Research

19 pages, 488 KiB

Open AccessArticle

The Simultaneous Confidence Interval for the Ratios of the Coefficients of Variation of Multiple Inverse Gaussian Distributions and Its Application to PM_2.5 Data

by Wasana Chankham, Sa-Aat Niwitpong and Suparat Niwitpong

Symmetry 2024, 16(3), 331; https://doi.org/10.3390/sym16030331 - 08 Mar 2024

Viewed by 669

Abstract

Due to slash/burn agricultural activity and frequent forest fires,

P M_{2.5}

has become a significant air pollution problem in Thailand, especially in the north and north east regions. Since its dispersion differs both spatially and temporally, estimating

P M_{2.5}

concentrations discretely [...] Read more.

Due to slash/burn agricultural activity and frequent forest fires,

P M_{2.5}

has become a significant air pollution problem in Thailand, especially in the north and north east regions. Since its dispersion differs both spatially and temporally, estimating

P M_{2.5}

concentrations discretely by area, for which the inverse Gaussian distribution is suitable, can provide valuable information. Herein, we provide derivations of the simultaneous confidence interval for the ratios of the coefficients of variation of multiple inverse Gaussian distributions using the generalized confidence interval, the Bayesian interval based on the Jeffreys’ rule prior, the fiducial interval, and the method of variance estimates recovery. The efficacies of these methods were compared by considering the coverage probability and average length obtained from simulation results of daily

P M_{2.5}

datasets. The findings indicate that in most instances, the fiducial method with the highest posterior density demonstrated a superior performance. However, in certain scenarios, the Bayesian approach using the Jeffreys’ rule prior for the highest posterior density yielded favorable results. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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19 pages, 1325 KiB

Open AccessArticle

The Testing Procedure for the Overall Lifetime Performance Index of Rayleigh Products in Multiple Production Lines Based on the Progressive Type I Interval Censored Sample

by Shu-Fei Wu and Pei-Tzu Huang

Symmetry 2024, 16(2), 195; https://doi.org/10.3390/sym16020195 - 06 Feb 2024

Viewed by 927

Abstract

The lifetime performance index is a process capability index that is commonly used for the evaluation of the durability of products in life testing and reliability analysis. In the context of multiple production lines, we introduce an overall lifetime performance index and explore [...] Read more.

The lifetime performance index is a process capability index that is commonly used for the evaluation of the durability of products in life testing and reliability analysis. In the context of multiple production lines, we introduce an overall lifetime performance index and explore the relationship between this comprehensive index and individual lifetime performance indices. For products with lifespans following the Rayleigh distribution in the ith production line, we delve into the maximum likelihood estimator and asymptotic distribution to derive both the individual and overall lifetime performance indices. By establishing a predetermined target for the overall lifetime performance index, we can determine the corresponding target for each individual lifetime performance index. The testing algorithmic procedure is proposed to ascertain whether the overall lifetime performance index has reached its target value based on the maximum likelihood estimator, accompanied by figures illustrating the analysis of test power. We found that there is a monotonic relationship for the test power with various structures of parameters. Finally, a practical illustration with one numeral example is presented to demonstrate how the testing procedure is employed to evaluate the capabilities of multiple production lines. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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13 pages, 3507 KiB

Open AccessArticle

Spectral Analysis of Anomalous Capacitance Measurements in Interleaving Structures: Study of Frequency Distribution in Photomultipliers

by Víctor Milián-Sánchez, Miguel E. Iglesias-Martínez, Jose Guerra Carmenate, Juan Carlos Castro-Palacio, Eduardo Balvis Outeiriño, Pedro Fernández de Córdoba, Francisco Misael Muñoz-Pérez, Juan Antonio Monsoriu and Sarira Sahu

Symmetry 2024, 16(1), 15; https://doi.org/10.3390/sym16010015 - 21 Dec 2023

Viewed by 655

Abstract

This study presents experimental results on capacitance fluctuations in several devices located within an interleaving structure. Specifically, it examines the behavior of the capacitance between the anode and cathode of a photomultiplier, comparing it with the characteristics of the ultra-stable capacitor analyzed in [...] Read more.

