Improvement in estimating the population mean in simple random sampling using information on auxiliary attribute

This paper proposes some estimators for the population mean by adapting the estimator in Singh et al. (2008) [5] to the ratio estimators presented in Kadilar and Cingi 2006 [2]. We obtain mean square error (MSE) equation for all proposed estimators, and show that all proposed estimators are always more efficient than ratio estimator in Naik and Gupta (1996) [3], and Singh et al. (2008) [5]. The results have been illustrated numerically by taking some empirical population considered in the literature.     

比类型估计量在数量特征随机化回答技术中的应用

本文在数量特征随机化回答技术中当变异系数、偏度系数、峰度系数已知时,对总体均值提出了一系列比类型估计量,并且在一定条件下,证明了这些估计量优于Gupta et al.提出的估计量。     

Estimation of two normal means which may be common

Consider the problem of estimating the mean of a normal population when independent samples from this as well as a second normal population are available. Pre-test estimators which combine the two sample means if a test of the hypothesis of equal population means accepts but otherwise use only the first sample mean, are compared to limited translation estimators which are derived in the spirit of Bickel (1984, Ann. Statist., 12, 864–879) (we also cover the cases of unknown variances). Our conclusion is that if the accuracy with which the second population mean can be estimated is of the same or better order of magnitude as teh accuracy with which the first can be estimated, then the limited translation estimators largely dominate the pre-test estimators in terms of mean square error loss.This research was supported by grants from the FRD of the CSIR of South Africa.     

Statistical inference for DEA estimators of directional distances

In productivity and efficiency analysis, the technical efficiency of a production unit is measured through its distance to the efficient frontier of the production set. The most familiar non-parametric methods use Farrell–Debreu, Shephard, or hyperbolic radial measures. These approaches require that inputs and outputs be non-negative, which can be problematic when using financial data. Recently, Chambers et al. (1998) have introduced directional distance functions which can be viewed as additive (rather than multiplicative) measures efficiency. Directional distance functions are not restricted to non-negative input and output quantities; in addition, the traditional input and output-oriented measures are nested as special cases of directional distance functions. Consequently, directional distances provide greater flexibility. However, until now, only free disposal hull (FDH) estimators of directional distances (and their conditional and robust extensions) have known statistical properties (Simar and Vanhems, 2012). This paper develops the statistical properties of directional d estimators, which are especially useful when the production set is assumed convex. We first establish that the directional Data Envelopment Analysis (DEA) estimators share the known properties of the traditional radial DEA estimators. We then use these properties to develop consistent bootstrap procedures for statistical inference about directional distance, estimation of confidence intervals, and bias correction. The methods are illustrated in some empirical examples.     

缺失数据下Jackknife方差估计量的渐近设计无偏性

在实际的调查中经常会出现缺失数据．如何处理这种情况下总体目标量的估计是一个重要问题．Zou等(2002)对缺失数据情况下的样本轮换方法证明了他们所提出的线性化．Jackknife方差估计量在均匀回答下是估计量方差的近似的设计无偏估计．这一性质对于．Jackknife方差估计量的使用提供了重要依据．对于其它情况下．Jackknife方差估计量是否也具有这一性质无疑是一个有意义的问题．作者旨在研究文献中已提出的若干．Jackknife方差估计量的渐近设计无偏性．我们的结果表明Zou等(2002)所注意到的Jackknife方差估计量的渐近设计无偏性具有一定的普遍性。     

Maximum likelihood estimation of covariance matrices under simple tree ordering

The closed-form maximum likelihood estimators for the completely balanced multivariate one-way random effect model are obtained by Anderson et al. (Ann. Statist. 14 (1986) 405). It remains open whether there exist the closed-form maximum likelihood estimators for the more general completely balanced multivariate multi-way random effects models. In this paper, a new parameterization technique for covariance matrices is used to grasp the inside structure of likelihood function so that the maximum likelihood equations can be dramatically simplified. As such we obtain the closed-form maximum likelihood estimators of covariance matrices for Wishart density functions over the simple tree ordering set, which can then be applied to get the maximum likelihood estimators for the completely balanced multivariate multi-way random effects models without interactions.     

Variance comparison between infinitesimal perturbation analysis and likelihood ratio estimators to stochastic gradient

We theoretically compare variances between the Infinitesimal Perturbation Analysis (IPA) estimator and the Likelihood Ratio (LR) estimator to Monte Carlo gradient for stochastic systems. The results presented in Cui et al. (2020) [2] on variance comparison between these two estimators are substantially improved. We also prove a practically interesting result that the IPA estimators to European vanilla and arithmetic Asian options' Delta, respectively, have smaller variance when the underlying asset's return process is independent with the initial price and square integrable.     

