Mean Integrated Squared Error Research Articles

Bayesian inference for Rayleigh Pareto distribution under progressively type-II right censored data

In this paper, we consider inference problems including estimation for a Rayleigh Pareto (RP) distribution under progressively type-II right censored data. We use two approaches, the classical maximum likelihood approach and the Bayesian approach for estimating the distribution parameters and the reliability characteristics. Bayes estimators and corresponding posterior risks (PR) have been derived using different loss functions (symmetric and asymmetric). The estimators cannot be obtained explicitly, so we use the method of Monte Carlo. Finally, we use the integrated mean square error (IMSE) and the Pitman closeness criterion to compare the results of the two methods.

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.

Hybrid threshold combining and maximum ratio combining model for satellite communication over composite Rayleigh and Rician fading channel

SummaryCommunication through satellite is of great task and importance especially in the world of telecommunication infrastructure. It plays a leading role for future communication systems. However, the performance of the satellite communication system is dominated by the channel conditions overwhelmed by multipath fading and ionospheric scintillation effects. Maximum ratio combining (MRC) previously used to tackle this problem is characterized with hardware complexity resulted in long processing time, while threshold combining (TC) with simple hardware display poor performance. Thus, evaluation of hybrid TC‐MRC, with closed‐form expression over composite Rayleigh and Rician fading channel is necessary. In this paper, TC and MRC are combined to improve the signal quality and performance. The data are modulated using M‐ary quadrature amplitude modulation and transmitted over the combined effects. The received signals at varying paths are scanned to select the best paths. A mathematical expression using the probability density function of composite Rayleigh and Rician fading channel for mean integrated square error (MISE) is derived. The model is efficiently simulated and the performance is estimated. The results show that the hybrid TC‐MRC model achieves lower MISE and processing time compared to threshold combiner and even maximum ratio combiner, and thereby enhance the performance of the satellite communication system.

Asymptotic properties of Bernstein estimators on the simplex

Bernstein estimators are well-known to avoid the boundary bias problem of traditional kernel estimators. The theoretical properties of these estimators have been studied extensively on compact intervals and hypercubes, but never on the simplex, except for the mean squared error of the density estimator in Tenbusch (1994) when d=2. The simplex is an important case as it is the natural domain of compositional data. In this paper, we make an effort to prove several asymptotic results (bias, variance, mean squared error (MSE), mean integrated squared error (MISE), asymptotic normality, uniform strong consistency) for Bernstein estimators of cumulative distribution functions and density functions on the d-dimensional simplex. Our results generalize the ones in Leblanc (2012a) and Babu et al. (2002), who treated the case d=1, and significantly extend those found in Tenbusch (1994). In particular, our rates of convergence for the MSE and MISE are optimal.

A Gibbs Sampling-based approach for parameter estimation of the EGK distribution

We present a novel approach for estimating the parameters of the extended generalized-K (EGK) distribution commonly used as a fading model in wireless and optical communications links. The proposed method is based on the Gibbs sampling technique and does not require solving nonlinear equations nor performing numerical integrations. Numerical and simulation results are presented showing that the estimated and original distributions are virtually indistinguishable and formal metrics like Kullback-Leibler (KL) divergence, the mean integrated squared bias (MISB), the mean integrated variance (MIV) and the mean integrated squared error (MISE) all show excellent agreement between the two as well.

Nonparametric Estimation of Multivariate Density and its Derivative by Dependent Data Using Gamma Kernels

We consider the nonparametric estimation of the multivariate probability density function and its partial derivative with a support on nonnegative axis by dependent data. We use the class of kernel estimators with asymmetric gamma kernel functions. The gamma kernels are nonnegative; they may change their shape depending on the position on the semi-axis and possess good boundary properties for a wide class of densities. Asymptotic estimates of the multivariate density and of its partial derivatives such as biases, variances, and covariances are derived. The optimal bandwidth of both estimates is obtained as a minimum of the mean integrated squared error (MISE) by dependent data with a strong mixing. Optimal convergence rates of the MISE both for the density and its derivative are found.

A novel adaptive control design method for stochastic nonlinear systems using neural network

This paper presents a novel method for designing an adaptive control system using radial basis function neural network. The method is capable of dealing with nonlinear stochastic systems in strict-feedback form with any unknown dynamics. The proposed neural network allows the method not only to approximate any unknown dynamic of stochastic nonlinear systems, but also to compensate actuator nonlinearity. By employing dynamic surface control method, a common problem that intrinsically exists in the back-stepping design, called “explosion of complexity”, is resolved. The proposed method is applied to the control systems comprising various types of the actuator nonlinearities such as Prandtl–Ishlinskii (PI) hysteresis, and dead-zone nonlinearity. The performance of the proposed method is compared to two different baseline methods: a direct form of backstepping method, and an adaptation of the proposed method, named APIC-DSC, in which the neural network is not contributed in compensating the actuator nonlinearity. It is observed that the proposed method improves the failure-free tracking performance in terms of the Integrated Mean Square Error (IMSE) by 25%/11% as compared to the backstepping/APIC-DSC method. This depression in IMSE is further improved by 76%/38% and 32%/49%, when it comes with the actuator nonlinearity of PI hysteresis and dead-zone, respectively. The proposed method also demands shorter adaptation period compared with the baseline methods.

