Classical Estimation Methods Research Articles

Comparison between classical and Bayesian estimation with joint Jeffrey’s prior to Weibull distribution parameters in the presence of large sample conditions

Weibull distribution has been considered one of the most common and valuable distributions for building and analyzing good models for lifetime data. Many researchers have studied the properties of Weibull distribution, also in search of the best method to estimate both parameters. In this paper, we proposed a comparison of Weibull distribution parameters under large sample conditions. We chose to study the classical estimation methods of Weibull distribution parameters, including the maximum likelihood estimator and moments estimation (ME). Next, we compared these methods with the Bayesian estimation method (BE) with Jeffrey’s prior function. We validated the proposed study via simulation using both small and large samples. We used mean square errors (MSE) to determine the best estimation method. Our simulation findings suggest that maximum likelihood estimators are reasonably effective when using small sample sizes. In addition, in cases where the sample size is larger, the BE performed more effectively for both scale and shape parameters of the Weibull distribution function.

The Marshall-Olkin Extended Gamma Lindley distribution: Properties, characterization and inference

Abstract In this research paper, we introduce a new three-parameter model called Marshall-Olkin Extended Gamma Lindley, which provides greater flexibility for modeling lifetime data. We investigate some of its structural properties, including moments and moment generating function. Furthermore, we prove that this model is identifiable. We corroborate that the Marshall-Olkin Extended Gamma Lindley distribution is characterized by its truncated moments of order statistics. Relying on this characterization and the least squares, we set forward a new method in order to estimate the unknown parameters of the proposed model. A comparative study is conducted between this approach and different classical methods of estimation. A real data set is applied to show the flexibility of the suggested model against other models.

Estimation of a~New Asymmetry Based Process Capability Index 𝐶𝑐 for Gamma Distribution

Abstract In this work, we provide five different classical estimation methods for the gamma distribution and unknown parameters, which are used to estimate the newly constructed process capability index (PCI) C c C_{c} . Next, we use the aforementioned estimation techniques to consider five bootstrap confidence intervals: standard bootstrap (SB), percentile bootstrap (PB), student’s 𝑡 bootstrap (STB), bias-corrected percentile bootstrap (BCPB), and bias-corrected accelerated bootstrap (BCAB). These allow us to construct confidence intervals of the PCI C c C_{c} . Using the suggested techniques of estimate for small, moderate, and large sample sizes, Monte Carlo simulations are run to compare the performances of the bootstrap confidence intervals’ average widths and coverage probabilities. Simulation results indicate that MLE gives the least AW for all BCIs (SB, PB, BCPB, TB, and BCAB) when compared to all other estimation approaches evaluated in this paper. Further, simulation results demonstrate that, for almost all sample sizes and all of the examined estimation approaches, BCAB CI generates smaller AW and larger CP than their counterparts. Finally, for illustrative purposes, three real data sets are analysed.

Handling Outliers in Panel Data Models: A Robust Approach

Real-world data often violate the conditions assumed by classical estimation methods. One reason for this failure may be the presence of observations with a low probability of belonging to the same distribution as the majority of the data, known as outliers. Outliers can appear in different forms, such as casewise and cellwise outliers. The results of classical estimation methods, particularly those based on least squares, can be seriously affected by the presence of any type of outlier. Panel data modeling is applied in various fields, including economics, finance, marketing, biology, environmental studies, healthcare, and more. The estimation of these models is typically performed using classical methods. In this paper, we consider the random effects panel data model and propose a robust method to estimate the parameters of this model. To evaluate the performance of the proposed robust estimation method compared to the classical estimation method, we conducted a Monte Carlo simulation study. Additionally, we illustrate the proposed methodology by applying it to estimate a model based on a real panel data set.

