Asymptotic properties of maximum-likelihood estimators for Heston models based on continuous time observations

We study asymptotic properties of maximum-likelihood estimators for Heston models based on continuous time observations of the log-price process. We distinguish three cases: subcritical (also called ergodic), critical and supercritical. In the subcritical case, asymptotic normality is proved for all the parameters, while in the critical and supercritical cases, non-standard asymptotic behaviour is described.     

Local asymptotic properties for Cox-Ingersoll-Ross process with discrete observations

In this paper, we consider a one-dimensional Cox-Ingersoll-Ross (CIR) process whose drift coefficient depends on unknown parameters. Considering the process discretely observed at high frequency, we prove the local asymptotic normality property in the subcritical case, the local asymptotic quadraticity in the critical case, and the local asymptotic mixed normality property in the supercritical case. To obtain these results, we use the Malliavin calculus techniques developed recently for CIR process together with the estimation for positive and negative polynomial moments of the CIR process. In this study, we require the same conditions of high frequency and infinite horizon as in the case of ergodic diffusions with globally Lipschitz coefficients studied earlier in the literature. However, in the non-ergodic cases, additional assumptions on the decreasing rate are required due to the fact that the square root diffusion coefficient of the CIR process is not regular enough.     

Fitting logistic regression models with contaminated case–control data

Errors in measurement frequently occur in observing responses. If case–control data are based on certain reported responses, which may not be the true responses, then we have contaminated case–control data. In this paper, we first show that the ordinary logistic regression analysis based on contaminated case–control data can lead to very serious biased conclusions. This can be concluded from the results of a theoretical argument, one example, and two simulation studies. We next derive the semiparametric maximum likelihood estimate (MLE) of the risk parameter of a logistic regression model when there is a validation subsample. The asymptotic normality of the semiparametric MLE will be shown along with consistent estimate of asymptotic variance. Our example and two simulation studies show these estimates to have reasonable performance under finite sample situations.     

Testing for residual correlation of any order in the autoregressive process

We are interested in the implications of a linearly autocorrelated driven noise on the asymptotic behavior of the usual least-squares estimator in a stable autoregressive process. We show that the least-squares estimator is not consistent and we suggest a sharp analysis of its almost sure limiting value as well as its asymptotic normality. We also establish the almost sure convergence and the asymptotic normality of the estimated serial correlation parameter of the driven noise. Then, we derive a statistical procedure enabling to test for correlation of any order in the residuals of an autoregressive modelling, giving clearly better results than the commonly used portmanteau tests of Ljung–Box and Box–Pierce, and appearing to outperform the Breusch–Godfrey procedure on small-sized samples.     

The MLE of Geometric Parameter for a Geometric Process

In this article, we provide the MLE of the ratio parameter of a geometric process and discuss its consistency and asymptotic normality.     

On asymptotic posterior normality for controlled branching processes

For a controlled branching process (CBP) with offspring distribution belonging to the power series family, the asymptotic normality of the posterior distribution of the basic parameter and the offspring mean is proved. As practical applications, we calculate asymptotic high probability density credibility sets for the offspring mean and we provide a rule to make inference about the value of this parameter. Moreover, the asymptotic posterior normality of the respective parameters of two classical branching models, namely the standard Galton–Watson process and the Galton–Watson process with immigration, is derived as particular cases of the CBP.     

The discretely observed immigration-death process: Likelihood inference and spatiotemporal applications

ABSTRACT

We consider a stochastic process, the homogeneous spatial immigration-death (HSID) process, which is a spatial birth-death process with as building blocks (i) an immigration-death (ID) process (a continuous-time Markov chain) and (ii) a probability distribution assigning iid spatial locations to all events. For the ID process, we derive the likelihood function, reduce the likelihood estimation problem to one dimension, and prove consistency and asymptotic normality for the maximum likelihood estimators (MLEs) under a discrete sampling scheme. We additionally prove consistency for the MLEs of HSID processes. In connection to the growth-interaction process, which has a HSID process as basis, we also fit HSID processes to Scots pine data.     

Pseudo maximum-likelihood estimation of the univariate GARCH (1,1) and asymptotic properties

One provides in this paper the pseudo-likelihood estimator (PMLE) and asymptotic theory for the GARCH (1,1) process. Strong consistency of the pseudo-maximum-likelihood estimator (MLE) is established by appealing to conditions given in Jeantheau (1998) concerning the existence of a stationary and ergodic solution to the multivariate GARCH (p, q) process. One proves the asymptotic normality of the PMLE by appealing to martingales' techniques.     

Estimation in the presence of incidental parameters

This paper considers the estimation of “structural” parameters when the number of unknown parameters increases with the sample size. Neyman and Scott (1948) had demonstrated that maximum likelihood estimators (MLE) of structural parameters may be inconsistent in this case. Patefield (1977) further observed that the asymptotic covariance matrix of the MLE is not equal to the inverse of the information matrix. In this paper we establish asymptotic properties of estimators (which include in particular the MLE) obtained via the usual likelihood approach when the incidental parameters are first replaced by their estimates (which are allowed to depend on the structural parameters). Conditions for consistency and asymptotic normality together with a proper formula for the asymptotic covariance matrix are given. The results are illustrated and applied to the problem of estimating linear functional relationships, and mild conditions on the incidental parameters for the MLE (or an adjusted MLE) to be consistent and asymptotically normal are obtained. These conditions are weaker than those imposed by previous authors.     