Integrated Square Error of Hazard Rate Estimation for Survival Data with Missing Censoring Indicators

The problem of hazard rate estimation under right-censored assumption has been investigated extensively. Integrated square error (ISE) of estimation is one of the most widely accepted measurements of the global performance for nonparametric kernel estimation. But there are no results available for ISE of hazard rate estimation under right-censored model with censoring indicators missing at random (MAR) so far. This paper constructs an imputation estimator of the hazard rate function and establish asymptotic normality of the ISE for the kernel hazard rate estimator with censoring indicators MAR. At the same time, an asymptotic representation of the mean integrated square error (MISE) is also presented. The finite sample behavior of the estimator is investigated via one simple simulation.

Generalized Birnbaum–Saunders kernel for hazard rate function estimation

This paper proposes the nonparametric kernel method for the hazard rate (HR) function estimation in the context of positively skewed data. The class of generalized Birnbaum-Saunders (GBS) kernels is considered because of its several interesting properties and flexibility. Some asymptotic properties, such as bias, variance and mean integrated squared error (MISE) are established for the proposed estimator. In addition, we demonstrate that, the GBS-HR estimator is strongly consistent and asymptotically normal. The choice of bandwidth is also investigated by rule of thumb and unbiased cross validation approaches. Finally, performances of the HR estimator based on GBS kernels and comparison of the two bandwidth selection methods are illustrated by a simulation study and real applications.

Density Estimation by Randomized Quasi-Monte Carlo

We consider the problem of estimating the density of a random variable $X$ that can be sampled exactly by Monte Carlo (MC). We investigate the effectiveness of replacing MC by randomized quasi Monte Carlo (RQMC) or by stratified sampling over the unit cube, to reduce the integrated variance (IV) and the mean integrated square error (MISE) for kernel density estimators. We show theoretically and empirically that the RQMC and stratified estimators can achieve substantial reductions of the IV and the MISE, and even faster convergence rates than MC in some situations, while leaving the bias unchanged. We also show that the variance bounds obtained via a traditional Koksma-Hlawka-type inequality for RQMC are much too loose to be useful when the dimension of the problem exceeds a few units. We describe an alternative way to estimate the IV, a good bandwidth, and the MISE, under RQMC or stratification, and we show empirically that in some situations, the MISE can be reduced significantly even in high-dimensional settings.

A Suggested Nonparametric Bivariate Logistic Density Estimator with Application on the Productivity of Egyptian Wheat during 2019/2020

In this study, the nonparametric standard logistic density estimator, introduced by Abo-El-Hadid (2018), is extended to the bivariate case. The multiplicative standard logistic distribution is used as a kernel function to derive the bivariate kernel estimator. The statistical properties of the resulting estimator are studied, which are: The asymptotic bias, variance, Mean Squared Error (MSE) and Integrated Mean Squared Error (IMSE); also, the optimal bandwidth is obtained. A simulation study is introduced to investigate the performance of the proposed estimator with other estimators. We also apply the proposed estimator to a real data set to estimate the bivariate density of the planted and productive areas of wheat in Egypt.

Density Estimation Using Gumbel Kernel Estimator

In this article, our proposed kernel estimator, named as Gumbel kernel, which broadened the class of non-negative, asymmetric kernel density estimators. Such kernel estimator can be used in nonparametric estimation of the probability density function (pdf). When the density functions have limited bounded support on [0, ∞) and they are liberated of boundary bias, always non-negative and obtain the optimal rate of convergence for the mean integrated squared error (MISE). The bias, variance and the optimal bandwidth of the proposed estimators are investigated on theoretical grounds as well as on simulation basis. Further, the applicability of the proposed estimator is compared to Weibull kernel estimator, where performance of newly proposed kernel is outstanding.

