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When a parameter of interest is nondifferentiable in the probability, the existing theory of semiparametric efficient estimation is not applicable, as it does not have an influence function. Song (2014) recently developed a local asymptotic minimax estimation theory for a parameter that is a nondifferentiable transform of a regular parameter, where the transform is a composite map of a continuous piecewise linear map with a single kink point and a translation-scale equivariant map. The contribution of this paper is twofold. First, this paper extends the local asymptotic minimax theory to nondifferentiable transforms that are a composite map of a Lipschitz continuous map having a finite set of nondifferentiability points and a translation-scale equivariant map. Second, this paper investigates the discontinuity of the local asymptotic minimax risk in the true probability and shows that the proposed estimator remains to be optimal even when the risk is locally robustified not only over the scores at the true probability, but also over the true probability itself. However, the local robustification does not resolve the issue of discontinuity in the local asymptotic minimax risk.

For this discussion, assume there are n sample observations of the dependent variable y at unique locations. In spatial samples, often each observation is uniquely associated with a particular location or region, so that observations and regions are equivalent. Spatial dependence arises when an observation at one location, say y i is dependent on “neighboring” observations y j, y j∈ϒi. We use ϒi to denote the set of observations that are “neighboring” to observation i, where some metric is used to define the set of observations that are spatially connected to observation i. For general definitions of the sets ϒi,i=1,…,n, typically at least one observation exhibits simultaneous dependence, so that an observation y j, also depends on y i. That is, the set ϒj contains the observation y i, creating simultaneous dependence among observations. This situation constitutes a difference between time series analysis and spatial analysis. In time series, temporal dependence relations could be such that a “one-period-behind relation” exists, ruling out simultaneous dependence among observations. The time series one-observation-behind relation could arise if spatial observations were located along a line and the dependence of each observation were strictly on the observation located to the left. However, this is not in general true of spatial samples, requiring construction of estimation and inference methods that accommodate the more plausible case of simultaneous dependence among observations.

Despite the difference in information sets, we are able to compare the asymptotic distribution of volatility estimators involving data sampled at different frequencies. To do so, we propose extensions of the continuous record asymptotic analysis for rolling sample variance estimators developed by Foster and Nelson (1996, Econometrica, 64, 139–174). We focus on traditional historical volatility filters involving monthly, daily and intradaily observations. Theoretical results are complemented with Monte Carlo simulations in order to assess the validity of the asymptotics for sample sizes and filters encountered in empirical studies.

A method of estimating lifetime parameters from doubly censored data is given on the basis of the maximum likelihood principle. Consistency and asymptotic normality of the proposed estimator is established. When the observation window is defined by the interval of time between the first and the pth events in a homogeneous Poisson process, asymptotic variances and efficiencies are analyzed assuming exponential lifetime distribution.

This chapter compares the performance of the maximum simulated likelihood (MSL) approach with the composite marginal likelihood (CML) approach in multivariate ordered-response situations. The ability of the two approaches to recover model parameters in simulated data sets is examined, as is the efficiency of estimated parameters and computational cost. Overall, the simulation results demonstrate the ability of the CML approach to recover the parameters very well in a 5–6 dimensional ordered-response choice model context. In addition, the CML recovers parameters as well as the MSL estimation approach in the simulation contexts used in this study, while also doing so at a substantially reduced computational cost. Further, any reduction in the efficiency of the CML approach relative to the MSL approach is in the range of nonexistent to small. When taken together with its conceptual and implementation simplicity, the CML approach appears to be a promising approach for the estimation of not only the multivariate ordered-response model considered here, but also for other analytically intractable econometric models.

Modelling and forecasting interval-valued time series (ITS) have received increasing attention in statistics and econometrics. An interval-valued observation contains more information than a point-valued observation in the same time period. The previous literature has mainly considered modelling and forecasting a univariate ITS. However, few works attempt to model a vector process of ITS. In this paper, we propose an interval-valued vector autoregressive moving average (IVARMA) model to capture the cross-dependence dynamics within an ITS vector system. A minimum-distance estimation method is developed to estimate the parameters of an IVARMA model, and consistency, asymptotic normality and asymptotic efficiency of the proposed estimator are established. A two-stage minimum-distance estimator is shown to be asymptotically most efficient among the class of minimum-distance estimators. Simulation studies show that the two-stage estimator indeed outperforms other minimum-distance estimators for various data-generating processes considered.

Compares optimum constant‐stress and simple step‐stress ALTs for the Weibull distribution. The purpose is to quantify the advantage of using step‐stress testing in comparison to constant‐stress testing when censoring is likely to occur at the lower levels of stress. Assumes that a log‐linear relationship exists between log scale parameter and stress and that the cumulative exposure model holds for the effect of changing stress in step‐stress test. The variance of the MLE of log scale parameter at design stress is used as the criterion for comparing ALTs. The efficiency of simple step‐stress tests relative to constant‐stress with Type I censoring is studied in terms of the ratio of these variances. Results are given for asymptotic variances and for finite sample sizes using simulation to estimate variances.

For kernel-based estimators, smoothness conditions ensure that the asymptotic rate at which the bias goes to zero is determined by the kernel order. In a finite sample, the leading term in the expansion of the bias may provide a poor approximation. We explore the relation between smoothness and bias and provide estimators for the degree of the smoothness and the bias. We demonstrate the existence of a linear combination of estimators whose trace of the asymptotic mean-squared error is reduced relative to the individual estimator at the optimal bandwidth. We examine the finite-sample performance of a combined estimator that minimizes the trace of the MSE of a linear combination of individual kernel estimators for a multimodal density. The combined estimator provides a robust alternative to individual estimators that protects against uncertainty about the degree of smoothness.

We show how to use a smoothed empirical likelihood approach to conduct efficient semiparametric inference in models characterized as conditional moment equalities when data are collected by variable probability sampling. Results from a simulation experiment suggest that the smoothed empirical likelihood based estimator can estimate the model parameters very well in small to moderately sized stratified samples.