Asymptotic Mean Integrated Square Error Research Articles

On the fourth-order hybrid beta polynomial kernels in kernel density estimation

This paper introduces a novel family of fourth-order hybrid beta polynomial kernels tailored for statistical analysis. The efficacy of these kernels is evaluated using two principal performance metrics: asymptotic mean integrated squared error (AMISE) and kernel efficiency. Comprehensive assessments were conducted using both simulated and real-world datasets, enabling a thorough comparison with conventional fourth-order polynomial kernels. The evaluation process entailed computing AMISE and efficiency metrics for both the hybrid and classical kernels. Consistently, the results illustrated the superior performance of the hybrid kernels over their classical counterparts across diverse datasets, underscoring the robustness and effectiveness of the hybrid approach. By leveraging these performance metrics and conducting evaluations on simulated and real world data, this study furnishes compelling evidence supporting the superiority of the proposed hybrid beta polynomial kernels. The heightened performance, evidenced by lower AMISE values and elevated efficiency scores, strongly advocates for the adoption of the proposed kernels in statistical analysis tasks, presenting a marked improvement over traditional kernels.

Generalised local polynomial estimators of smooth functionals of a distribution function with nonnegative support

This paper introduces generalised smooth asymmetric kernel estimators for smooth functionals with non-negative support. More precisely, for x ∈ [ 0 , ∞ ) , and for a functional, Φ ( x , F ) of the distribution function F, we develop estimators of the functional Φ and its derivatives. The proposed estimator can be seen as the solution to a minimisation problem in the polynomial space L 2 ( q ) , where q is an asymmetric density function. The framework presented here covers several classical nonparametric functional estimators and is linked with estimation using hierarchical kernels. We establish the asymptotic properties of the proposed estimators in the general framework. Furthermore, special attention is paid to comparing the asymptotic mean integrated square error (AMISE) of the proposed estimator with that of other classical symmetric/asymmetric density estimators. Additionally, a comparison of finite sample behaviour is conducted for both density estimation and hazard rate estimation via simulation and real data application.

On the convergence rate of d-dimensional fourth-order beta polynomial kernels

This article focuses on formulating the Asymptotic Mean Integrated Squared Error (AMISE) scheme for d-dimensional fourth-order beta polynomial kernels in the context of kernel density estimation. The primary objective is to assess how this scheme influences the convergence rate, which directly impacts the speed at which the estimated density converges to the true density, contributing to bias reduction. The article employs AMISE as a metric to quantify the overall dissimilarity between the estimated density fˆ and the true density f . Quantitatively, the proposed convergence scheme is compared to existing ones by Deheuvels and Jones et al. across various scenarios of sample sizes and dimensionalities. The study’s findings provide compelling evidence that fourth-order beta polynomial kernels exhibit a significantly faster convergence rate compared to the rates documented in the literature. This accelerated convergence rate implies a substantial improvement in bias reduction capabilities. The results underscore the potential effectiveness of fourth-order beta polynomial kernels as a powerful tool for enhancing the accuracy of kernel density estimation tasks.

TAKDE: Temporal Adaptive Kernel Density Estimator for Real-Time Dynamic Density Estimation.

Real-time density estimation is ubiquitous in many applications, including computer vision and signal processing. Kernel density estimation is arguably one of the most commonly used density estimation techniques, and the use of "sliding window" mechanism adapts kernel density estimators to dynamic processes. In this article, we derive the asymptotic mean integrated squared error (AMISE) upper bound for the "sliding window" kernel density estimator. This upper bound provides a principled guide to devise a novel estimator, which we name the temporal adaptive kernel density estimator (TAKDE). Compared to heuristic approaches for "sliding window" kernel density estimator, TAKDE is theoretically optimal in terms of the worst-case AMISE. We provide numerical experiments using synthetic and real-world datasets, showing that TAKDE outperforms other state-of-the-art dynamic density estimators (including those outside of kernel family). In particular, TAKDE achieves a superior test log-likelihood with a smaller run-time.