An Experimental Study on the Effects of Nonuniform Vane Spacing on Forced Response Reduction in a Mistuned Rotor Blisk in a Multistage Axial Compressor

Abstract Stators with nonuniform vane spacing (NUVS) are known to effectively reduce the forced response of adjacent rotors by spreading the primary engine order (EO) excitation at a certain speed to a series of weaker excitations at nearby EOs over a wider speed range. To be used in a real gas turbine engine, it is essential to know the forced response vibratory reduction that can be achieved for a specific NUVS configuration at different operating conditions. The classical estimation method to predict the blade response reduction is to quantify the reduction of the forcing function at a potential resonance crossing by conducting a circumferential Fourier analysis of the asymmetric flow field. However, besides excitation reduction, the blade forced response also depends on other factors, such as damping and blade-to-blade interactions due to mistuning. To study the blade forced response reduction in a realistic environment, a comprehensive experimental study was conducted in the Purdue three-stage axial research compressor at three different loading conditions. The vibratory response of the first torsion (1 T) mode forced response of rotor 2 was measured by strain gages (SGs) for two different upstream stator 1 configurations: the symmetric 38-vaned stator 1 configuration and the NUVS stator 1 configuration with 18–20 vane halves. As expected, the strong 38EO-1T response from the symmetric stator reduced to a series of weaker responses from 35EO to 41EO in the NUVS configuration. The overlap of adjacent EO responses caused considerable beating phenomena in the blade SG time–history data. There was substantial blade-to-blade variation in blade response for both symmetric and asymmetric stator 1 configurations due to the inherent nonintentional mistuning in the rotor 2 blisk. This, in turn, causes a large blade-to-blade variation in the reduction factors. While higher responding blades tend to have larger reduction factors, some blades show almost no forced response reduction when switching to the NUVS stator. In addition, the reduction factors for each blade and the maximum reduction factor for the whole blisk also change significantly with loading conditions. The measured reduction factors and the peak response EO are not well predicted by the classical estimation method. This indicates that both damping and mistuning effects need to be considered in the NUVS reduction factor prediction.

An Analytical Study on the Effects of Non-Uniform Vane Spacing on Forced Response Reduction in a Mistuned Blisk Rotor in a Multistage Axial Compressor

Abstract Stators with non-uniform vane spacing (NUVS) have been sought as an effective way to reduce the rotor-forced response at certain resonance crossing of concern. A comprehensive experimental study has been conducted in Purdue three-stage axial research compressor to evaluate the effectiveness of the NUVS stator on reducing the forced response of the downstream rotor using strain gauge measurement. The experimental results showed that the classical estimation method failed to predict the reduction factor correctly, mainly due to the lack of consideration for the damping and mistuning effects. To better understand the experimental results, and to provide an efficient and more accurate analysis tool for the rotor forced response under NUVS stator excitation, a detailed analytical study was conducted in this paper. The blade-forced response was solved using an efficient, steady-state linearized approach. A single-blade analysis was done first to study the effect of damping. The case studies show that higher damping causes more overlap of the blade forced response from adjacent engine order excitations and, thus, increases the total normalized blade response under NUVS stator excitation. A mistuned blisk aeroelastic model was introduced next, with the mistuning effect modeled using the fundamental mistuning model. Both structural coupling and aerodynamic coupling effects can be included in the aeroelastic model. Similar to the experimental results, the mistuned blisk case study shows a large blade-to-blade variation in the blade-forced response under both symmetric and asymmetric NUVS stator excitation, even at a low, non-intentional mistuning level. A statistical study with 100 randomly generated mistuned blisks shows that the forced response reduction effect of a specific NUVS design could vary significantly with a small change of the mistuning pattern, which suggests that mistuning has to be carefully considered in the design, evaluation, and optimization of the NUVS stator.

In silico data-based comparison of the accuracy and error source of various methods for noninvasively estimating central aortic blood pressure

