Density estimation in the uniform deconvolution model

We consider the problem of estimating a probability density function based on data that are corrupted by noise from a uniform distribution. The (nonparametric) maximum likelihood estimator for the corresponding distribution function is well defined. For the density function this is not the case. We study two nonparametric estimators for this density. The first is a type of kernel density estimate based on the empirical distribution function of the observable data. The second is a kernel density estimate based on the MLE of the distribution function of the unobservable (uncorrupted) data.     

Asymptotically optimal estimation of smooth functionals for interval censoring, part 1

Estimation in the interval censoring model is considered. A class of smooth functionals is introduced, of which the mean is an example. The asymptotic information lower bound for such functionals can be represented as an inner product of two functions. In case 1, i.e. one observation time per unobservable event time, both functions can be given explicitly. We mainly consider case 2, with two observation times for each unobservable event time, in the situation that the observation times can not become arbitrarily close to each other. For case 2, one of the functions in the inner product can only be given implicitly as solution to a Fredholm integral equation. We study properties of this solution and, in a sequel to this paper, prove that the nonparametric maximum likelihood estimator of the functional asymptotically reaches the information lower bound.     

Likelihood-based specification analysis of continuous-time models of the short-term interest rate

An extensive collection of continuous-time models of the short-term interest rate is evaluated over data sets that have appeared previously in the literature. The analysis, which uses the simulated maximum likelihood procedure proposed by Durham and Gallant (2002), provides new insights regarding several previously unresolved questions. For single factor models, I find that the volatility, not the drift, is the critical component in model specification. Allowing for additional flexibility beyond a constant term in the drift provides negligible benefit. While constant drift would appear to imply that the short rate is nonstationary, in fact, stationarity is volatility-induced. The simple constant elasticity of volatility model fits weekly observations of the three-month Treasury bill rate remarkably well but is easily rejected when compared with more flexible volatility specifications over daily data. The methodology of Durham and Gallant can also be used to estimate stochastic volatility models. While adding the latent volatility component provides a large improvement in the likelihood for the physical process, it does little to improve bond-pricing performance.     

Robustness issues in multilevel regression analysis

A multilevel problem concerns a population with a hierarchical structure. A sample from such a population can be described as a multistage sample. First, a sample of higher level units is drawn (e.g. schools or organizations), and next a sample of the sub‐units from the available units (e.g. pupils in schools or employees in organizations). In such samples, the individual observations are in general not completely independent. Multilevel analysis software accounts for this dependence and in recent years these programs have been widely accepted. Two problems that occur in the practice of multilevel modeling will be discussed. The first problem is the choice of the sample sizes at the different levels. What are sufficient sample sizes for accurate estimation? The second problem is the normality assumption of the level‐2 error distribution. When one wants to conduct tests of significance, the errors need to be normally distributed. What happens when this is not the case? In this paper, simulation studies are used to answer both questions. With respect to the first question, the results show that a small sample size at level two (meaning a sample of 50 or less) leads to biased estimates of the second‐level standard errors. The answer to the second question is that only the standard errors for the random effects at the second level are highly inaccurate if the distributional assumptions concerning the level‐2 errors are not fulfilled. Robust standard errors turn out to be more reliable than the asymptotic standard errors based on maximum likelihood.     

Progressive stress accelerated life tests under finite mixture models

In this paper, progressive stress accelerated life tests are considered when the lifetime of a product under use condition follows a finite mixture of distributions. The experiment is performed when each of the components in the mixture follows a general class of distributions which includes, among others, the Weibull, compound Weibull, power function, Gompertz and compound Gompertz distributions. It is assumed that the scale parameter of each component satisfies the inverse power low, the progressive stress is directly proportional to time and the cumulative exposure model for the effect of changing stress holds. Based on type-I censoring, the maximum likelihood estimates (MLEs) of the parameters under consideration are obtained. A special attention is paid to a mixture of two Rayleigh components. Simulation results are carried out to study the precision of the MLEs and to obtain confidence intervals for the parameters involved.     

ON THE FIRST–ORDER EFFICIENCY AND ASYMPTOTIC NORMALITY OF MAXIMUM LIKELIHOOD ESTIMATORS OBTAINED FROM DEPENDENT OBSERVATIONS

Abstract. In this paper we study the first–order efficiency and asymptotic normality of the maximum likelihood estimator obtained from dependent observations. Our conditions are weaker than usual, in that we do not require convergences in probability to be uniform or third–order derivatives to exist.
The paper builds on Witting and Nolle's result concerning the asymptotic normality of the maximum likelihood estimator obtained from independent and identically distributed observations, and on a martingale theorem by McLeish.     

Pseudo empirical likelihood method in the presence of missing data

In this paper, we propose an estimator for the population mean when some observations on the study and auxiliary variables are missing from the sample. The proposed estimator is valid for any unequal probability sampling design, and is based upon the pseudo empirical likelihood method. The proposed estimator is compared with other estimators in a simulation study.     

Convenient estimators for the panel probit model: Further results

Bertschek and Lechner (1998) propose several variants of a GMM estimator based on the period specific regression functions for the panel probit model. The analysis is motivated by the complexity of maximum likelihood estimation and the possibly excessive amount of time involved in maximum simulated likelihood estimation. But, for applications of the size considered in their study, full likelihood estimation is actually straightforward, and resort to GMM estimation for convenience is unnecessary. In this note, we reconsider maximum likelihood based estimation of their panel probit model then examine some extensions which can exploit the heterogeneity contained in their panel data set. Empirical results are obtained using the data set employed in the earlier study. Helpful comments and suggestions by Irene Bertschek and Michael Lechner are gratefully acknowledged. This paper has also benefited from comments by two anonymous referees and from seminar participants at the Center for Health Economics at the University of York. Any remaining errors are the responsibility of the author.     

Estimation methods for stochastic volatility models: a survey

Abstract.  Although stochastic volatility (SV) models have an intuitive appeal, their empirical application has been limited mainly due to difficulties involved in their estimation. The main problem is that the likelihood function is hard to evaluate. However, recently, several new estimation methods have been introduced and the literature on SV models has grown substantially. In this article, we review this literature. We describe the main estimators of the parameters and the underlying volatilities focusing on their advantages and limitations both from the theoretical and empirical point of view. We complete the survey with an application of the most important procedures to the S&P 500 stock price index.     

The British low-wage sector and the employment prospects of the unemployed

Are low wages a way for the unemployed to switch to higher-paying jobs? Using data from the British Household Panel Survey, the labour market dynamics of unemployed, low-paid and higher-paid employed men are analysed. Moreover, the respective (un)employment duration and occupational skill level are accounted for. Results show that in general low wages significantly reduce the risk of future unemployment and increase the chances of ascending the salary ladder, especially in the case of long-term unemployment (>360 days). Furthermore, the occupational skill level has a substantial influence on the upward mobility of low-paid jobs: individuals working in the initial period in a low-paid and higher-skilled occupation have on average an 11 percentage points higher probability of entering higher pay compared to when working in a low-paid and low-skilled occupation.