On sequential estimation for branching processes with immigration

Consider a Galton–Watson process with immigration. The limiting distributions of the nonsequential estimators of the offspring mean have been proved to be drastically different for the critical case and subcritical and supercritical cases. A sequential estimator, proposed by Sriram et al. (Ann. Statist. 19 (1991) 2232), was shown to be asymptotically normal for both the subcritical and critical cases. Based on a certain stopping rule, we construct a class of two-stage estimators for the offspring mean. These estimators are shown to be asymptotically normal for all the three cases. This gives, without assuming any prior knowledge, a unified estimation and inference procedure for the offspring mean.     

Parameter Estimation of Multivariate Regression Model with Fuzzy Random Errors

In many real-world problems, observations are usually described by approximate values due to fuzzy uncertainty, unlikeprobabilistic uncertainty that has nothing to do with experimentation. The combination of statistical model and fuzzy set theory is helpful to improve the identification and analysis of complex systems. As an extension ofstatistical techniques, this study is an investigation of the relationship between fuzzy multiple explanatory variables and fuzzy response with numeric coefficients and the fuzzy random error term. In this work we describe a parameter estimation procedure carrying out the least-squares method in a complete metric space of fuzzy numbers to determine the coefficients based on the extension principle. We demonstrate how the fuzzy least squares estimators present large sample statistical properties, including asymptotic normality, strong consistency and confidence region. The estimators are also examined via asymptotic relative efficiency concerning traditional least squares estimators. Different from the construction of error term in Kim et al.\cite{21}, it is more reasonable in the proposed model since the problems of inconsistency in referring to fuzzy variable and producing the negative spreads may be avoided. The experimental study verifies that the proposed fuzzy least squares estimators achieve the meaning consistent with the theory identification for large sample data set and better generalization regarding one single variable model.     

Vitesses de convergence uniforme presque sûre d'estimateurs non-paramétriques de la régression

We obtain rates of strong uniform consistency for some nonparametric regression estimators, including the local linear regression and some wavelet estimators. Our method of proof relies on recent empirical process theory developed by Deheuvels and Mason (Statist. Inference Stoch. Process. 7 (3) (2004) 225–277). To cite this article: D. Blondin et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Asymptotic properties of parallel Bayesian kernel density estimators

Annals of the Institute of Statistical Mathematics - In this article, we perform an asymptotic analysis of Bayesian parallel kernel density estimators introduced by Neiswanger et al. (in:...     

Interval estimation of the mean of a normal distribution based on quantized observations

We consider the problem of making statistical inference about the mean of a normal distribution based on a random sample of quantized (digitized) observations. This problem arises, for example, in a measurement process with errors drawn from a normal distribution and with a measurement device or process with a known resolution, such as the resolution of an analog-to-digital converter or another digital instrument. In this paper we investigate the effect of quantization on subsequent statistical inference about the true mean. If the standard deviation of the measurement error is large with respect to the resolution of the indicating measurement device, the effect of quantization (digitization) diminishes and standard statistical inference is still valid. Hence, in this paper we consider situations where the standard deviation of the measurement error is relatively small. By Monte Carlo simulations we compare small sample properties of the interval estimators of the mean based on standard approach (i.e. by ignoring the fact that the measurements have been quantized) with some recently suggested methods, including the interval estimators based on maximum likelihood approach and the fiducial approach. The paper extends the original study by Hannig et al. (2007).     

Non-periodic averaging principles for measure functional differential equations and functional dynamic equations on time scales involving impulses

We present a non-periodic averaging principle for measure functional differential equations and, using the correspondence between solutions of measure functional differential equations and solutions of functional dynamic equations on time scales (see Federson et al., 2012 [8]), we obtain a non-periodic averaging result for functional dynamic equations on time scales. Moreover, using the relation between measure functional differential equations and impulsive measure functional differential equations, we get a non-periodic averaging theorem for these equations. Also, it is a known fact that we can relate impulsive measure functional differential equations and impulsive functional dynamic equations on time scales (see Federson et al., 2013 [9]). Therefore, applying this correspondence to our averaging principle, we obtain a non-periodic averaging theorem for impulsive functional dynamic equations on time scales.     

On Exponential Representations of Log-Spacings of Extreme Order Statistics

In Beirlant et al. (1999) and Feuerverger and Hall (1999) an exponential regression model (ERM) was introduced on the basis of scaled log-spacings between subsequent extreme order statistics from a Pareto-type distribution. This lead to the construction of new bias-corrected estimators for the tail index. In this note, under quite general conditions, asymptotic justification for this regression model is given as well as for resulting tail index estimators. Also, we discuss diagnostic methods for adaptive selection of the threshold when using the Hill (1975) estimator which follow from the ERM approach. We show how the diagnostic presented in Guillou and Hall (2001) is linked to the ERM, while a new proposal is suggested. We also provide some small sample comparisons with other existing methods.     