Background and objectivesThe higher clinical significance of central aortic blood pressure (CABP) compared to peripheral blood pressures has been extensively demonstrated. Accordingly, many methods for noninvasively estimating CABP have been proposed. However, there still lacks a systematic comparison of existing methods, especially in terms of how they differ in the ability to tolerate individual differences or measurement errors. The present study was designed to address this gap. MethodsA large-scale ‘virtual subject’ dataset (n = 600) was created using a computational model of the cardiovascular system, and applied to examine several classical CABP estimation methods, including the direct method, generalized transfer function (GTF) method, n-point moving average (NPMA) method, second systolic pressure of periphery (SBP2) method, physical model-based wave analysis (MBWA) method, and suprasystolic cuff-based waveform reconstruction (SCWR) method. The errors of CABP estimation were analyzed and compared among methods with respect to the magnitude/distribution, correlations with physiological/hemodynamic factors, and sensitivities to noninvasive measurement errors. ResultsThe errors of CABP estimation exhibited evident inter-method differences in terms of the mean and standard deviation (SD). Relatively, the estimation errors of the methods adopting pre-trained algorithms (i.e., the GTF and SCWR methods) were overall smaller and less sensitive to variations in physiological/hemodynamic conditions and random errors in noninvasive measurement of brachial arterial blood pressure (used for calibrating peripheral pulse wave). The performances of all the methods worsened following the introduction of random errors to peripheral pulse wave (used for deriving CABP), as characterized by the enlarged SD and/or increased mean of the estimation errors. Notably, the GTF and SCWR methods did not exhibit a better capability of tolerating pulse wave errors in comparison with other methods. ConclusionsClassical noninvasive methods for estimating CABP were found to differ considerably in both the accuracy and error source, which provided theoretical evidence for understanding the specific advantages and disadvantages of each method. Knowledge about the method-specific error source and sensitivities of errors to different physiological/hemodynamic factors may contribute as theoretical references for interpreting clinical observations and exploring factors underlying large estimation errors, or provide guidance for optimizing existing methods or developing new methods.

Continuous-Time Multivariate Analysis

The starting point for much of multivariate analysis (MVA) is an n × p data matrix whose n rows represent observations and whose p columns represent variables. Some multivariate datasets, however, may be best conceptualized not as n discrete p-variate observations, but as p curves or functions defined on a common time interval. Here we introduce a framework for extending techniques of multivariate analysis to such settings. The proposed continuous-time multivariate analysis (CTMVA) framework rests on the assumption that the curves can be represented as linear combinations of basis functions such as B-splines, as in the Ramsay-Silverman representation of functional data; but whereas functional data analysis extends MVA to the case of observations that are curves rather than vectors—heuristically, n × p data with p infinite—we are instead concerned with what happens when n is infinite. We present continuous-time extensions of the classical MVA methods of covariance and correlation estimation, principal component analysis, Fisher’s linear discriminant analysis, and k-means clustering. We show that CTMVA can improve on the performance of classical MVA, in particular for correlation estimation and clustering, and can be applied in some settings where classical MVA cannot, including variables observed at disparate time points. CTMVA is illustrated with a novel perspective on a well-known Canadian weather dataset, and with applications to datasets involving international development, brain signals, and air quality. The proposed methods are implemented in the publicly available R package ctmva. Supplementary materials for this article are available online.

The Weighted Xgamma Model: Estimation, Risk Analysis and Applications

The weighted xgamma distribution, a new weighted two-parameter lifespan distribution, is introduced in this study. Theoretical characteristics of this model are deduced and thoroughly examined, including quantile function, extreme value, moments, moment generating function, cumulative entropy, and residual cumulative. Some classical estimation methods such as the the maximum likelihood, weighted least square, Anderson Darling and Cramer-von-Mises are considered. A simulation experiments are performed to compare the estimation methods. Four real-life data sets is finally examined to demonstrate the viability of this model. Four key risk indicators are defined and analyzed under the maximum likelihood method. A risk analysis for the exceedances of flood peaks is presented.

Carbon market risk estimation using quantum conditional generative adversarial network and amplitude estimation

AbstractAccurately and efficiently estimating the carbon market risk is paramount for ensuring financial stability, promoting environmental sustainability, and facilitating informed decision‐making. Although classical risk estimation methods are extensively utilized, the implicit pre‐assumptions regarding distribution are predominantly contained and challenging to balance accuracy and computational efficiency. A quantum computing‐based carbon market risk estimation framework is proposed to address this problem with the quantum conditional generative adversarial network‐quantum amplitude estimation (QCGAN‐QAE) algorithm. Specifically, quantum conditional generative adversarial network (QCGAN) is employed to simulate the future distribution of the generated return rate, whereas quantum amplitude estimation (QAE) is employed to measure the distribution. Moreover, the quantum circuit of the QCGAN improved by reordering the data interaction layer and data simulation layer is coupled with the introduction of the quantum fully connected layer. The binary search method is incorporated into the QAE to bolster the computational efficiency. The simulation results based on the European Union Emissions Trading System reveals that the proposed framework markedly enhances the efficiency and precision of Value‐at‐Risk and Conditional Value‐at‐Risk compared to original methods.

