Average Coverage Probability Research Articles

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.

Estimating the Confidence Intervals for the Coefficients of Variation Delta-Gamma Distributions

A common application of the coefficient of variation (CV), which is the ratio of the population standard deviation to the population mean, is frequently used to assess quality control and economic processes, among others. The fiducial quantity approach, Bayesian confidence intervals (CIs) using the Jeffreys, uniform, or normal-gamma-beta (NGB) priors, and highest posterior density (HPD) intervals using the Jeffreys, uniform, or NGB priors were used to provide estimators for the CI for the ratio of CV of two delta-gamma distributions. An evaluation of their performance in terms of average length and coverage probability was carried out using Monte Carlo simulations. The results of this study indicate that the HPD using the Jeffreys prior and fiducial quantity methods are the best for estimating the CI for the ratio of the CV of two delta-gamma distributions. Rainfall data from Mae Hong Son province in Thailand was used to illustrate their practicability when analyzing real-life processes.

Classical and Bayesian inference of Cpc$\mathcal {C}_{pc}$ for Wilson–Hilferty distribution under progressively first‐failure type‐II censoring samples

AbstractThis study uses two frequentist approaches and the Bayesian method of estimation using progressively first‐failure type‐II censored data to estimate process capability index (PCI), , for the Wilson–Hilferty (WH) distribution. A competitive maximum product of spacing (MPS) method for estimation of is proposed in the frequentist method as an alternative to conventional likelihood (LK)‐based estimation. We have also taken into account the PS function in the Bayesian setup as an alternative to the conventional LK function, and both LK and PS‐based Bayesian estimates are given for . Further, the Bayes estimates are evaluated by applying Tierney–Kadane (TK) approximation method and Markov chain Monte Carlo (MCMC) method with the help of both LK and PS functions and Jeffrey's prior. Additionally, the approximate confidence intervals based on the suggested frequentist techniques as well as the Bayes highest posterior density (HPD) credible intervals are built for the index . In addition, two bootstrap confidence intervals are obtained. In the simulation exercise, the performance of the Bayes and traditional estimates of is assessed in terms of their mean squared errors, and the average width and coverage probabilities of the CIs and HPD intervals are compared. An actual data set from the electronic industries is reanalyzed in order to show the efficacy of the proposed index and estimation methodology.

Application of wavelet and seasonal-based emotional ANN (EANN) models to predict solar irradiance

This study models solar irradiance at six stations in Iran and the USA on an hourly scale. We explored two seasonal emotional artificial neural networks (EANN): sequence-EANN (SEANN) and wavelet EANN (WEANN). Analyzing ten years of climatic and solar data, we evaluated uncertainty using prediction intervals (PIs) computed via the bootstrap method based on artificial neural networks (ANNs). Unlike standalone EANNs, the proposed seasonal models effectively captured seasonal information and leveraged time series processing advantages. Utilizing Wavelet and Fourier transforms, these models captured long-short autoregressive dependencies in solar irradiance, addressing extended seasonal dependencies. Results showed that the seasonal EANN models outperformed the classic EANN model by approximately 15 % and the classic feed-forward neural network (FFNN) by about 25 % in both training and testing. The WEANN model demonstrated the highest performance in PIs, with an average normalized mean PI width (NMPIW) of 0.8 and an average PI coverage probability (PICP) of 0.96.

Different Statistical Inference Algorithms for the New Pareto Distribution Based on Type-II Progressively Censored Competing Risk Data with Applications

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.

Fast Bayesian Functional Regression for Non-Gaussian Spatial Data

Functional generalized linear models (FGLM) have been widely used to study the relations between non-Gaussian response and functional covariates. However, most existing works for FGLM assume independence among observations and therefore they are of limited applicability for correlated data. A particularly important example is spatial functional data, where we observe functions over spatial domains, such as the age population curve or temperature curve at each areal unit. In this paper, we extend FGLM by incorporating spatial random effects. Especially, we study the relationship between the non-Gaussian response variable and functional covariates that are spatially observed. However, such model has computational and inferential challenges. The high-dimensional spatial random effects cause the slow mixing of Markov chain Monte Carlo (MCMC) algorithms. Furthermore, spatial confounding can lead to bias in parameter estimates and inflate their variances. To address these issues, we propose an efficient Bayesian method using a sparse reparameterization of high-dimensional random effects. We also study the average coverage probabilities of the credible intervals of functional parameters. We apply our methods to simulated and real data examples, including malaria incidence data and US COVID-19 data. The proposed method is fast while providing accurate functional estimates.

