusion coefficients. Consistency and asymptotic normality of the resulting estimators is investigated. Power transforms are
Bartlett Identities and Large Deviations in Likelihood Theory
 Ann. Statist
, 1999
"... this paper that these two angles of research can be uni ed, at least for smooth families. The connection between small and large deviation expansions is, at least heuristically, clear cut. They both rely on cumulants, though the cumulants show up in terms of dierent order in the two types of expans ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
this paper that these two angles of research can be uni ed, at least for smooth families. The connection between small and large deviation expansions is, at least heuristically, clear cut. They both rely on cumulants, though the cumulants show up in terms of dierent order in the two types of expansion. For an example of a rigorous study concerning this connection, see, e.g., Robinson, Hoglund, Holst and Quine (1990)
Projected Partial Likelihood and Its Application to Longitudinal Data
, 1995
"... this paper we provide a method for constructing such estimating functions. When applied to longitudinal data, this handles covariates and random dropout satisfactorily, as does partial likelihood; it avoids distributional assumptions, as does generalized estimating equations. It is obtained by proj ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
this paper we provide a method for constructing such estimating functions. When applied to longitudinal data, this handles covariates and random dropout satisfactorily, as does partial likelihood; it avoids distributional assumptions, as does generalized estimating equations. It is obtained by projecting the partial score function onto a collection of Hilbert spaces with inner product specified by conditional moments, conditioned on nested events. We also demonstrate, within a prequential frame of reference (Dawid 1984, 1991), that the estimating function is optimal among the largest collection of estimating functions that can be described by the postulated conditional moments.
Fisher's Method Of Scoring
 Int. Stat. Rev
, 1992
"... . An analysis is given of the computational properties of Fisher's method of scoring for maximizing likelihoods and solving estimating equations based on quasilikelihoods. Consistent estimation of the true parameter vector is shown to be important if a fast rate of convergence is to be achieved, bu ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
. An analysis is given of the computational properties of Fisher's method of scoring for maximizing likelihoods and solving estimating equations based on quasilikelihoods. Consistent estimation of the true parameter vector is shown to be important if a fast rate of convergence is to be achieved, but if this condition is met then the algorithm is very attractive. This link between the performance of the scoring algorithm and the adequacey of the underlying problem modelling is stressed. The effect of linear constraints on performance is discussed, and examples of likelihood and quasilikelihood calculations are presented. 1. Introduction Two basic paradigms play important roles in the material developed in this paper. These are: (1) Newton's method for function minimization, and (2) the method of maximum likelihood for parameter estimation in data analysis problems. The main aim is to examine aspects of the structure and performance of Fisher's method of scoring, a minimization techniq...