Smooth estimation of a monotone density

We investigate the interplay of smoothness and monotonicity assumptions when estimating a density from a sample of observations. The nonparametric maximum likelihood estimator of a decreasing density on the positive half line attains a rate of convergence of [Formula: See Text] at a fixed point t if the density has a negative derivative at t. The same rate is obtained by a kernel estimator of bandwidth [Formula: See Text], but the limit distributions are different. If the density is both differentiable at t and known to be monotone, then a third estimator is obtained by isotonization of a kernel estimator. We show that this again attains the rate of convergence [Formula: See Text], and compare the limit distributions of the three types of estimators. It is shown that both isotonization and smoothing lead to a more concentrated limit distribution and we study the dependence on the proportionality constant in the bandwidth. We also show that isotonization does not change the limit behaviour of a kernel estimator with a bandwidth larger than [Formula: See Text], in the case that the density is known to have more than one derivative.     

Optimal estimation of a finite population mean under generalized random permutation models

Employing certain generalized random permutation models and a general class of linear estimators of a finite population mean, it is shown that many of the conventional estimators are “optimal” in the sense of minimum average mean square error. Simple proofs are provided by using a well-known theorem on UMV estimation. The results also cover certain simple response error situations.     

A class of Shrunken estimators for normal mean

This paper presents a class of estimators for the mean of a normal population and determines the conditions on characterizing scalars under which the class of estimators uniformly dominates over the conventional sample mean according to the mean-square-error criterion.     

Estimation of the mean of the exponential distribution using moving extremes ranked set sampling

Moving Extremes Ranked Set Sampling (MERSS) is a useful modification of Ranked Set Sampling (RSS). Unlike RSS, MERSS allows for an increase of set size without introducing too much ranking error. The method is considered parametrically under exponential distribution. Maximum likelihood estimator (MLE), and a modified MLE are considered and their properties are studied. The method is studied under both perfect and imperfect ranking (with error in ranking). It appears that these estimators can be real competitors to the MLE using the usual simple random sampling (SRS).     

Asymptotic properties of the complex valued non-linear regression model

The non-linear regression model, when the parameters are complex valued is considered here. Jennrich(1969) considered the non-linear regression model when the parameters are real valued. He first rigorously proved the existence of the least square estimator and showed its consistency properties and asymptotic normality. In this paper we generalise the idea for the com-plex parameters case. Large sample properties of the proposed estimator has been studied.     

Edge Estimation in the Population of a Binary Tree Using Node-Sampling

Suppose a finite population of several vertices, each connected to single or multiple edges. This constitutes a structure of graphical population of vertices and edges. As a special case, the graphical population like a binary tree having only two child vertices associated to parent vertex is taken into consideration. The entire binary tree is divided into two sub-graphs such as a group of left-nodes and a group of right-nodes. This paper takes into account a mixture of graph structured and population sampling theory together and presents a methodology for mean-edge-length estimation of left sub-graph using right edge sub-graph as an auxiliary source of information. A node-sampling procedure is developed for this purpose and a class of estimators is proposed containing several good estimators. Mathematical conditions for minimum bias and optimum mean squared error of the class are derived and theoretical results are numerically supported with a test of 99% confidence intervals. It is shown that suggested class has a sub-class of optimum estimators, and sample-based estimates are closer to the true value of the population parameter.     

Finite sample properties of ml and reml estimators in time series regression models with long memory noise

Finite sample properties of ML and REML estimators in time series regression models with fractional ARIMA noise are examined. In particular, theoretical approximations for bias of ML and REML estimators of the noise parameters are developed and their accuracy is assessed through simulations. The impact of noise parameter estimation on performance of t -statistics and likelihood ratio statistics for testing regression parameters is also investigated.     

A note on properties of spatial yule-walker estimators

For two-dimensional spatial autoregressive (AR) models, asymptotic properties of the spatial Yule-Walker (YW) estimators (Tjøstheim, 1978) are studied. These estimators although consistent, are shown to be asymptotically biased. Estimators from the first-order spatial bilateral AR model are looked at in more detail and the spatial YW estimators for this model are compared with the exact maximum likelihood estimators. Small sample properties of both estimators are also discussed briefly and some simulation results are presented.     

On the mean L1-error in the heteroscedastic deconvolution problem with compactly supported noises

We study the heteroscedastic deconvolution problem when random noises have compactly supported densities. In this context, the Fourier transforms of the densities can vanish on the real line. We propose a truncated type of estimator for target density and derive the convergence rate of the mean L¹-error uniformly over a class of target densities. A lower bound for the mean L¹-error is also established. Some simulations will be given to illustrate the performance of the proposed estimator.     

异质性大数据的分布式估计

随着物联网技术的进步,大数据给网络带宽和计算机存储能力带来巨大挑战,传统的集中式数据处理难以实现,客观上促进了分布式统计学习的发展。在无迭代算法研究中,Zhang等(2013)证明了当数据集个数s=O(N) 时,基于局部经验风险最小化的分治(DC)简单平均估计量具有O(N-1)均方误差收敛速度,Huang和Huo(2019)在M估计框架下进一步提出分布式一步估计量,但上述方法均未考虑海量数据可能存在的异质性对分治估计效果的影响。本文在线性模型框架下提出海量异质数据的分治一步加权估计,证明了估计量的渐近性质并考虑了异质性检验问题。将本文提出的方法应用于美国医疗保险实际数据分析,结果表明该方法能更好地拟合数据的线性趋势且显著提高了计算效率。