Classical Inference of a New PCI CNpmkc for Logistic-Exponential Process Distribution

The process capability index (PCI) has been developed as a useful instrument for measuring how well a product matches customer expectations while also assessing the performance of a process. We are aware that when the quality characteristics of the processes follow normal distribution, conventional PCIs produces superior results. However, these traditional indices could not yield reliable findings when assessing nonnormally distributed processes, which could make decision-making more difficult. In this paper, we take into account the [Formula: see text] process capacity index, which may be used for both normally distributed and nonnormally distributed processes. We have employed 10 traditional methods of estimation to estimate the PCI [Formula: see text] when the process has a logistic-exponential distribution, and the performances of these traditional estimates of the index [Formula: see text] are compared in terms of their mean squared errors through a simulation exercise. Then, we create five PCI [Formula: see text] bootstrap confidence intervals and contrast them based on their average widths and related coverage probabilities. In order to demonstrate the applicability of the suggested methods of estimation, two data sets pertaining to the electronic industries are re-analyzed.

Objective Bayesian estimation for multistate stress‐strength model's reliability with various kernel functions

AbstractThe paper discusses the objective Bayesian estimation of the reliability of a multistate stress‐strength model (MSSM) based on different kernel functions. For the MSSM, we first derive the reliability function and Fisher information matrix. The Jeffreys prior, reference prior, and probability matching prior for the reliability function of the MSSM are constructed based on the objective Bayesian paradigm. Subsequently, we demonstrated that these priors are improper density, then evaluated the effects of these priors on Bayes estimates for MSSM's reliability based on a complete sample. The Bayesian estimates are calculated using random walk Metropolis–Hastings techniques. We employ Monte Carlo simulation to examine the effectiveness of Bayes estimates for MSSM's reliability in terms of average bias and mean squared error, meanwhile the highest posterior density credible intervals are investigated in terms of average length and coverage probability. Finally, two real datasets were examined, demonstrating the viability of the objective Bayes technique for small sample data.

Applications of a new loss and cost-based process capability index to electronic industries

In this article, we propose a new process capability index called C pmc ′ which is based on asymmetric loss function (linear exponential) and tolerance cost function for a normal process which provides a tailored way of incorporating the loss and tolerance cost in capability analysis. Next, we estimate the proposed PCI C pmc ′ when the process follows the normal distribution using six classical methods of estimation and we compare the performance of the considered methods of estimation in terms of their mean squared errors through simulation study. Besides, five bootstrap methods are employed for constructing the confidence intervals for the index C pmc ′ . The performance of the bootstrap confidence intervals (BCIs) are compared in terms of average width and coverage probabilities using Monte Carlo simulation. Finally, for illustrating the effectiveness of the proposed index and methods of estimation and BCIs, two real data sets from electronic industries are re-analyzed.

RIS Selection Scheme for UAV-Based Multi-RIS-Aided Multiuser Downlink Network With Imperfect and Outdated CSI

