Semiparametric jump-preserving estimation for single-index models

Estimation of the single-index model with a discontinuous unknown link function is considered in this paper. Existed refined minimum average variance estimation (rMAVE) method can estimate the single-index parameter and unknown link function simultaneously by minimising the average pointwise conditional variance, where the conditional variance can be estimated using the local linear fit method with centred kernel function. When there are jumps in the link function, big biases around jumps can appear. For this reason, we embed the jump-preserving technique in the rMAVE method, then propose an adaptive jump-preserving estimation procedure for the single-index model. Concretely speaking, the conditional variance is obtained by the one among local linear fits with centred, left-sided and right-sided kernel functions who has minimum weighted residual mean squares. The resulting estimators can preserve the jumps well and also give smooth estimates of the continuity parts. Asymptotic properties are established under some mild conditions. Simulations and real data analysis show the proposed method works well.     

Estimation of entropy-type integral functionals

ABSTRACT

Entropy-type integral functionals of densities are widely used in mathematical statistics, information theory, and computer science. Examples include measures of closeness between distributions (e.g., density power divergence) and uncertainty characteristics for a random variable (e.g., Rényi entropy). In this paper, we study U-statistic estimators for a class of such functionals. The estimators are based on ε-close vector observations in the corresponding independent and identically distributed samples. We prove asymptotic properties of the estimators (consistency and asymptotic normality) under mild integrability and smoothness conditions for the densities. The results can be applied in diverse problems in mathematical statistics and computer science (e.g., distribution identification problems, approximate matching for random databases, two-sample problems).     

A Note on Collapsibility in DAG Models of Contingency Tables

Abstract.  Necessary and sufficient conditions for collapsibility of a directed acyclic graph (DAG) model for a contingency table are derived. By applying the conditions, we can easily check collapsibility over any variable in a given model either by using the joint probability distribution or by using the graph of the model structure. It is shown that collapsibility over a set of variables can be checked in a sequential manner. Furthermore, a DAG is compared with its moral graph in the context of collapsibility.     

Consistency of Generalized Maximum Spacing Estimates

General methods for the estimation of distributions can be derived from approximations of certain information measures. For example, both the maximum likelihood (ML) method and the maximum spacing (MSP) method can be obtained from approximations of the Kullback–Leibler information. The ideas behind the MSP method, whereby an estimation method for continuous univariate distributions is obtained from an approximation based on spacings of an information measure, were used by Ranneby & Ekstrom (1997) (using simple spacings) and Ekstrom (1997b) (using high order spacings) to obtain a class of methods, called generalized maximum spacing (GMSP) methods. In the present paper, GMSP methods will be shown to give consistent estimates under general conditions, comparable to those of Bahadur (1971) for the ML method, and those of Shao & Hahn (1999) for the MSP method. In particular, it will be proved that GMSP methods give consistent estimates in any family of distributions with unimodal densities, without any further conditions on the distributions.     

Inference in the presence of redundant moment conditions and the impact of government health expenditure on health outcomes in England

In his 1999 article with Breusch, Qian, and Wyhowski in the Journal of Econometrics, Peter Schmidt introduced the concept of “redundant” moment conditions. Such conditions arise when estimation is based on moment conditions that are valid and can be divided into two subsets: one that identifies the parameters and another that provides no further information. Their framework highlights an important concept in the moment-based estimation literature, namely, that not all valid moment conditions need be informative about the parameters of interest. In this article, we demonstrate the empirical relevance of the concept in the context of the impact of government health expenditure on health outcomes in England. Using a simulation study calibrated to this data, we perform a comparative study of the finite performance of inference procedures based on the Generalized Method of Moment (GMM) and info-metric (IM) estimators. The results indicate that the properties of GMM procedures deteriorate as the number of redundant moment conditions increases; in contrast, the IM methods provide reliable point estimators, but the performance of associated inference techniques based on first order asymptotic theory, such as confidence intervals and overidentifying restriction tests, deteriorates as the number of redundant moment conditions increases. However, for IM methods, it is shown that bootstrap procedures can provide reliable inferences; we illustrate such methods when analysing the impact of government health expenditure on health outcomes in England.     

Inference with a contrast-based posterior distribution and application in spatial statistics

The likelihood function is often used for parameter estimation. Its use, however, may cause difficulties in specific situations. In order to circumvent these difficulties, we propose a parameter estimation method based on the replacement of the likelihood in the formula of the Bayesian posterior distribution by a function which depends on a contrast measuring the discrepancy between observed data and a parametric model. The properties of the contrast-based (CB) posterior distribution are studied to understand what the consequences of incorporating a contrast in the Bayes formula are. We show that the CB-posterior distribution can be used to make frequentist inference and to assess the asymptotic variance matrix of the estimator with limited analytical calculations compared to the classical contrast approach. Even if the primary focus of this paper is on frequentist estimation, it is shown that for specific contrasts the CB-posterior distribution can be used to make inference in the Bayesian way.The method was used to estimate the parameters of a variogram (simulated data), a Markovian model (simulated data) and a cylinder-based autosimilar model describing soil roughness (real data). Even if the method is presented in the spatial statistics perspective, it can be applied to non-spatial data.     

Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH processes

We consider estimation of a class of power-transformed threshold GARCH models. When the power of the transformation is known, the asymptotic properties of the quasi-maximum likelihood estimator (QMLE) are established under mild conditions. Two sequences of least-squares estimators are also considered in the pure ARCH case, and it is shown that they can be asymptotically more accurate than the QMLE for certain power transformations. In the case where the power of the transformation has to be estimated, the asymptotic properties of the QMLE are proven under the assumption that the noise has a density. The finite-sample properties of the proposed estimators are studied by simulation.     

A new lifetime distribution for a series-parallel system: properties,applications and estimations under progressive type-II censoring

A compound class of zero truncated Poisson and lifetime distributions is introduced. A specialization is paved to a new three-parameter distribution, called doubly Poisson-exponential distribution, which may represent the lifetime of units connected in a series-parallel system. The new distribution can be obtained by compounding two zero truncated Poisson distributions with an exponential distribution. Among its motivations is that its hazard rate function can take different shapes such as decreasing, increasing and upside-down bathtub depending on the values of its parameters. Several properties of the new distribution are discussed. Based on progressive type-II censoring, six estimation methods [maximum likelihood, moments, least squares, weighted least squares and Bayes (under linear-exponential and general entropy loss functions) estimations] are used to estimate the involved parameters. The performance of these methods is investigated through a simulation study. The Bayes estimates are obtained using Markov chain Monte Carlo algorithm. In addition, confidence intervals, symmetric credible intervals and highest posterior density credible intervals of the parameters are obtained. Finally, an application to a real data set is used to compare the new distribution with other five distributions.     

Dynamic Conditional Correlation: On Properties and Estimation

This article addresses some of the issues that arise with the Dynamic Conditional Correlation (DCC) model. It is proven that the DCC large system estimator can be inconsistent, and that the traditional interpretation of the DCC correlation parameters can result in misleading conclusions. Here, we suggest a more tractable DCC model, called the cDCC model. The cDCC model allows for a large system estimator that is heuristically proven to be consistent. Sufficient stationarity conditions for cDCC processes of interest are established. The empirical performances of the DCC and cDCC large system estimators are compared via simulations and applications to real data.     

An On-Line Estimation Algorithm for Periodic Autoregressive Models

This paper develops an on-line estimation algorithm for periodic autoregressive models (PAR). Indeed, we provide an adaptation of the well known recursive least squares algorithm (RLS), which has been successfully applied to classical autoregressive models (AR), to deal with PAR models. The obtained estimators are shown to be asymptotically efficient under mild conditions. Moreover, the performance of the periodic least squares algorithm (PRLS) is assessed via an intensive simulation study.     

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.     

概化理论两侧面设计方差分量及其变异量估计方法比较

文章针对正态分布数据,对比Traditional方法、Bootstrap方法和MCMC方法在两侧面交叉设计(p×i×h)和两侧面嵌套设计(p×(i:h))下各个方差分量的估计精度,为实际应用提供参考。使用R软件模拟1000批数据,并在R软件上实现三种方法的方差分量及其变异量估计。结果表明:(1)相较于Traditional方法和MCMC方法,相同条件下,Bootstrap方法估计的方差分量及其变异量结果更为理想;(2)对于两侧面交叉设计和两侧面嵌套设计,在正态分布数据下,建议优先使用Bootstrap方法。     

Correction of Density Estimators that are not Densities

Abstract. Several old and new density estimators may have good theoretical performance, but are hampered by not being bona fide densities; they may be negative in certain regions or may not integrate to 1. One can therefore not simulate from them, for example. This paper develops general modification methods that turn any density estimator into one which is a bona fide density, and which is always better in performance under one set of conditions and arbitrarily close in performance under a complementary set of conditions. This improvement-for-free procedure can, in particular, be applied for higher-order kernel estimators, classes of modern h ⁴ bias kernel type estimators, superkernel estimators, the sinc kernel estimator, the k -NN estimator, orthogonal expansion estimators, and for various recently developed semi-parametric density estimators.     