A statistical study of wind speed and its connectivity with relative humidity and temperature in Ughelli, Delta State, Nigeria

One of the vital climatic parameters with significant roles in many natural phenomena is wind. The importance of wind cannot be overemphasized due to its role as a source of renewable energy. The understanding of wind is of great importance particularly for the purpose of prediction and management of severe weather events. However, wind as a climatic parameter depends on relative humidity and temperature as well as other weather parameters and several statistical approaches such as time series analysis, extreme value analysis and spatial analysis have been used to analyze wind speed data. This study uses the kernel density method in analyzing wind speed in Ughelli, Delta State and its connection with relative humidity and temperature using the Gaussian kernel function for a period of five consecutive years from 2018 to 2022. The performance measure employ is the asymptotic mean integrated squared error (AMISE) with the Pearson R test that measures the strength of the relationship that exists between parameters. The results of the investigation with regards to the AMISE shows that 2018 recorded best performance with wind speed and relative humidity while 2021 recorded best performance for wind speed and temperature but 2019 recorded unsatisfactory outcomes for wind speed and the two parameters. This implies that human activities that depend on these parameters for their performance did best in 2018 and 2021 respectively. Furthermore, in terms of connectivity, wind speed and relative humidity are negatively correlated in 2018 and 2022 but positively correlated in 2019, 2020 and 2021 while wind speed and temperature are negatively correlated which implies that as temperature increases, wind speed decreases.

Methodology for nonparametric bias reduction in kernel regression estimation

Abstract In this paper, we propose and investigate two new kernel regression estimators based on a bias reduction transformation technique. We study the properties of these estimators and compare them with Nadaraya–Watson’s regression estimator and Slaoui’s (2016) regression estimator. It turns out that, with an adequate choice of the parameters of the two proposed estimators, the rate of convergence of two estimators will be faster than the two classical estimators, and the asymptotic MISE (mean integrated squared error) will be smaller than the two classical estimators. We corroborate these theoretical results through simulations and a real Malaria dataset.

Multivariate locally adaptive kernel density estimation

When the underlying density exhibits multiple modes with different scales and orientations, density estimators with locally adaptive smoothing parameters show substantial gains over those with fixed bandwidths. However, it is a concern that the local smoothing matrices may not be well parameterized, and the corresponding optimization problems will be difficult. In this paper, we build a more promising and practical algorithm. The local bandwidth factors are chosen through clustering, and the global smoothing parameter is achieved by optimizing the Asymptotic Mean Integrated Squared Error. Most importantly, our locally adaptive estimator involves optimizing a scalar rather than solving a costly multivariate optimization problem. Our method, which can also be applied to manifold density estimation, is an improvement and generalization of the binned version estimator of Sain.

The Interactions between Temperature and Relative Humidity: Results for Benin City, Nigeria using Statistical Analysis

The interactions between temperature and relative humidity are the main focus of this article. These two meteorological observations are of great significance due to their direct effects on humans and their environment. A series of studies on meteorological variables was carried out over a decade with emphasis on their effects on humans and their environment. Similar studies are going on due to the constant changes in the globe because of climate change. Appropriate measures can be provided for weather experts to curb some of the adverse effects of these variables. This article addresses the connectivity of temperature and relative humidity as well as their joint effects in Benin City, Nigeria for the period of ten years from 2010 to 2019 using nonparametric techniques that employ the Gaussian kernel estimator as analytic tool. The statistical analysis of the relationship between temperature and relative humidity vividly revealed that human activities were more successful in 2016 and 2017 under the period been studied with the asymptotic mean integrated squared error (AMISE) used as the performance measure.

A new family of kernels from the beta polynomial kernels with applications in density estimation