In this paper, we explore the use of reconfigurable intelligent surface (RIS) in unmanned aerial vehicle (UAV) based multiuser downlink communications, where a flying UAV serves multiple single antenna users through multiple RISs mounted on various buildings. More specifically, we consider the selection of RISs based on the outdated and imperfect channel state information (CSI) of the composite UAV-RIS-User channels at the UAV. After selection process, the UAV communicates to the user via the selected RISs and also with the direct link. Particularly, we derive an infinite series based expression for selection probability of RISs under both the outdated and imperfect CSI of composite channels based selection scheme. We also derive the statistical distribution of instantaneously received signal-to-noise ratio (SNR) under outdated and imperfect CSI conditions of both the direct and composite links at the user. Next, using the derived statistics, we analyze the network’s performance in terms of the average coverage probability (ACP) and average bit error rate (ABER) over the complete UAV flight time. Moreover, we discuss the behavior of ACP and ABER for very small and very large values of UAV transmit power, respectively. It is depicted through numerical results that selecting more RISs from a group of small-sized RISs may not be as advantageous as selecting fewer RISs from a group of large-sized RISs. Moreover, we also demonstrate the effect of several system parameters such as number of RIS reflecting elements, number of selected RISs, the severity of UAV-RIS and RIS-User links, and the severity of imperfect and outdated CSI on the network’s performance. The analytical results are corroborated with Monte-Carlo simulations.

Skyward bound: Empowering disaster resilience with multi-UAV-assisted B5G networks for enhanced connectivity and energy efficiency

It is not uncommon that cellular communication infrastructures will likely be in a failure state when disaster strikes, e.g., floods, earthquakes, and forest fires. Therefore, developing an effective disaster management system is paramount to facing these critical circumstances. As a solution, unmanned aerial vehicles (UAVs) can be deployed as temporary aerial base stations to assist beyond fifth-generation (B5G) networks during and after the disaster. UAVs are increasingly used to aid the search and rescue (SAR) teams in multiple disaster management operations to improve disaster response effectiveness. However, maintaining connectivity and improving energy efficiency are indispensable to ensure continuous communication between the UAVs and SAR and prolong the network’s lifetime. This paper proposes a multi-UAV and SAR device-to-device communication model to maintain connectivity and enhance energy efficiency in multiple disaster scenarios. A clustering approach for UAV placement is utilized to improve energy efficiency, network connectivity, and coverage. We evaluate the performance of the proposed model in terms of the average capacity, energy efficiency, coverage probability and outage probability. The simulation results demonstrate an improved performance of multi-UAV and SAR collaborations and their appropriateness to support efficient and reliable disaster recovery in hard-to-reach and remote disaster areas.

Impacts of Antenna Downtilt and Backhaul Connectivity on the UAV-Enabled Heterogeneous Networks

The performance of unmanned aerial vehicle (UAV)-enabled networks is generally bottlenecked by severe inter-cell interference and limited backhaul connectivity. Against this backdrop, we analyze the performance of the UAV-enabled heterogeneous networks, where antenna downtilt at each UAV-mounted BS is employed to mitigate inter-cell interference, and the tethered UAV (TUAV)-mounted BS is deployed to provide backhaul connectivity for multiple spatially randomly distributed untethered UAV-mounted BSs, while they are cooperatively serving the terrestrial users. Through leveraging stochastic geometry, the user association probability and conditional distance distributions of serving links are derived. Then, the compact expressions of coverage probability and ergodic rate of the UAV-enabled heterogeneous networks are derived, and the impacts of antenna downtilt and backhaul connectivity are investigated. Numerical results validate our analysis and show the effectiveness of antenna downtilt and the TUAV-enabled backhaul connectivity. Moreover, we find that a larger antenna downtilt angle is required for a higher deployment altitude. We show that the optimal size of the network area that maximizes the average coverage probability reduces, when the backhaul connectivity is considered. We also demonstrate that the average network performance can be significantly improved by judiciously selecting the number of untethered UAVs, which is closely related to the the size of the network area. Finally, the advantage of the proposed UAV-enabled heterogeneous network on the average ergodic rate has been validated by comparing it with two benchmarks under the 3GPP channel model.

Simultaneous confidence intervals for quantile differences of several heterogeneous normal populations: With application to vitamin D supplement treatment on colorectal cancer patients.