Classical and Bayesian estimation of multicomponent stress–strength reliability with power Lindley distribution under progressive first-failure censored samples

This research article presents a methodology for estimating the multicomponent reliability based on progressively first-failure censored samples, where the underlying distribution of stress and strength is modelled using the power Lindley distribution. The classical and Bayesian estimation methods are utilized for estimating the multicomponent reliability. In the Bayesian estimation, the Markov Chain Monte Carlo approximation method is employed to obtain the posterior mean under a generalized entropy loss function. Various intervals including asymptotic confidence, bootstrap-p confidence, bootstrap-t confidence, Bayesian credible, and highest posterior density credible intervals are computed. A simulation study is conducted to evaluate the performance of the proposed methodology. Also, two different real data applications are presented to illustrate the practicality of the approach.

Data analysis with applications in physics and engineering using XLindley model with improved adaptive Type-II progressively censored samples

Recently, a novel improved adaptive Type-II progressive censoring strategy has been suggested in order to obtain adequate data from trials that require a lengthy amount of time. Considering this scheme, this paper focuses on various classical and Bayesian estimation challenges for parameter and some reliability metrics for the XLindley distribution. Two classical estimation methods are considered from the classical perspective to get the point and interval estimations of the model parameter as well as reliability and hazard rate functions. In addition to the usual approaches, the Bayesian methodology is looked at by leveraging the squared error loss function and the Markov chain Monte Carlo technique. The Bayes point and credible intervals are obtained based on two forms of the posterior distribution. A simulation examination is implemented adopting multiple circumstances to distinguish between the conventional and Bayesian estimations. The simulation results demonstrate that the Bayesian approach using the likelihood function is superior for estimating the model parameter when compared with the other methods. In contrast, when estimating reliability metrics, it is advisable to utilize the Bayesian method with the spacings function. Two real-world data sets are analyzed to integrate the proposed approaches into practice, and the ideal progressive censoring strategy is chosen using some optimality criteria.

Fitting data to a multiple structural measurement errors model

This paper proposes two new estimation methods to fit a multiple structural measurement error model when all variables are subject to errors. The new estimation methods were extensions of the Wald estimation method, one is the weighted grouping method, and the other is the iterative method. A Monte Carlo experiment is performed to investigate the performance of the new estimators compared with the classical estimation methods; the Maximum Likelihood Estimator and Method of Moment, in terms of root mean square error and its bias. The simulation outcomes demonstrated that the suggested estimators are more effective than conventional estimators. In addition, real data analysis is discussed to examine the relationship between the national gross domestic product, unemployment rate, and human development index after applying the two proposed estimation methods.

A statistical study for the impact of REMS and nuclear energy on carbon dioxide emissions reductions in G20 countries

Renewable and nuclear energy are seen as clean and eco-friendly substitutes for energy derived from fossil fuels. Which of the two is better, though, is still up for debate. In order to meet their combined global energy objectives and perhaps become carbon neutral or even carbon negative in the near future, a shift to additional renewable energy sources and nuclear energy will be necessary. One such method of improving energy usage through the implementation of a renewable energy management system (REMS) that will help attain a continuous and reliable source of energy, allowing for the proactive and effective management of renewable assets and dependable supply. This article investigates attitudes and opinions about nuclear and renewable energy production technologies by analyzing the dependence of carbon dioxide (CO2) emissions on non-renewable energy sources, renewable energy sources, and nuclear energy in the group of twenty (G20) countries, over the period from 2000 to 2021. This study employs advanced panel data econometric techniques in the presence of outliers in the G20 dataset, utilizing a robust random coefficient regression (RRCR) model via the M-estimation approach to study the impacts of non-renewable, renewable, and nuclear energy use with respect to CO2 emission mitigation in G20 countries. The overall conclusions of the econometric analysis confirm renewable and nuclear energy's importance in reducing CO2 emissions. We can conclude that the robust M-estimate approach is the best estimation technique for the G20 dataset, as compared to the classical estimation methods in the presence of outliers. The results show that for every percentage point of energy consumption that falls into the category of non-renewable energy, CO2 emissions will increase by 2.5737% for all G20 countries. Also, A 1% rise in renewable energy consumption reduces CO2 emissions by 1.0566% for all G20 countries. Similarly, a 1% increase in nuclear energy consumption will reduce CO2 emissions by 5.5703% for all G20 countries. Therefore, we can conclude that energy derived from renewable sources and nuclear energy greatly cuts CO2 emissions in all G20 countries. Total energy consumption, on the other hand, increases CO2 emissions in the long run. According to these findings, it is pivotal for the G20 countries to increase nuclear energy use and employ REMS to reduce the majority of G20 country's reliance on fossil fuels in order to minimize CO2 emissions. These findings are vital for understanding the relationship between these variables and the implementation of new or revised policies and initiatives supporting carbon neutrality at the G20 countries and national levels.

