scholarly journals Infill Asymptotics and Bandwidth Selection for Kernel Estimators of Spatial Intensity Functions

2019 ◽  
Vol 22 (3) ◽  
pp. 995-1008
Author(s):  
M. N. M. van Lieshout
Keyword(s):  
Point Process ◽  
Mean Squared Error ◽  
Bandwidth Selection ◽  
Intensity Function ◽  
Kernel Estimators ◽  
Infill Asymptotics ◽  
Squared Error ◽  
Selection For ◽  
Asymptotic Mean Squared Error ◽  
Intensity Functions

AbstractWe investigate the asymptotic mean squared error of kernel estimators of the intensity function of a spatial point process. We derive expansions for the bias and variance in the scenario that n independent copies of a point process in $\mathbb {R}^{d}$ ℝ d are superposed. When the same bandwidth is used in all d dimensions, we show that an optimal bandwidth exists and is of the order n− 1/(d+ 4) under appropriate smoothness conditions on the true intensity function.

scholarly journals New Bandwidth Selection for Kernel Quantile Estimators

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Ali Al-Kenani ◽  
Keming Yu
Keyword(s):  
Cross Validation ◽  
Kernel Smoothing ◽  
Bandwidth Selection ◽  
Unbiased Estimation ◽  
Optimal Bandwidth ◽  
Squared Error ◽  
Bandwidth Parameter ◽  
Integrated Squared Error ◽  
Selection For ◽  
Quantile Estimators

We propose a cross-validation method suitable for smoothing of kernel quantile estimators. In particular, our proposed method selects the bandwidth parameter, which is known to play a crucial role in kernel smoothing, based on unbiased estimation of a mean integrated squared error curve of which the minimising value determines an optimal bandwidth. This method is shown to lead to asymptotically optimal bandwidth choice and we also provide some general theory on the performance of optimal, data-based methods of bandwidth choice. The numerical performances of the proposed methods are compared in simulations, and the new bandwidth selection is demonstrated to work very well.

scholarly journals OPTIMAL BANDWIDTH SELECTION FOR ROBUST GENERALIZED METHOD OF MOMENTS ESTIMATION

2014 ◽  
Vol 31 (5) ◽  
pp. 1054-1077 ◽  
Author(s):  
Daniel Wilhelm
Keyword(s):  
Method Of Moments ◽  
Mean Squared Error ◽  
Order Approximation ◽  
Generalized Method Of Moments ◽  
Bandwidth Selection ◽  
Estimation Procedure ◽  
Point Estimate ◽  
Optimal Bandwidth ◽  
Squared Error ◽  
Method Of Moments Estimation

A two-step generalized method of moments estimation procedure can be made robust to heteroskedasticity and autocorrelation in the data by using a nonparametric estimator of the optimal weighting matrix. This paper addresses the issue of choosing the corresponding smoothing parameter (or bandwidth) so that the resulting point estimate is optimal in a certain sense. We derive an asymptotically optimal bandwidth that minimizes a higher-order approximation to the asymptotic mean-squared error of the estimator of interest. We show that the optimal bandwidth is of the same order as the one minimizing the mean-squared error of the nonparametric plugin estimator, but the constants of proportionality are significantly different. Finally, we develop a data-driven bandwidth selection rule and show, in a simulation experiment, that it may substantially reduce the estimator’s mean-squared error relative to existing bandwidth choices, especially when the number of moment conditions is large.

A Class of Estimators of the Population Mean Using Multi­Auxiliary Information

1983 ◽  
Vol 32 (1-2) ◽  
pp. 47-56 ◽  
Author(s):  
S. K. Srivastava ◽  
H. S. Jhajj
Keyword(s):  
Mean Squared Error ◽  
Broad Class ◽  
Auxiliary Variable ◽  
Population Parameters ◽  
Auxiliary Variables ◽  
Sample Mean ◽  
Squared Error ◽  
Class Of Estimators ◽  
The Mean ◽  
Asymptotic Mean Squared Error

For estimating the mean of a finite population, Srivastava and Jhajj (1981) defined a broad class of estimators which we information of the sample mean as well as the sample variance of an auxiliary variable. In this paper we extend this class of estimators to the case when such information on p(> 1) auxiliary variables is available. The estimators of the class involve unknown constants whose optimum values depend on unknown population parameters. When these population parameters are replaced by their consistent estimates, the resulting estimators are shown to have the same asymptotic mean squared error. An expression by which the mean squared error of such estimators is smaller than those which use only the population means of the auxiliary variables, is obtained.

scholarly journals CONSISTENCY OF KERNEL-TYPE ESTIMATORS FOR THE FIRST AND SECOND DERIVATIVES OF A PERIODIC POISSON INTENSITY FUNCTION

2007 ◽  
Vol 6 (2) ◽  
pp. 47
Author(s):  
I W. MANGKU ◽  
S. SYAMSURI ◽  
H. HERNIWAT
Keyword(s):  
Mean Squared Error ◽  
Intensity Function ◽  
Parametric Form ◽  
Bounded Interval ◽  
Squared Error ◽  
Single Realization ◽  
Second Derivatives ◽  
The Mean ◽  
Derivatives Of ◽  
Kernel Type

<p>We construct and investigate consistent kernel-type estimators for the first and second derivatives of a periodic Poisson intensity function when the period is known. We do not assume any particular parametric form for the intensity function. More- over, we consider the situation when only a single realization of the Poisson process is available, and only observed in a bounded interval. We prove that the proposed estimators are consistent when the length of the interval goes to infinity. We also prove that the mean-squared error of the estimators converge to zero when the length of the interval goes to infinity.<br />1991 Mathematics Subject Classification: 60G55, 62G05, 62G20.</p>

Rates of Expansions for Functional Estimators

2021 ◽  
Author(s):  
Yulia Kotlyarova ◽  
Marcia M. A. Schafgans ◽  
Victoria Zinde-Walsh
Keyword(s):  
Convergence Rates ◽  
Mean Squared Error ◽  
Model Averaging ◽  
Finite Sample ◽  
Squared Error ◽  
Semiparametric Estimators ◽  
Asymptotic Mean Squared Error ◽  
The Impact ◽  
Nonconvex Model

AbstractIn this paper, we summarize results on convergence rates of various kernel based non- and semiparametric estimators, focusing on the impact of insufficient distributional smoothness, possibly unknown smoothness and even non-existence of density. In the presence of a possible lack of smoothness and the uncertainty about smoothness, methods of safeguarding against this uncertainty are surveyed with emphasis on nonconvex model averaging. This approach can be implemented via a combined estimator that selects weights based on minimizing the asymptotic mean squared error. In order to evaluate the finite sample performance of these and similar estimators we argue that it is important to account for possible lack of smoothness.

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