On the Parameters Estimators for a Discrete Analog of the Generalized Exponential Distribution

Nekoukhou et. al (Commun. Statist. Th. Meth., 2012) introduced a two-parameters discrete probability distribution so-called Discrete Analog of the Generalized Exponential Distribution (in short, DGED). We shall attempt to derive conditions under which a solution for the system of likelihood equations exists and coincides with the maximum likelihood (ML) estimators of the DGED. This kind of ML estimators are coincided with some moment estimators. An approximate computation based on Fisher’s accumulation method is presented in order for the ML estimations of the unknown parameters. Simulation study is also illustrated. Meanwhile, in the sequel two special cases of the DGED are considered. Some statistical properties for such special cases of the DGED are provided. We also propose a linear regression-type model for estimation of the parameter. Finally, we fit the DGED to a real data set and compare it with two other discrete distributions.     

Hybrid estimators for stochastic differential equations from reduced data

We treat parametric inference for unknown parameters of stochastic differential equations from discrete observations from the viewpoint of computational cost. Following Kamatani et al. (Bull Inf Cybern 48:19–35, 2016) and Kaino and Uchida (Hybrid estimators for ergodic diffusion processes from thinned data, 2018), we present the asymptotic results of the multi-step estimators with the initial Bayes type estimators for both ergodic and non-ergodic diffusion type processes. The initial Bayes type estimators are constructed by means of both the reduced data and the thinned data obtained from the full data. Some examples and simulation results are also given.     

Semi-parametric estimation of partially linear single-index models

One of the most difficult problems in applications of semi-parametric partially linear single-index models (PLSIM) is the choice of pilot estimators and complexity parameters which may result in radically different estimators. Pilot estimators are often assumed to be root-n consistent, although they are not given in a constructible way. Complexity parameters, such as a smoothing bandwidth are constrained to a certain speed, which is rarely determinable in practical situations.In this paper, efficient, constructible and practicable estimators of PLSIMs are designed with applications to time series. The proposed technique answers two questions from Carroll et al. [Generalized partially linear single-index models, J. Amer. Statist. Assoc. 92 (1997) 477-489]: no root-n pilot estimator for the single-index part of the model is needed and complexity parameters can be selected at the optimal smoothing rate. The asymptotic distribution is derived and the corresponding algorithm is easily implemented. Examples from real data sets (credit-scoring and environmental statistics) illustrate the technique and the proposed methodology of minimum average variance estimation (MAVE).     

Asymptotic Properties of Backfitting Estimators

When additive models with more than two covariates are fitted with the backfitting algorithm proposed by Buja et al. [2], the lack of explicit expressions for the estimators makes study of their theoretical properties cumbersome. Recursion provides a convenient way to extend existing theoretical results for bivariate additive models to models of arbitrary dimension. In the case of local polynomial regression smoothers, recursive asymptotic bias and variance expressions for the backfitting estimators are derived. The estimators are shown to achieve the same rate of convergence as those of univariate local polynomial regression. In the case of independence between the covariates, non-recursive bias and variance expressions, as well as the asymptotically optimal values for the bandwidth parameters, are provided.     

Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming

Sample average approximation (SAA) is one of the most popular methods for solving stochastic optimization and equilibrium problems. Research on SAA has been mostly focused on the case when sampling is independent and identically distributed (iid) with exceptions (Dai et al. (2000) [9], Homem-de-Mello (2008) [16]). In this paper we study SAA with general sampling (including iid sampling and non-iid sampling) for solving nonsmooth stochastic optimization problems, stochastic Nash equilibrium problems and stochastic generalized equations. To this end, we first derive the uniform exponential convergence of the sample average of a class of lower semicontinuous random functions and then apply it to a nonsmooth stochastic minimization problem. Exponential convergence of estimators of both optimal solutions and M-stationary points (characterized by Mordukhovich limiting subgradients (Mordukhovich (2006) [23], Rockafellar and Wets (1998) [32])) are established under mild conditions. We also use the unform convergence result to establish the exponential rate of convergence of statistical estimators of a stochastic Nash equilibrium problem and estimators of the solutions to a stochastic generalized equation problem.     

Large and moderate deviations of realized covolatility

In this note, we consider the large and moderate deviation principle of the estimators of the integrated covariance of two-dimensional diffusion processes when they are observed only at discrete times in a synchronous manner. The proof is extremely simple. It is essentially an application of the contraction principle for the results given in the case of the volatility by Djellout et al. (1999).