A Relationship between Classical and Bayesian Estimation Procedures through Fisher Information

The aim of this article is to investigate the association between the classical and Bayesian approaches through Fisher information. For any particular distribution, the computation of Fisher information is quite significant, as it provides the amount of information about the unknown parameter inferred from the observed data and is related to classical methods of estimation. Also, in the light of some prior knowledge, we may estimate the unknown parameter through Bayesian approach. Specifically, we want to see a relationship between information and Bayes estimation. In this article, the scale parameter of the one-parameter exponential distribution is estimated under the weighted squared error and Kullback-Leibler distance loss functions. The information acquired from both the classical and Bayesian methodologies have been connected through the risk intensity and error intensity which have been introduced in this article. The results of extensive simulation studies using these intensity measures show that the Bayes estimator performs more intensely as the amount of Fisher information increases. It is seen that the Fisher information, which is pivotal to many classical estimation methods, has a relationship with the Bayesian method depending on prior distribution, at least in this case, as the intensity measures of the Bayes estimator decrease with the increase in information. Further, to comprehend the theoretical notion of association, two real-life datasets have been included to show usefulness in practical field.

Efficient Convex-Hull-Based Vehicle Pose Estimation Method for 3D LiDAR

Vehicle pose estimation with light detection and ranging (LiDAR) is essential in the perception technology of autonomous driving. However, because of incomplete observation measurements and sparsity of the LiDAR point cloud, it is challenging to achieve satisfactory pose extraction based on 3D LiDAR with the existing pose estimation methods. In addition, the demand for real-time performance further increases the difficulty of the pose estimation task. In this paper, we propose a novel vehicle pose estimation method based on the convex hull. The extracted 3D cluster is reduced to the convex hull, reducing the subsequent computation burden while preserving essential contour information. Subsequently, a novel criterion based on the minimum occlusion area is developed for the search-based algorithm, enabling accurate pose estimation. Additionally, this criterion renders the proposed algorithm particularly well-suited for obstacle avoidance. The proposed algorithm is validated on the KITTI dataset and a manually labeled dataset acquired at an industrial park. The results demonstrate that our proposed method can achieve better accuracy than the classical pose estimation method while maintaining real-time speed.

Rhythm-adaptive statistical estimation methods of probabilistic characteristics of cyclic random processes

The paper is devoted to the rhythm-adaptive statistical estimation methods of cyclic random processes probabilistic characteristics. The main goal of the article is development and research of rhythm-adaptive methods of cyclic random processes statistical processing. Based on the fundamental fact of invariance of probabilistic characteristics of a cyclic random process to time shifts determined by its rhythm function, rhythm-adaptive statistical estimations of initial, central and mixed moment functions of cyclic random processes of discrete parameter were obtained. The article analytically proves the unbiasedness and consistency of rhythm-adaptive statistical estimates of the mathematical expectation and the initial moment functions of the higher order of cyclic random process. By conducting a series of computer simulation experiments, it has been demonstrated, that the absolute errors of rhythm-adaptive methods of statistical estimation are significantly smaller than the corresponding errors for classical non-rhythm adaptive methods of statistical estimation of cyclic stochastic signals probabilistic characteristics. The effectiveness of using rhythm-adaptive methods of statistical estimation of probabilistic characteristics of ECG signals has been demonstrated. The results obtained in the work clearly demonstrate the essential advantages of rhythm-adaptive methods of statistical processing of cyclic stochastic signals with a variable (irregular) rhythm in comparison with non-rhythm-adaptive analogues.