Estimation of Stigmatized Characteristics of a Hidden Gang in a Finite Population

This paper considers the problem of estimating the size and mean value of a stigmatized quantitative character of a hidden gang in a finite population. The proposed method may be applied to solve domestic problems in a particular country or across countries: for example, a government may be interested in estimating the average income of victims or perpetrators of domestic violence. The proposed method is based on the technique introduced by Warner (1965) to estimate the proportion of a sensitive attribute in a finite population without threatening the privacy of the respondents. Expressions for the bias and variance of the proposed estimators are given, to a first order of approximation. Circumstances in which the method can be applied are studied and illustrated using a numerical example.     

A nonparametric estimation procedure for the Hawkes process: comparison with maximum likelihood estimation

In earlier work, Kirchner [An estimation procedure for the Hawkes process. Quant Financ. 2017;17(4):571–595], we introduced a nonparametric estimation method for the Hawkes point process. In this paper, we present a simulation study that compares this specific nonparametric method to maximum-likelihood estimation. We find that the standard deviations of both estimation methods decrease as power-laws in the sample size. Moreover, the standard deviations are proportional. For example, for a specific Hawkes model, the standard deviation of the branching coefficient estimate is roughly 20% larger than for MLE – over all sample sizes considered. This factor becomes smaller when the true underlying branching coefficient becomes larger. In terms of runtime, our method clearly outperforms MLE. The present bias of our method can be well explained and controlled. As an incidental finding, we see that also MLE estimates seem to be significantly biased when the underlying Hawkes model is near criticality. This asks for a more rigorous analysis of the Hawkes likelihood and its optimization.     

Robust estimators for additive models using backfitting

Additive models provide an attractive setup to estimate regression functions in a nonparametric context. They provide a flexible and interpretable model, where each regression function depends only on a single explanatory variable and can be estimated at an optimal univariate rate. Most estimation procedures for these models are highly sensitive to the presence of even a small proportion of outliers in the data. In this paper, we show that a relatively simple robust version of the backfitting algorithm (consisting of using robust local polynomial smoothers) corresponds to the solution of a well-defined optimisation problem. This formulation allows us to find mild conditions to show Fisher consistency and to study the convergence of the algorithm. Our numerical experiments show that the resulting estimators have good robustness and efficiency properties. We illustrate the use of these estimators on a real data set where the robust fit reveals the presence of influential outliers.     

Efficiently weighted estimating equations with application to proportional excess hazards

A general approach to estimation, that can lead to efficient estimation in two stages, is presented. The method will not always be available, but sufficient conditions for efficiency are provided together with four examples of its use: (1) estimation of the odds ratio in 1:M matched case-control studies with a dichotomous exposure variable; (2) estimation of the relative hazard in a two-sample survival setting; (3) estimation of the regression parameters in the proportional excess hazards model; and (4) estimation in a partly linear parametric additive hazards model. The method depends upon finding a family of weighted estimating equations, which includes a simple initial equation yielding a consistent estimate and also an equation that yields an efficient estimate, provided the optiomal weights are used.     

On the Estimation of Parameters of Kumaraswamy-G Distributions

Cordeiro and de Castro proposed a new family of generalized distributions based on the Kumaraswamy distribution (denoted as Kw-G). Nadarajah et al. showed that the density function of the new family of distributions can be expressed as a linear combination of the density of exponentiated family of distributions. They derived some properties of Kw-G distributions and discussed estimation of parameters using the maximum likelihood (ML) method. Cheng and Amin and Ranneby introduced a new method of estimating parameters based on Kullback–Leibler divergence (the maximum spacing (MSP) method). In this article, the estimates of parameters of Kw-G distributions are obtained using the MSP method. For some special Kw-G distributions, the new estimators are compared with ML estimators. It is shown by simulations and a real data application that MSP estimators have better properties than ML estimators.     

Recursive estimation of the mode of a multivariate density

Let f be an unknown possibly multimodal density on R^d and let X₁, X₂, … be a sequence of independent random vectors with density f. Several recursive estimates of the mode of f are proposed, and sufficient conditions ensuring their weak and strong consistency are established.     

ASYMPTOTIC PROPERTIES OF RESTRICTED MAXIMUM LIKELIHOOD (REML) ESTIMATES FOR HIERARCHICAL MIXED LINEAR MODELS

This paper explores the asymptotic distribution of the restricted maximum likelihood estimator of the variance components in a general mixed model. Restricting attention to hierarchical models, central limit theorems are obtained using elementary arguments with only mild conditions on the covariates in the fixed part of the model and without having to assume that the data are either normally or spherically symmetrically distributed. Further, the REML and maximum likelihood estimators are shown to be asymptotically equivalent in this general framework, and the asymptotic distribution of the weighted least squares estimator (based on the REML estimator) of the fixed effect parameters is derived.