This study presents experimental results on capacitance fluctuations in several devices located within an interleaving structure. Specifically, it examines the behavior of the capacitance between the anode and cathode of a photomultiplier, comparing it with the characteristics of the ultra-stable capacitor analyzed in via measurements inside and outside a modified Faraday cage. The results cover spectral and correlation analyses both inside and outside the box, confirming differences in the spectrum using the periodograms. In particular, the confidence intervals for the mean capacitance values show significant changes between the two scenarios, from the inside to the outside of the enclosure. In the case of the ultra-stable capacitor, there is an increase from 0.004 to 0.008 nF. On the other hand, a symmetry analysis is conducted for all measurements taken both outside and inside the modified Faraday cage. It is observed that in all cases, there is clear non-symmetric behavior in the data. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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15 pages, 424 KiB

Open AccessArticle

A New Cosine-Originated Probability Distribution with Symmetrical and Asymmetrical Behaviors: Repetitive Acceptance Sampling with Reliability Application

by Huda M. Alshanbari, Gadde Srinivasa Rao, Jin-Taek Seong, Sultan Salem and Saima K. Khosa

Symmetry 2023, 15(12), 2187; https://doi.org/10.3390/sym15122187 - 12 Dec 2023

Viewed by 778

Abstract

Several new acceptance sampling plans using various probability distribution methods have been developed in the literature. However, there is no published work on the design of new sampling plans using trigonometric-based probability distributions. In order to cover this amazing and fascinating research gap, [...] Read more.

Several new acceptance sampling plans using various probability distribution methods have been developed in the literature. However, there is no published work on the design of new sampling plans using trigonometric-based probability distributions. In order to cover this amazing and fascinating research gap, we first introduce a novel probabilistic method called a new modified cosine-G method. A special member of the new modified cosine-G method, namely, a new modified cosine-Weibull distribution, is examined and implemented. The density function of the new model possesses symmetrical as well as asymmetrical behaviors. The usefulness and superior fitting power of the new modified cosine-Weibull distribution are demonstrated by analyzing an asymmetrical data set. Furthermore, based on the new modified cosine-Weibull distribution, we develop a new repetitive acceptance sampling strategy for attributes with specified shape parameters. Finally, a real-world application is presented to illustrate the proposed repetitive acceptance sampling strategy. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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19 pages, 352 KiB

Open AccessArticle

Insight into Spatially Colored Stochastic Heat Equation: Temporal Fractal Nature of the Solution

by Wensheng Wang

Symmetry 2023, 15(12), 2181; https://doi.org/10.3390/sym15122181 - 10 Dec 2023

Viewed by 640

Abstract

In this paper, the solution to a spatially colored stochastic heat equation (SHE) is studied. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time has exact, dimension-dependent, global continuity moduli, [...] Read more.

In this paper, the solution to a spatially colored stochastic heat equation (SHE) is studied. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time has exact, dimension-dependent, global continuity moduli, and laws of the iterated logarithm (LILs). It is obtained that the set of fast points at which LILs fail in this process, and occur infinitely often, is a random fractal, the size of which is evaluated by its Hausdorff dimension. These points of this process are everywhere dense with the power of the continuum almost surely, and their hitting probabilities are determined by the packing dimension

{dim}_{P} (E)

of the target set E. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

17 pages, 618 KiB

Open AccessArticle

Simultaneous Confidence Intervals for All Pairwise Differences between Means of Weibull Distributions

by Manussaya La-ongkaew, Sa-Aat Niwitpong and Suparat Niwitpong

Symmetry 2023, 15(12), 2142; https://doi.org/10.3390/sym15122142 - 01 Dec 2023

Cited by 1 | Viewed by 1162

Abstract

The Weibull distribution is a continuous probability distribution that finds wide application in various fields for analyzing real-world data. Specifically, wind speed data often adhere to the Weibull distribution. In our study, our aim is to compare the mean wind speed datasets from [...] Read more.

