Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract. We consider N independent stochastic processes (X _i (t), t ∈ [0,T _i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ _i . The distribution of the random effect φ _i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ _i and φ _i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T _i =T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.     

On sufficient conditions for the strong consistency of least-squares estimates

The strong consistency of the least-squares estimates in regression models is obtained when the errors are i.i.d. with absolute moment of order r, 0<r? 2. The assumptions presented for the random error sequence will permit us to obtain improvements of the conditions on the regressors in order to obtain the strong consistency of the least-squares estimates in linear and nonlinear regression models.     

Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications

In this paper, the Rosenthal-type maximal inequalities and Kolmogorov-type exponential inequality for negatively superadditive-dependent (NSD) random variables are presented. By using these inequalities, we study the complete convergence for arrays of rowwise NSD random variables. As applications, the Baum–Katz-type result for arrays of rowwise NSD random variables and the complete consistency for the estimator of nonparametric regression model based on NSD errors are obtained. Our results extend and improve the corresponding ones of Chen et al. [On complete convergence for arrays of rowwise negatively associated random variables. Theory Probab Appl. 2007;52(2):393–397] for arrays of rowwise negatively associated random variables to the case of arrays of rowwise NSD random variables.     

Strong consistency of least squares estimates with i.i.d. errors with mean values not necessarily defined

We establish strong consistency of the least squares estimates in multiple regression models discarding the usual assumption of the errors having null mean value. Thus, we required them to be i.i.d. with absolute moment of order r, 0<r<2, and null mean value when r>1. Only moderately restrictive conditions are imposed on the model matrix. In our treatment, we use an extension of the Marcinkiewicz–Zygmund strong law to overcome the errors mean value not being defined. In this way, we get a unified treatment for the case of i.i.d. errors extending the results of some previous papers.     

Asymptotic properties of a rank estimate in linear regression with symmetric non-identically distributed errors

In this article, a simple linear regression model with independent and symmetric but non-identically distributed errors is considered. Asymptotic properties of the rank regression estimate defined in Jaeckel [Estimating regression coefficients by minimizing the dispersion of the residuals, Ann. Math. Statist. 43 (1972), pp. 1449–1458] are studied. We show that the studied estimator is consistent and asymptotically normally distributed. The cases of bounded and unbounded score functions are examined separately. The regularity conditions of the article are exemplified for finite mixture distributions.     

Simultaneous sparse model selection and coefficient estimation for heavy-tailed autoregressive processes

We propose a sparse coefficient estimation and automated model selection procedure for autoregressive processes with heavy-tailed innovations based on penalized conditional maximum likelihood. Under mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator satisfies a strong consistency, O_P(N^?1/2) consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator and smoothly clipped average deviation, are compared. The proposed method provides a distribution-based penalized inference to AR models, which is especially useful when the other estimation methods fail or under perform for AR processes with heavy-tailed innovations [Feigin, Resnick. Pitfalls of fitting autoregressive models for heavy-tailed time series. Extremes. 1999;1:391–422]. A simulation study confirms our theoretical results. At the end, we apply our method to a historical price data of the US Industrial Production Index for consumer goods, and obtain very promising results.     

Simultaneous structure estimation and variable selection in partial linear varying coefficient models for longitudinal data

Partial linear varying coefficient models (PLVCM) are often considered for analysing longitudinal data for a good balance between flexibility and parsimony. The existing estimation and variable selection methods for this model are mainly built upon which subset of variables have linear or varying effect on the response is known in advance, or say, model structure is determined. However, in application, this is unreasonable. In this work, we propose a simultaneous structure estimation and variable selection method, which can do simultaneous coefficient estimation and three types of selections: varying and constant effects selection, relevant variable selection. It can be easily implemented in one step by employing a penalized M-type regression, which uses a general loss function to treat mean, median, quantile and robust mean regressions in a unified framework. Consistency in the three types of selections and oracle property in estimation are established as well. Simulation studies and real data analysis also confirm our method.     

On the consistency and the robustness in model selection criteria

Abstract

In the model selection problem, the consistency of the selection criterion has been often discussed. This paper derives a family of criteria based on a robust statistical divergence family by using a generalized Bayesian procedure. The proposed family can achieve both consistency and robustness at the same time since it has good performance with respect to contamination by outliers under appropriate circumstances. We show the selection accuracy of the proposed criterion family compared with the conventional methods through numerical experiments.     

Morality rules: Understanding the role of prior reputation in consequences of scansis

A favorable reputation casts positive influences over organizations’ short- and long-term development, which can be especially important when organizations grapple with unpredictable challenges such as organizational crises. Yet, keeping a good track record might also backfire to cause more reputational damages, which is often observed when organizations deal with value-oriented moral crises. Against this backdrop, drawing from the literature on crisis communication and moral psychology, we situated our study in the scansis context, a particular type of morality-focused negative situation featuring characteristics of both crisis and scandal, and explored effects of prior reputation on people’s responses to an organizational scansis. Through an online experiment (N = 293), we found severe backlash towards the scansis-stricken organization in both the control condition (i.e., no indication of reputation) and the good reputation condition, whereas such prominent difference was not observed in the bad reputation condition. The findings thus implicated the need to take the unique role of morality in scansis into account in both pertinent research and practice.     

An Adaptive,Rate‐Optimal Test of a Parametric Mean‐Regression Model Against a Nonparametric Alternative

We develop a new test of a parametric model of a conditional mean function against a nonparametric alternative. The test adapts to the unknown smoothness of the alternative model and is uniformly consistent against alternatives whose distance from the parametric model converges to zero at the fastest possible rate. This rate is slower than n^−1/2. Some existing tests have nontrivial power against restricted classes of alternatives whose distance from the parametric model decreases at the rate n^−1/2. There are, however, sequences of alternatives against which these tests are inconsistent and ours is consistent. As a consequence, there are alternative models for which the finite‐sample power of our test greatly exceeds that of existing tests. This conclusion is illustrated by the results of some Monte Carlo experiments.