In many applications, comparing the q-quantiles of several normal populations are more advantageous than comparing their means. In this paper, we consider the problem of constructing simultaneous confidence intervals (SCIs) for quantile differences of several heterogeneous normal distributions. To the best of our knowledge, this problem remains unsolved. We propose a novel method for constructing a set of SCI. We propose two new sets of SCI by using the proposed technique and discuss two classic and two simulation-based SCIs. We show that the proposed classic SCIs are conservative for all population parameters configuration. We also show that the simulation-based SCIs have correct coverage probability asymptotically. We then compare these six sets of SCI in terms of average volume and coverage probability via an extensive simulation study. Results show that one of the proposed classic SCI can be recommended. Finally, the proposed methods are illustrated by a real example that is a vitamin D study on colorectal cancer patients.

Confidence Intervals of a Loss Based PCI S′pmk Using Exponentiated-Exponential Distributed Quality Characteristics

Process capability indices (PCIs) are most effective devices/techniques used in industries for determining the quality of products and performance of manufacturing processes. In this article, we consider the process capability index which is based on an asymmetric loss function (linear exponential) and is applicable to normally as well as non-normally distributed processes. In order to estimate the PCI when the process follows exponentiated exponential distribution, we have used ten classical methods of estimation and the performances of these classical estimates for the index are compared in terms of mean squared errors (MSEs) through simulation study. Also, the confidence intervals for the index are constructed based on four bootstrap confidence interval (BCIs) methods. A simulation study is performed in order to compare the performance of these four BCIs in terms of average width and coverage probabilities. We use two published data sets related to electronic and food industries to illustrate the performance of the proposed methods of estimation and BCIs. All the data sets show that width of bias-corrected accelerated bootstrap interval is the lowest among all other considered BCIs.

Conformal asymmetric multi-quantile generative transformer for day-ahead wind power interval prediction

With the rapid increase in the installed capacity of wind power, day-ahead wind power interval prediction is becoming more and more important. To solve such a challenging problem and provide intervals of higher quality, this paper proposes a prediction method based on conformal asymmetric multi-quantile generative transformer. Herein, the multi-quantile generative transformer is a deep learning model that generates multi-quantile forecast results for next day through one forward propagation process. Then, two quantiles, whose width is the smallest while satisfying the nominal confidence constraint, are selected from the predicted sequence as the upper bound and lower bound of the asymmetric interval. Furthermore, we introduce the conformal quantile regression to calibrate the bounds of the prediction interval to ensure that its coverage rate is as close as nominal confidence. The experiments show that the proposed method surpasses the benchmarks by providing narrower prediction intervals with more accurate empirical coverage probability. Under nominal confidence 90%, it gives prediction intervals with average empirical coverage probability of 90.50% and normalized average width of 0.44 on four wind farms. Compared with symmetric prediction intervals given by common benchmark quantile long short term memory network, the average width is reduced by 19.6%.

Forecasting with a panel Tobit model

We use a dynamic panel Tobit model with heteroskedasticity to generate forecasts for a large cross‐section of short time series of censored observations. Our fully Bayesian approach allows us to flexibly estimate the cross‐sectional distribution of heterogeneous coefficients and then implicitly use this distribution as prior to construct Bayes forecasts for the individual time series. In addition to density forecasts, we construct set forecasts that explicitly target the average coverage probability for the cross‐section. We present a novel application in which we forecast bank‐level loan charge‐off rates for small banks.

Estimation and Confidence Intervals of Modified Process Capability Index Using Robust Measure of Variability

Abstract The process capability index (PCI), denoted by 𝐼, is a well-known characteristic in quality control analysis. Using Gini’s mean difference, we construct a new PCI, I G I_{G} say, assuming the two-parameter Weibull distribution (WD). In order to estimate the proposed I G I_{G} when the process follows the WD, we use five classical methods of estimation and compare the performance of the obtained estimators with respect to their mean squared errors (MSEs) through a simulation study. Confidence intervals for the proposed PCI are constructed based on five bootstrap confidence intervals (BCIs) methods. Monte Carlo simulation study has been carried out to compare the performance of these five BCIs in terms of average widths and coverage probabilities. Finally, three real data sets from electronic and food industries are employed for illustrating the effectiveness of the proposed study. All these data sets show that the width of bias-corrected accelerated bootstrap interval is minimum among all other considered BCIs.