One of the fundamental data analytics tools in statistical estimation is the non-parametric kernel method that involves probability estimates production. The method uses the observations to obtain useful statistical information to aid the practicing statistician in decision making and further statistical investigations. The kernel techniques primarily examine essential characteristics in a data set, and this research aims to introduce new kernel functions that can easily detect inherent properties in any given observations. However, accurate application of kernel estimator as data analytics apparatus requires the kernel function and smoothing parameter that regulates the level of smoothness applied to the estimates. A plethora of kernel functions of different families and smoothing parameter selectors exist in the literature, but no one method is universally acceptable in all situations. Hence, more kernel functions with smoothing parameter selectors have been propounded customarily in density estimation. This article proposes a distinct kernel family from the beta polynomial kernel family using the exponential progression in its derivation. The newly proposed kernel family was evaluated with simulated and life data. The outcomes clearly indicated that this kernel family could compete favorably well with other kernel families in density estimation. A further comparison of numerical results of the new family and the existing beta family revealed that the new family outperformed the classical beta kernel family with simulation and real data examples with the aid of asymptotic mean integrated squared error (AMISE) as criterion function. The information obtained from the data analysis of this research could be used for decision making in an organization, especially when human and material resources are to be considered. In addition, Kernel functions are vital tools for data analysis and data visualization; hence the newly proposed functions are vital exploratory tools.

A Discourse on Smoothing Parameterizations using Hypothetical Dataset

The univariate kernel estimator usually requires a smoothing parameter, unlike the multi-dimensional estimators that necessarily require more smoothing parameters. The smoothing parameter(s) of kernels with a higher dimension may be called smoothing matrices. Kernels of higher dimensions have three kinds of parameterizations as estimators viz: constant, diagonal, and full parameterizations. Unlike the full parameterization, the diagonal parameterization exhibit some levels of restrictions. This study attempts to reconnoiter the coherence exhibited by kernel estimators especially where smoothing parameterizations are employed. In this discourse, asymptotic mean-integrated squared error(AMISE) is used as a criterion function and bivariate cases alone are considered. With some hypothetical data, the results show that full smoothing parameterization outperformed the constant and diagonal parameterizations in respect of the asymptotic mean-integrated squared error’s value and the kernel estimate’s ability to retain the true characteristics of the affected distribution.

Data Dependent Asymmetric Kernels for Estimating the Density Function

In the nonparametric setup, smooth density estimators are obtained by using a kernel function and it is used to account for contribution of each observation to the estimator. Generally, these kernel functions are symmetric about zero. We propose an estimator by using data dependent asymmetric kernels being generated by an unimodal symmetric kernel. To account for the contribution of the ith order statistic to the estimator, a suitable asymmetric kernel is used. Asymptotic Bias, Mean Square Error (MSE) and Mean Integrated Square Error (MISE) of the estimator are obtained with higher accuracies by using some properties of the densities of order statistics. The convergence rates depend on both n, the sample size and hn, the band width. Expressions for the optimum band width minimizing the asymptotic MISE and the rate of convergence of the optimum MISE are obtained. Simulation study and data analysis indicate that the proposed estimator not only reduces bias outside the support significantly but also it is closer to the true density function as compared to that of the usual estimator based on a common symmetric kernel. Scope for the multivariate generalization of the method is given. Some methods of generating multivariate skew symmetric kernels are described. As an illustration one of them has been used in the simulation study and estimation of density in bivariate case.

Bias reduction in kernel density estimation

ABSTRACTIn this paper, we propose two kernel density estimators based on a bias reduction technique. We study the properties of these estimators and compare them with Parzen–Rosenblatt's density estimator and Mokkadem, A., Pelletier, M., and Slaoui, Y. (2009, ‘The stochastic approximation method for the estimation of a multivariate probability density’, J. Statist. Plann. Inference, 139, 2459–2478) is density estimators. It turns out that, with an adequate choice of the parameters of the two proposed estimators, the rate of convergence of two estimators will be faster than the two classical estimators and the asymptotic MISE (Mean Integrated Squared Error) will be smaller than the two classical estimators. We corroborate these theoretical results through simulations.

BOOSTING AND BAGGING IN KERNEL DENSITY ESTIMATION

Boosting is a bias reduction technique while bagging is a variance reduction method. These two methods aim at reducing the asymptotic mean integrated square error (AMISE). This study aims to show that bagging is a boosting algorithm in kernel density estimation since both techniques use large smoothing parameter(s). This relationship was verified by real and simulated data

ON THE EXACT AND THE APPROXIMATE MEAN INTEGRATED SQUARE ERROR FOR THE KERNEL DISTRIBUTION FUNCTION ESTIMATOR

The asymptotic mean integrated square error (AMISE) is used as an approximate measure of error for the mean integrated square error (MISE). The exact MISE for kernel density estimator is discussed by Marron and Wand [1]. In this investigation we discuss the exact MISE and the AMISE for the cumulative distribution function estimate. Also, we compare between the optimal bandwidth that minimize the AMISE and that minimize the MISE. In addition, through simulations these optimal bandwidths are compared with the bandwidth selectors using the least square cross - validation (LSCV), biased cross - validation (BCV), and direct plug -in (DPI) techniques.