A New Kernel Estimator Based on Scaled Inverse Chi-Squared Density Function

In this work, a new kernel estimator based on scaled inverse chi-squared distribution is proposed to estimate densities having nonnegative support. The optimal rates of convergence for the mean squared error (MSE) and the mean integrated squared error (MISE) are obtained. Adaptive Bayesian bandwidth selection method with Lindley approximation is used for heavy tailed distributions. Simulation studies are performed to compare the performance of the average integrated square error (ISE) by using the bandwidths obtained from the global least squares cross-validation bandwidth selection method and the bandwidths obtained from adaptive Bayesian method with Lindley approximation. Finally, real data sets are presented to illustrate the findings.

Density deconvolution for generalized skew-symmetric distributions

The density deconvolution problem is considered for random variables assumed to belong to the generalized skew-symmetric (GSS) family of distributions. The approach is semiparametric in that the symmetric component of the GSS distribution is assumed known, and the skewing function capturing deviation from the symmetric component is estimated using a deconvolution kernel approach. This requires the specification of a bandwidth parameter. The mean integrated square error (MISE) of the GSS deconvolution estimator is derived, and two bandwidth estimation methods based on approximating the MISE are also proposed. A generalized method of moments approach is also developed for estimation of the underlying GSS location and scale parameters. Simulation study results are presented including a comparing the GSS approach to the nonparametric deconvolution estimator. For most simulation settings considered, the GSS estimator is seen to have performance superior to the nonparametric estimator.

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.

Robust Bayesian Estimators For Survival Function Under Prior Data Conflict With Practical Application in the Health Side

The analysis of survival functions is concerned with knowing how long humans will survive, This means studying and analyzing the time from the beginning of the disease to the end point of death. For example, the survival function of patients with heart attacks was studied and analyzed at Al Manathira General Hospital. The Weibull distribution was used to match the real data. The scale parameter & survival function have been estimated for Weibull distribution with have two parameters, this distribution was used in two cases, prior data unconflict & prior data conflict. A regular Bayes method & robust Bayes were used for estimation. We used inverse gamma distribution as a prior where it is a conjugate prior for Weibull distribution. Two simulation experiments have been used; the first experiment used was prior data unconflict where the regular Bayes method is the best for estimating the scale & survival function by using the integrated mean square error (IMSE) as a criterion for comparing. The second experiment is in the case of prior data conflict. The results showed that the robust Bayes method is the best for estimation of the scale parameter & survival function by using (IMSE).

False-Data-Injection Attacks on Remote Distributed Consensus Estimation.

This article studies a security issue in remote distributed consensus estimation where sensors transmit their measurements to remote estimators via a wireless communication network. The relative entropy is utilized as a stealthiness metric to detect whether the data transmitted through the wireless network are attacked. The performance degradation induced by an attacker that attempts to be stealthy or undetected is analyzed, and the corresponding false-data attack strategy is characterized, which can achieve the maximal integrated mean-square error (IMSE). Finally, the tradeoff between the performance degradation and attack stealthiness level is evaluated through an example.

Nonparametric relative recursive regression

Abstract In this paper, we propose the problem of estimating a regression function recursively based on the minimization of the Mean Squared Relative Error (MSRE), where outlier data are present and the response variable of the model is positive. We construct an alternative estimation of the regression function using a stochastic approximation method. The Bias, variance, and Mean Integrated Squared Error (MISE) are computed explicitly. The asymptotic normality of the proposed estimator is also proved. Moreover, we conduct a simulation to compare the performance of our proposed estimators with that of the two classical kernel regression estimators and then through a real Malaria dataset.

MISE of wavelet estimators for regression derivatives with biased strong mixing data

Using a wavelet basis, this article considers the mean integrated squared error (MISE) of linear and non linear wavelet estimators for regression derivatives based on biased strong mixing data. It turns out that the convergence rates coincide with those of Chesneau and Shirazi’s (Communication in Statistics-Theory and Methods, 2014), when d = 0 and the random sample is independent.

Fault Diagnosis of Complex Chemical Processes Based on Enhanced Naive Bayesian Method

Fault diagnosis is crucial for the stable and reliable operation of chemical processes. However, faced with the complexity of chemical processes, conventional diagnosis methods suffer from the expertise of feature extraction and classifier design. They also lack extracting effective features from raw data whose attributes are correlated and coupled. This article proposes an enhanced naive Bayesian classifier (ENBC) for fault diagnosis of the complex chemical processes. Assuming that all attributes are related, ENBC can utilize a joint probability density function (pdf) based on multivariate Gaussian kernel function and adaptively extract the related information from the raw data. ENBC can estimate the class-conditional pdf based on the optimal smoothing parameter, which is the crucial part of the joint pdf estimation. ENBC seeks an optimal smoothing parameter by minimizing the mean integrated squared error (MISE) between the true pdf and the estimated pdf. The benchmark Tennessee Eastman (TE) process is used to illustrate the performance of ENBC. The experimental results demonstrate that ENBC is more effective than other methods.