The Weibull distribution is a continuous probability distribution that finds wide application in various fields for analyzing real-world data. Specifically, wind speed data often adhere to the Weibull distribution. In our study, our aim is to compare the mean wind speed datasets from different areas in Thailand. To achieve this, we proposed simultaneous confidence intervals for all pairwise differences between the means of Weibull distributions. The generalized confidence interval (GCI), method of variance estimates recovery (MOVER), and a Bayesian approach, utilizing both gamma and uniform prior distributions, are proposed to construct simultaneous confidence intervals. Through simulations, we find that the Bayesian highest posterior density (HPD) interval using a gamma prior distribution demonstrates satisfactory performance, while the GCI proves to be a viable alternative. Finally, we applied these proposed approaches to real wind speed data in northeastern and southern Thailand to illustrate their effectiveness and practicality. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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25 pages, 381 KiB

Open AccessArticle

Model Selection in Generalized Linear Models

by Abdulla Mamun and Sudhir Paul

Symmetry 2023, 15(10), 1905; https://doi.org/10.3390/sym15101905 - 11 Oct 2023

Cited by 1 | Viewed by 1015

Abstract

The problem of model selection in regression analysis through the use of forward selection, backward elimination, and stepwise selection has been well explored in the literature. The main assumption in this, of course, is that the data are normally distributed and the main [...] Read more.

The problem of model selection in regression analysis through the use of forward selection, backward elimination, and stepwise selection has been well explored in the literature. The main assumption in this, of course, is that the data are normally distributed and the main tool used here is either a t test or an F test. However, the properties of these model selection procedures are not well-known. The purpose of this paper is to study the properties of these procedures within generalized linear regression models, considering the normal linear regression model as a special case. The main tool that is being used is the score test. However, the F test and other large sample tests, such as the likelihood ratio and the Wald test, the AIC, and the BIC, are included for the comparison. A systematic study, through simulations, of the properties of this procedure was conducted, in terms of level and power, for symmetric and asymmetric distributions, such as normal, Poisson, and binomial regression models. Extensions for skewed distributions, over-dispersed Poisson (the negative binomial), and over-dispersed binomial (the beta-binomial) regression models, are also given and evaluated. The methods are applied to analyze two health datasets. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

22 pages, 562 KiB

Open AccessArticle

A New Sine-Based Distributional Method with Symmetrical and Asymmetrical Natures: Control Chart with Industrial Implication

by Huda M. Alshanbari, Gadde Srinivasa Rao, Jin-Taek Seong and Saima K. Khosa

Symmetry 2023, 15(10), 1892; https://doi.org/10.3390/sym15101892 - 09 Oct 2023

Cited by 1 | Viewed by 900

Abstract

Control charts are widely used in quality control and industrial sectors. Because of their important role, researchers are focusing on the development of new control charts. According to our study, there is no significant amount of published work on control charts using trigonometrically [...] Read more.

Control charts are widely used in quality control and industrial sectors. Because of their important role, researchers are focusing on the development of new control charts. According to our study, there is no significant amount of published work on control charts using trigonometrically generated distribution methods. In this paper, we contribute to this interesting research gap by developing a new control chart using a sine-based distributional method. The proposed distributional method (or family of probability distributions) may be called a new modified sine-G family of distributions. Based on the new modified sine-G method, a novel modification of the Weibull distribution, namely, a new modified sine-Weibull distribution, is introduced. The new modified sine-Weibull distribution is flexible enough to capture symmetrical and asymmetrical behaviors of its density function. An industrial application is considered to show the importance and implacability of the proposed distribution in quality control. Based on the proposed model, an attribute control chart is developed under a truncated life test. The control chart limits (ARLs) are also computed for the proposed model. The ARLs of the proposed control chart are compared with the attribute control chart of the Weibull distribution. The results show that the developed chart is more efficient than the existing attribute control chart for the Weibull distribution. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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26 pages, 511 KiB

Open AccessArticle

Bayesian Analysis Using Joint Progressive Type-II Censoring Scheme

by Mohamed G. M. Ghazal, Mustafa M. Hasaballah, Rashad M. EL-Sagheer, Oluwafemi Samson Balogun and Mahmoud E. Bakr

Symmetry 2023, 15(10), 1884; https://doi.org/10.3390/sym15101884 - 07 Oct 2023

Cited by 1 | Viewed by 732

Abstract

The joint censoring technique becomes crucial when the study’s aim is to assess the comparative advantages of products concerning their service times. In recent years, there has been a growing interest in progressive censoring as a means to reduce both cost and experiment [...] Read more.

