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.     

On Empirical Processes for Ergodic Diffusions and Rates of Convergence of M-estimators

Abstract. This paper contributes to the development of empirical process theory for ergodic diffusions. We prove an entropy‐type maximal inequality for the increments of the empirical process of an ergodic diffusion. The inequality is used to study the rate of convergence of M‐estimators.     

Maximum likelihood drift estimation for a threshold diffusion

We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called drifted oscillating Brownian motion. For this continuously observed diffusion, the maximum likelihood estimator coincides with a quasi-likelihood estimator with constant diffusion term. We show that this estimator is the limit, as observations become dense in time, of the (quasi)-maximum likelihood estimator based on discrete observations. In long time, the asymptotic behaviors of the positive and negative occupation times rule the ones of the estimators. Differently from most known results of the literature, we do not restrict ourselves to the ergodic framework: indeed, depending on the signs of the drift, the process may be ergodic, transient, or null recurrent. For each regime, we establish whether or not the estimators are consistent; if they are, we prove the convergence in long time of the properly rescaled difference of the estimators towards a normal or mixed normal distribution. These theoretical results are backed by numerical simulations.     

Understanding and Addressing the Unbounded “Likelihood” Problem

The joint probability density function, evaluated at the observed data, is commonly used as the likelihood function to compute maximum likelihood estimates. For some models, however, there exist paths in the parameter space along which this density-approximation likelihood goes to infinity and maximum likelihood estimation breaks down. In all applications, however, observed data are really discrete due to the round-off or grouping error of measurements. The “correct likelihood” based on interval censoring can eliminate the problem of an unbounded likelihood. This article categorizes the models leading to unbounded likelihoods into three groups and illustrates the density-approximation breakdown with specific examples. Although it is usually possible to infer how given data were rounded, when this is not possible, one must choose the width for interval censoring, so we study the effect of the round-off on estimation. We also give sufficient conditions for the joint density to provide the same maximum likelihood estimate as the correct likelihood, as the round-off error goes to zero.     

On asymptotic normality of likelihood and conditional analysis

The likelihood function from a large sample is commonly assumed to be approximately a normal density function. The literature supports, under mild conditions, an approximate normal shape about the maximum; but typically a stronger result is needed: that the normalized likelihood itself is approximately a normal density. In a transformation-parameter context, we consider the likelihood normalized relative to right-invariant measure, and in the location case under moderate conditions show that the standardized version converges almost surely to the standard normal. Also in a transformation-parameter context, we show that almost sure convergence of the normalized and standardized likelihood to a standard normal implies that the standardized distribution for conditional inference converges almost surely to a corresponding standard normal. This latter result is of immediate use for a range of estimating, testing, and confidence procedures on a conditional-inference basis.     

MAXIMUM LIKELIHOOD ESTIMATION IN LINEAR MODELS WITH EQUI-CORRELATED RANDOM ERRORS

Necessary and sufficient conditions for the existence of maximum likelihood estimators of unknown parameters in linear models with equi‐correlated random errors are presented. The basic technique we use is that these models are, first, orthogonally transformed into linear models with two variances, and then the maximum likelihood estimation problem is solved in the environment of transformed models. Our results generalize a result of Arnold, S. F. (1981) [The theory of linear models and multivariate analysis. Wiley, New York]. In addition, we give necessary and sufficient conditions for the existence of restricted maximum likelihood estimators of the parameters. The results of Birkes, D. & Wulff, S. (2003) [Existence of maximum likelihood estimates in normal variance‐components models. J Statist Plann. Inference. 113 , 35–47] are compared with our results and differences are pointed out.     

Goodness‐of‐Fit based on Downsampling with Applications to Linear Drift Diffusions

Abstract. A goodness‐of‐fit test for continuous‐time models is developed that examines if the parameter estimates are consistent with another for different sampling frequencies. The test compares parameter estimates obtained from estimating functions for downsamples of the data. We prove asymptotic results for stationary and ergodic processes, and apply the downsampling test to linear drift diffusions. Simulations indicate that the test is quite powerful in detecting non‐Markovian deviations from the linear drift diffusions.     