Estimation of system reliability for exponential distributions based on L ranked set sampling

Abstract

In the case where strength and stress both follow exponential distributions, this paper considers the maximum likelihood estimator (MLE) of the system reliability based on L ranked set sampling (LRSS). The proposed MLE is shown to have existence, uniqueness and asymptotic normality, and its asymptotic variance is obtained by the Fisher information matrix of LRSS. The values of asymptotic relative efficiencies show that the proposed MLE is always more efficient than the MLE using simple random sampling (SRS). However, the MLE using LRSS cannot be written in closed form. Therefore, the modified MLE is proposed using the technique replaced some terms in the maximum likelihood equations by their expectations. The newly modified MLE using LRSS is shown to be superior to the MLE using SRS. Finally, the proposed method is applied to a real data set on metastatic renal carcinoma study.     

Approximation of distributions by using the Anderson Darling statistic

ABSTRACT

In practice, it is often not possible to find an appropriate family of distributions which can be used for fitting the sample distribution with high precision. In these cases, it seems to be opportune to search for the best approximation by a family of distributions instead of an exact fit. In this paper, we consider the Anderson–Darling statistic with plugged-in minimum distance estimator for the parameter vector. We prove asymptotic normality of the Anderson–Darling statistic which is used for a test of goodness of approximation. Moreover, we introduce a measure of discrepancy between the sample distribution and the model class.     

LAN, LAM and convergence of Iterative ML estimates; optimal design in Gaussian one-way mixed model

For a one-way mixed Gaussian ANOVA model we prove local asymptotic normality and local asymptotic minimaxity of maximum likelihood estimates (MLE) and of its certain iterative approximations. A geometric rate of convergence in probability is proved for these iterative estimates to MLE. Asymptotically optimal designs for large samples are studied.     

Asymptotic properties of the maximum-likelihood estimator in zero-inflated binomial regression

The zero-inflated binomial (ZIB) regression model was proposed to account for excess zeros in binomial regression. Since then, the model has been applied in various fields, such as ecology and epidemiology. In these applications, maximum-likelihood estimation (MLE) is used to derive parameter estimates. However, theoretical properties of the MLE in ZIB regression have not yet been rigorously established. The current paper fills this gap and thus provides a rigorous basis for applying the model. Consistency and asymptotic normality of the MLE in ZIB regression are proved. A consistent estimator of the asymptotic variance–covariance matrix of the MLE is also provided. Finite-sample behavior of the estimator is assessed via simulations. Finally, an analysis of a data set in the field of health economics illustrates the paper.     

Estimation for Continuous Branching Processes

The maximum-likelihood estimator for the curved exponential family given by continuous branching processes with immigration is investigated. These processes originated from population biology but also model the dynamics of interest rates and development of the state of technology in economics. It is proved that in contrast to branching processes with discrete space and/or time the MLE gives a unified approach to the inference. In order to include singular subdomains of the parameter space we modify the MLE slightly. Consistency and asymptotic normality for the MLE are considered. Concerning the asymptotic theory of the experiments, all three properties LAQ, LAN, and LAMN occur for different submodels     

Mixed Liu estimator in linear measurement error models

In this paper, we introduce mixed Liu estimator (MLE) for the vector of parameters in linear measurement error models by unifying the sample and the prior information. The MLE is a generalization of the mixed estimator (ME) and Liu estimator (LE). In particular, asymptotic normality properties of the estimators are discussed, and the performance of the MLE over the LE and ME are compared based on mean squared error matrix (MSEM). Finally, a Monte Carlo simulation and a numerical example are also presented for analysis.     

The efficiency of a test based on the asymptotic distribution of the mle for a linear functional relationship

A rigorous derivation is given of the asymptotic normality of the MLE of a linear functional relationship. Using these results, it is shown that the test proposed by VILLEGAS (1964) has Pitman efficiency zero w.r.t, a test based on the asymptotic distribution of the MLE.     

Rounded data analysis based on ranked set sample

In this paper we investigate the Fisher information matrix of a rounded ranked set sampling (RSS) sample and show that the sample is always more informative than a rounded simple random sampling (SRS) sample of the same size. On the other hand, we propose a new method to approximate maximum likelihood estimates (MLE) of unknown parameters for this model and further establish the strong consistency and asymptotic normality of the proposed estimators. Simulation experiments show that the approximated MLE based on rounded RSS is always more efficient than those based on rounded SRS.     

Asymptotic Properties of Generalized Multivariate Rank Statistics

This article discusses generalization of the well-known multivariate rank statistics under right-censored data case. Empirical process representation used to get the generalization. The marginal distribution functions are estimated by Kaplan–Meier estimators. Sufficient conditions for asymptotic normality of the generalized multivariate rank statistics under independently right censored data are specified. Several auxiliary results on sup-norm convergence of Kaplan–Meier estimators in randomly exhausting regions are given too.     

Asymptotic properties of one-step M-estimators

We study the asymptotic behavior of one-step M-estimators based on not necessarily independent identically distributed observations. In particular, we find conditions for asymptotic normality of these estimators. Asymptotic normality of one-step M-estimators is proven under a wide spectrum of constraints on the exactness of initial estimators. We discuss the question of minimal restrictions on the exactness of initial estimators. We also discuss the asymptotic behavior of the solution to an M-equation closest to the parameter under consideration. As an application, we consider some examples of one-step approximation of quasi-likelihood estimators in nonlinear regression.     

Asymptotic properties of MLE for partially observed fractional diffusion system with dependent noises

The paper studies long time asymptotic properties of the maximum likelihood estimator (MLE) for the signal drift parameter in a partially observed fractional diffusion system with dependent noise. Using the method of weak convergence of likelihoods due to Ibragimov and Khasminskii [1981. Statistics of Random Processes. Springer, New-York], consistency, asymptotic normality and convergence of the moments are established for MLE. The proof is based on Laplace transform computations which was introduced in Brouste and Kleptsyna [2008. Asymptotic properties of MLE for partially observed fractional diffusion system, preprint].