The joint censoring technique becomes crucial when the study’s aim is to assess the comparative advantages of products concerning their service times. In recent years, there has been a growing interest in progressive censoring as a means to reduce both cost and experiment duration. This article delves into the realm of statistical inference for the three-parameter Burr-XII distribution using a joint progressive Type II censoring approach applied to two separate samples. We explore both maximum likelihood and Bayesian methods for estimating model parameters. Furthermore, we derive approximate confidence intervals based on the observed information matrix and employ four bootstrap methods to obtain confidence intervals. Bayesian estimators are presented for both symmetric and asymmetric loss functions. Since closed-form solutions for Bayesian estimators are unattainable, we resort to the Markov chain Monte Carlo method to compute these estimators and the corresponding credible intervals. To assess the performance of our estimators, we conduct extensive simulation experiments. Finally, to provide a practical illustration, we analyze a real dataset. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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14 pages, 342 KiB

Open AccessArticle

Investigation of Exponential Distribution Utilizing Randomly Censored Data under Balanced Loss Functions and Its Application to Clinical Data

by Mustafa M. Hasaballah, Oluwafemi Samson Balogun and Mahmoud E. Bakr

Symmetry 2023, 15(10), 1854; https://doi.org/10.3390/sym15101854 - 02 Oct 2023

Viewed by 748

Abstract

In this research, random censoring is employed as a methodology for parameter estimation within the context of an exponential distribution. These parameter estimations are conducted using both the Bayesian and maximum likelihood approaches. In the Bayesian framework, Lindley’s approximation method is applied to [...] Read more.

In this research, random censoring is employed as a methodology for parameter estimation within the context of an exponential distribution. These parameter estimations are conducted using both the Bayesian and maximum likelihood approaches. In the Bayesian framework, Lindley’s approximation method is applied to derive estimates, which are subsequently assessed under three distinct balanced loss functions. To gauge the efficacy of different estimation techniques, simulation-based investigations are conducted. Additionally, a real-world data analysis is executed to illustrate the practical applicability of these methodologies. The findings consistently underscore the superiority of Bayesian parameter estimates in comparison with their maximum likelihood counterparts across all analyzed methodologies. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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20 pages, 727 KiB

Open AccessArticle

A New Bivariate Family Based on Archimedean Copulas: Simulation, Regression Model and Application

by Gabriela M. Rodrigues, Edwin M. M. Ortega, Roberto Vila and Gauss M. Cordeiro

Symmetry 2023, 15(9), 1778; https://doi.org/10.3390/sym15091778 - 18 Sep 2023

Cited by 1 | Viewed by 1163

Abstract

We use the Clayton and Frank copulas and the exponentiated odd log-logistic family to define a new flexible bivariate model to fit bimodal and asymmetry data. The copulas allow different distributions for the response variable, thus making analysis more suitable. We present some [...] Read more.

We use the Clayton and Frank copulas and the exponentiated odd log-logistic family to define a new flexible bivariate model to fit bimodal and asymmetry data. The copulas allow different distributions for the response variable, thus making analysis more suitable. We present some structural properties of the new model and describe a simulation study to show the consistency of the estimators. We construct a bivariate regression model based on the new family to fit oak lettuce plant data for different concentrations of silicon dioxide and organosilicon compounds. We check the response variables fresh weight and plant height together in order to verify the existing correlation between them. These variables exhibit a bimodal form, and the family used is able to model this behavior. Different marginal distributions are selected, which is an interesting point of the copula methodology. The variables have strong positive dependence, and the experiment is carried out comparing the control treatment with others leading to the following results: (i) the treatment 1-ethoxysilatrane (with concentrations 5 × 10