On maximum likelihood estimators for a threshold autoregression

For a stationary ergodic self-exciting threshold autoregressive model with single threshold parameter, Chan (1993) obtained the consistency and limiting distribution of the least-squares estimator for the underlying true parameters. In this paper, we derive the similar results for the maximum likelihood estimators of the same model under some regularity conditions on the error density, not necessarily Gaussian.     

Nonparametric Bayesian inference for ergodic diffusions

The problem of nonparametric drift estimation for ergodic diffusions is studied from a Bayesian perspective. In particular, Gaussian process priors are exhibited that yield optimal contraction rates if the drift function belongs to a smoothness class.     

On the calculations of the maximum likelihood estimates for the polynomial spline regression model with unknown knots and AR(1) errors

Asymptotic distributions of the maximum likelihood estimators of the regression coefficients and knot points for the polynomial spline regression models with unknown knots and AR(1) errors have been derived by Chan (1989). Chan showed that under some mild conditions the maximum likelihood estimators, after suitable standardization, asymptotically follow normal distributions as n diverges to infinity. For the calculations of the maximum likelihood estimators, iterative methods must be applied. But this is not easy to implement for the model considered. In this paper, we suggested an alternative method to compute the estimates of the regression parameters and knots. It is shown that the estimates obtained by this method are asymptotically equivalent to the maximum likelihood estimates considered by Chan.     

Empirical Likelihood Ratio Test for a Change-Point in Linear Regression Model

A nonparametric method based on the empirical likelihood is proposed to detect the change-point in the coefficient of linear regression models. The empirical likelihood ratio test statistic is proved to have the same asymptotic null distribution as that with classical parametric likelihood. Under some mild conditions, the maximum empirical likelihood change-point estimator is also shown to be consistent. The simulation results show the sensitivity and robustness of the proposed approach. The method is applied to some real datasets to illustrate the effectiveness.     

Estimating Spatial Autocorrelation With Sampled Network Data

Spatial autocorrelation is a parameter of importance for network data analysis. To estimate spatial autocorrelation, maximum likelihood has been popularly used. However, its rigorous implementation requires the whole network to be observed. This is practically infeasible if network size is huge (e.g., Facebook, Twitter, Weibo, WeChat, etc.). In that case, one has to rely on sampled network data to infer about spatial autocorrelation. By doing so, network relationships (i.e., edges) involving unsampled nodes are overlooked. This leads to distorted network structure and underestimated spatial autocorrelation. To solve the problem, we propose here a novel solution. By temporarily assuming that the spatial autocorrelation is small, we are able to approximate the likelihood function by its first-order Taylor’s expansion. This leads to the method of approximate maximum likelihood estimator (AMLE), which further inspires the development of paired maximum likelihood estimator (PMLE). Compared with AMLE, PMLE is computationally superior and thus is particularly useful for large-scale network data analysis. Under appropriate regularity conditions (without assuming a small spatial autocorrelation), we show theoretically that PMLE is consistent and asymptotically normal. Numerical studies based on both simulated and real datasets are presented for illustration purpose.     

Bias Reduction of Likelihood Estimators in Semiparametric Frailty Models

Abstract. Frailty models with a non‐parametric baseline hazard are widely used for the analysis of survival data. However, their maximum likelihood estimators can be substantially biased in finite samples, because the number of nuisance parameters associated with the baseline hazard increases with the sample size. The penalized partial likelihood based on a first‐order Laplace approximation still has non‐negligible bias. However, the second‐order Laplace approximation to a modified marginal likelihood for a bias reduction is infeasible because of the presence of too many complicated terms. In this article, we find adequate modifications of these likelihood‐based methods by using the hierarchical likelihood.     

A profile likelihood method for normal mixture with unequal variance

It is well known that the normal mixture with unequal variance has unbounded likelihood and thus the corresponding global maximum likelihood estimator (MLE) is undefined. One of the commonly used solutions is to put a constraint on the parameter space so that the likelihood is bounded and then one can run the EM algorithm on this constrained parameter space to find the constrained global MLE. However, choosing the constraint parameter is a difficult issue and in many cases different choices may give different constrained global MLE. In this article, we propose a profile log likelihood method and a graphical way to find the maximum interior mode. Based on our proposed method, we can also see how the constraint parameter, used in the constrained EM algorithm, affects the constrained global MLE. Using two simulation examples and a real data application, we demonstrate the success of our new method in solving the unboundness of the mixture likelihood and locating the maximum interior mode.     