Nonparametric smooth estimation of the expected inactivity time function

The expected inactivity time (EIT) function (also known as the mean past lifetime function) is a well known reliability function which has application in many disciplines such as survival analysis, actuarial studies and forensic science, to name but a few. In this paper, we use a fixed design local polynomial fitting technique to obtain estimators for the EIT function when the lifetime random variable has an unknown distribution. It will be shown that the proposed estimators are asymptotically unbiased, consistent and also, when standardized, has an asymptotic normal distribution. An optimal bandwidth, which minimizes the AMISE (asymptotic mean integrated squared error) of the estimator, is derived. Numerical examples based on simulated samples from various lifetime distributions common in reliability studies will be presented to evaluate the performances of these estimators. Finally, three real life applications will also be presented to further illustrate the wide applicability of these estimators.

On bandwidth selection in kernel density estimation

In this paper, we suggest a new method of bandwidth selection in kernel density estimation. The new selector is less subject to the undersmoothing effect than the AMISE (asymptotic mean integrated square error) optimal bandwidth.

Polynomial Histograms for Multivariate Density and Mode Estimation

Abstract. We consider the problem of efficiently estimating multivariate densities and their modes for moderate dimensions and an abundance of data. We propose polynomial histograms to solve this estimation problem. We present first‐ and second‐order polynomial histogram estimators for a general d‐dimensional setting. Our theoretical results include pointwise bias and variance of these estimators, their asymptotic mean integrated square error (AMISE), and optimal binwidth. The asymptotic performance of the first‐order estimator matches that of the kernel density estimator, while the second order has the faster rate of O(n−6/(d+6)). For a bivariate normal setting, we present explicit expressions for the AMISE constants which show the much larger binwidths of the second order estimator and hence also more efficient computations of multivariate densities. We apply polynomial histogram estimators to real data from biotechnology and find the number and location of modes in such data.

Nonlinear wavelet density estimation with censored dependent data

In this paper, we provide an asymptotic expansion for the mean integrated squared error (MISE) of nonlinear wavelet estimator of survival density for a censorship model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary and α‐mixing sequence. This asymptotic MISE expansion, when the density is only piecewise smooth, is same. However, for the kernel estimators, the MISE expansion fails if the additional smoothness assumption is absent. Also, we establish the asymptotic normality of the nonlinear wavelet estimator. Copyright © 2011 John Wiley & Sons, Ltd.

K-nearest neighbour kernel density estimation, the choice of optimal k

ABSTRACT The k-nearest neighbour kernel density estimationmethod is a special type of the kernel density estimation method with the local choice of the bandwidth. An advantage of this estimator is that smoothing varies according to the number of observations in a particular region. The crucial problem is how to estimate the value of the parameter k. In the paper we discuss the problem of choosing the parameter k in a way that minimizes the value of the asymptotic mean integrated square error (AMISE). We define the class of the modified cosine densities that meet the requirements given by the AMISE. The results are compared in a simulation study.

Bandwidth adaption for kernel particle filter

A novel particle filter bandwidth adaption for kernel particle filter (BAKPF) is proposed. Selection of the kernel bandwidth is a critical issue in kernel density estimation (KDE). The plug-in method is adopted to get the global fixed bandwidth byoptimizing the asymptotic mean integrated squared error (AMISE) firstly. Then, particle-driven bandwidth selection is invoked in the KDE. To get a more effective allocation of the particles, the KDE with adaptive bandwidth in the BAKPF is used to approximate the posterior probability density function (PDF) by moving particles toward the posterior. A closed-form expression of the true distribution is given. The simulation results show that the proposed BAKPF performs better than the standard particle filter (PF), unscented particle filter (UPF) and the kernel particle filter (KPF) both in efficiency and estimation precision.