Bootstrap-t Confidence Interval on Local Polynomial Regression Prediction

In local polynomial regression, prediction confidence interval estimation using standard theory will give coverage probability close to exact coverage probability. However, if the normality assumption is not met, the bootstrap method makes it possible to apply it. The working principle of the bootstrap method uses the resampling method where the sample data becomes a population and there is no need to know the distribution of the sample data is normal or not. Indiscriminate selection of smoothing parameters allows scatterplot results from local polynomial regressions to be rough and can even lead to misleading statistical conclusions. It is necessary to consider the optimal smoothing parameters to get local polynomial regression predictions that are not overfitting or underfitting. We offer two new algorithms based on the nested bootstrap resampling method to determine the bootstrap-t confidence interval in predicting local polynomial regression. Both algorithms consider the search for optimal smoothing parameters. The first algorithm performs paired and residual bootstrap samples, and the second algorithm performs based on residuals with residuals. The first algorithm provides a scatterplot and reasonable coverage probability on relatively large sample data. In contrast, the second algorithm is more powerful for each data size, including for relatively small sample data sizes. The mean of the bootstrap-t confidence interval coverage probability shows that the second algorithm for second-degree local polynomial regression is better than the other three. However, the larger the sample data size gives, the closer the closer the average coverage probability of the two algorithms is to the nominal coverage probability.

Event-related microstate dynamics represents working memory performance

In recent years, EEG microstate analysis has attracted much attention as a tool for characterizing the spatial and temporal dynamics of large-scale electrophysiological activities in the human brain. Canonical 4 states (classes A, B, C, and D) have been widely reported, and they have been pointed out for their relationships with cognitive functions and several psychiatric disorders such as schizophrenia, in particular, through their static parameters such as average duration, occurrence, coverage, and transition probability. However, the relationships between event-related microstate changes and their related cognitive functions, as is often analyzed in event-related potentials under time-locked frameworks, is still not well understood. Furthermore, not enough attention has been paid to the relationship between microstate dynamics and static characteristics.To clarify the relationships between the static microstate parameters and dynamic microstate changes, and between the dynamics and working memory (WM) function, we first examined the temporal profiles of the microstates during the N-back task. We found significant event-related microstate dynamics that differed predominantly with WM loads, which were not clearly observed in the static parameters. Furthermore, in the 2-back condition, patterns of state transitions from class A to C in the high- and low-performance groups showed prominent differences at 50–300 ms after stimulus onset. We also confirmed that the transition patterns of the specific time periods were able to predict the performance level (low or high) in the 2-back condition at a significant level, where a specific transition between microstates, namely from class A to C with specific polarity, contributed to the prediction robustly. Taken together, our findings indicate that event-related microstate dynamics at 50–300 ms after onset may be essential for WM function. This suggests that event-related microstate dynamics can reflect more highly-refined brain functions.

Modified estimation and confidence intervals of an asymmetric loss‐based process capability index Cpm′$\mathcal {C}^{\prime }_{pm}$

AbstractIn this article, we are interested to estimate a new capability index , which is based on asymmetric loss function (linear‐exponential) when the underlying process follows normal distribution. We estimate the parameters of the model using maximum likelihood method, bias‐corrected maximum likelihood method and bootstrap bias‐corrected maximum likelihood method and subsequently the process capability index (PCI) using the cited methods. Through extensive simulation studies, we compare the performances of the aforementioned methods of estimation for the PCI in terms of their absolute bias (AB) and MSEs. Besides, four bootstrap methods are employed for constructing the confidence intervals for the index by using the considered methods of estimation. The performances of the bootstrap confidence intervals (BCIs) are also compared in terms of average widths (AWs) and coverage probabilities (CPs) using Monte Carlo simulation. Finally, for illustrating the effectiveness of the proposed methods of estimation and BCIs, two real data sets from electronic industries are analyzed. All these data sets show that width of bias‐corrected accelerated bootstrap interval is minimum among all other considered BCIs.