Locally quadratic log likelihood and data-based transformations

The investigation of multi-parameter likelihood functions is simplified if the log likelihood is quadratic near the maximum, as then normal approximations to the likelihood can be accurately used to obtain quantities such as likelihood regions. This paper proposes that data-based transformations of the parameters can be employed to make the log likelihood more quadratic, and illustrates the method with one of the simplest bivariate likelihoods, the normal two-parameter likelihood.     

Bias Corrected Maximum Likelihood Estimator Under the Generalized Linear Model for a Binary Variable

Under the generalized linear models for a binary variable, an approximate bias of the maximum likelihood estimator of the coefficient, that is a special case of linear parameter in Cordeiro and McCullagh (1991), is derived without a calculation of the third-order derivative of the log likelihood function. Using the obtained approximate bias of the maximum likelihood estimator, a bias-corrected maximum likelihood estimator is defined. Through a simulation study, we show that the bias-corrected maximum likelihood estimator and its variance estimator have a better performance than the maximum likelihood estimator and its variance estimator.     

Full likelihood inference in the Cox model with LTRC data when covariates are discrete

For the regression parameter β in the Cox model, there have been several estimates based on different types of approximated likelihood. For right-censored data, Ren and Zhou [Full likelihood inferences in the Cox model: an empirical approach. Ann Inst Statist Math. 2011;63:1005–1018] derive the full likelihood function for (β, F₀), where F₀ is the baseline distribution function in the Cox model. In this article, we extend their results to left-truncated and right-censored data with discrete covariates. Using the empirical likelihood parameterization, we obtain the full-profile likelihood function for β when covariates are discrete. Simulation results indicate that the maximum likelihood estimator outperforms Cox's partial likelihood estimator in finite samples.     

Asymptotics for REML estimation of spatial covariance parameters

In agricultural field trials, restricted maximum likelihood estimation (REML) of the spatial covariance parameters is often preferred to maximum likelihood. Although it has either been conjectured or assumed that REML estimators are asymptotically Gaussian, conditions under which such asymptotic results hold are clearly needed. This article gives checkable conditions for spatial regression when sampling locations are either on a rectangular grid or are irregularly spaced but satisfy certain growth conditions.     

Maximum Likelihood Estimation for the 4-Parameter Beta Distribution

This paper addresses the problem of obtaining maximum likelihood estimates for the parameters of the Pearson Type I distribution (beta distribution with unknown end points and shape parameters). Since they do not seem to have appeared in the literature, the likelihood equations and the information matrix are derived. The regularity conditions which ensure asymptotic normality and efficiency are examined, and some apparent conflicts in the literature are noted. To ensure regularity, the shape parameters must be greater than two, giving an (assymmetrical) bell-shaped distribution with high contact in the tails. A numerical investigation was carried out to explore the bias and variance of the maximum likelihood estimates and their dependence on sample size. The numerical study indicated that only for large samples (n ≥ 1000) does the bias in the estimates become small and does the Cramér-Rao bound give a good approximation for their variance. The likelihood function has a global maximum which corresponds to parameter estimates that are inadmissable. Useful parameter estimates can be obtained at a local maximum, which is sometimes difficult to locate when the sample size is small.     

GMM Estimation of Count-Panel-Data Models With Fixed Effects and Predetermined Instruments

The “traditional” approach to the estimation of count-panel-data models with fixed effects is the conditional maximum likelihood estimator. The pseudo maximum likelihood principle can be used in these models to obtain orthogonality conditions that generate a robust estimator. This estimator is inconsistent, however, when the instruments are not strictly exogenous. This article proposes a generalized method of moments estimator for count-panel-data models with fixed effects, based on a transformation of the conditional mean specification, that is consistent even when the explanatory variables are predetermined. Two applications are discussed, the relationship between patents and research and development expenditures and the explanation of technology transfer.