Improved Underwater Single-Vector Acoustic DOA Estimation via Vector Convolution Preprocessing

Remote passive sonar detection with underwater acoustic vector sensor (UAVS) has attracted increasing attention due to its merit in measuring the full sound field information. However, the accurate estimation of the direction-of-arrival (DOA) remains a challenging problem, especially under low signal-to-noise ratio (SNR) conditions. In this paper, a novel convolution (COV)-based single-vector acoustic preprocessing method is proposed on the basis of the single-vector acoustic preprocessing model. In view of the theoretical analysis of the classical single-vector acoustic DOA estimation method, the principle of preprocessing can be described as “to achieve an improved denoising performance in the constraint of equivalent amplitude gain and phase response.” This can be naturally guaranteed by our proposed COV method. In addition, the upper bound with matched filtering (MF) preprocessing is provided in the consideration of the optimal linear signal processing for weak signal detection under Gaussian noise. Numerical analyses demonstrate the effectiveness of our proposed preprocessing method with both vector array signal processing-based and intensity-based methods. Experimental verification conducted in South China Sea further verifies the effectiveness of our approach for practical applications. This work can lay a solid foundation in improving underwater remote vector acoustic DOA estimation under low SNR, and can provide important guidance for future research work.

Adaptive Kalman filter with LSTM network assistance for abnormal measurements

A classic state estimation method, the Kalman filter integrates prior information, system dynamics models, and measurement data to achieve posterior state estimations. However, measurements often encounter various unknown disturbances, leading to abnormal or inaccurate measurements and subsequently impacting the performance of Kalman filtering. In response to this challenge, this paper introduces a novel adaptive Kalman filter approach aided by posterior state estimations using the Long short-term memory (LSTM) networks. The proposed method begins by utilizing prior residuals to construct a chi-square distribution model, which facilitates the detection of abnormal measurements within data. Upon identifying abnormal measurements, an LSTM network is employed to generate alternative predictive measurements, replacing the original inaccurate measurement. This approach enhances the model’s capability to handle complex relationship and measurement uncertainties and improves filtering estimation accuracy. A noise covariance adjustment method is introduced in extreme cases where alternative predictive measurements alone are insufficient for filter requirements. This method mitigates the adverse effects of abnormal measurements on posterior estimation. Throughout the entire adaptive assistance process involving the LSTM network, deep learning maintains a stable structure of the filter while enhancing adaptability. This strategy ensures the filter’s resilience in dynamic environments with unknown factors by providing predicted measurements conforming to the chi-square distribution and corresponding measurement noise covariance. Ultimately, the efficacy of the proposed algorithm is validated through simulations and experiments involving vehicle positioning with inertial sensors.

Learning‐based monocular visual‐inertial odometry with SE2(3) $S{E}_{2}(3)$‐EKF

AbstractLearning‐based visual odometry (VO) becomes popular as it achieves a remarkable performance without manually crafted image processing and burdensome calibration. Meanwhile, the inertial navigation can provide a localization solution to assist VO when the VO produces poor state estimation under challenging visual conditions. Therefore, the combination of learning‐based technique and classical state estimation method can further improve the performance of pose estimation. In this paper, we propose a learning‐based visual‐inertial odometry (VIO) algorithm, which consists of an end‐to‐end VO network and an ‐Extended Kalman Filter (EKF). The VO network mainly combines a convolutional neural network with a recurrent neural network, taking advantage of two consecutive monocular images to produce relative pose estimation with associated uncertainties. The ‐EKF, which is proved to overcome the inconsistency issues of VIO, propagates inertial measurement unit kinematics‐based states, and fuses relative measurements and uncertainties from the VO network in its update step. The extensive experimental results on the KITTI data set and the EuRoC data set demonstrate the superior performance of the proposed method compared to other related methods.