^{- 4}

mL·L

^{- 1}

and 10

^{- 3}

mL·L

^{- 1}

) is not significant for the response variables; (ii) the treatment amorphous silicon dioxide (with concentrations 50 mg·L

^{- 1}

and 100 mg·L

^{- 1}

) and the same treatment (with concentrations 5 × 10

^{- 3}

mL·L

^{- 1}

and 10

^{- 2}

mL·L

^{- 1}

) are significant and have positive effects on both responses; (iii) the treatment amorphous silicon dioxide (with concentrations 200 mg·L

^{- 1}

and 300 mg·L

^{- 1}

) are significant and have negative effects on the response variables. Overall, the proposed bivariate model is suitable for the current data and can be useful in other applications. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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15 pages, 3128 KiB

Open AccessArticle

Modeling of System Availability and Bayesian Analysis of Bivariate Distribution

by Muhammad Farooq, Ahtasham Gul, Huda M. Alshanbari and Saima K. Khosa

Symmetry 2023, 15(9), 1698; https://doi.org/10.3390/sym15091698 - 04 Sep 2023

Viewed by 687

Abstract

To meet the desired standard, it is important to monitor and analyze different engineering processes to obtain the desired output. The bivariate distributions have received a significant amount of attention in recent years due to their ability to describe randomness of natural as [...] Read more.

To meet the desired standard, it is important to monitor and analyze different engineering processes to obtain the desired output. The bivariate distributions have received a significant amount of attention in recent years due to their ability to describe randomness of natural as well as artificial mechanisms. In this article, a bivariate model is constructed by compounding two independent asymmetric univariate distributions and by using the nesting approach to study the effect of each component on reliability for better understanding. Furthermore, the Bayes analysis of system availability is studied by considering prior parametric variations in the failure time and repair time distributions. Basic statistical characteristics of marginal distribution like mean median and quantile function are discussed. We used inverse Gamma prior to study its frequentist properties by conducting a Monte Carlo Markov Chain (MCMC) sampling scheme. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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21 pages, 2923 KiB

Open AccessArticle

Estimation and Prediction for Alpha-Power Weibull Distribution Based on Hybrid Censoring

by Ehab M. Almetwally, Refah Alotaibi and Hoda Rezk

Symmetry 2023, 15(9), 1687; https://doi.org/10.3390/sym15091687 - 02 Sep 2023

Cited by 3 | Viewed by 674

Abstract

This work discusses the issues of estimation and prediction when lifespan data following alpha-power Weibull distribution are observed under Type II hybrid censoring. We calculate point and related interval estimates for both issues using both non-Bayesian and Bayesian methods. Using the Newton–Raphson technique [...] Read more.

This work discusses the issues of estimation and prediction when lifespan data following alpha-power Weibull distribution are observed under Type II hybrid censoring. We calculate point and related interval estimates for both issues using both non-Bayesian and Bayesian methods. Using the Newton–Raphson technique under the classical approach, we compute maximum likelihood estimates for point estimates in the estimation problem. Under the Bayesian approach, we compute Bayes estimates under informative and non-informative priors using the symmetric loss function. Using the Fisher information matrix under classical and Bayesian techniques, the corresponding interval estimates are derived. Additionally, using the best unbiased and conditional median predictors under the classical approach, as well as Bayesian predictive and associated Bayesian predictive interval estimates in the prediction approach, the predictive point estimates and associated predictive interval estimates are computed. We compare several suggested approaches of estimation and prediction using real data sets and Monte Carlo simulation studies. A conclusion is provided. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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24 pages, 773 KiB

Open AccessArticle

Theoretical Aspects for Bayesian Predictions Based on Three-Parameter Burr-XII Distribution and Its Applications in Climatic Data

by Mustafa M. Hasaballah, Abdulhakim A. Al-Babtain, Md. Moyazzem Hossain and Mahmoud E. Bakr

Symmetry 2023, 15(8), 1552; https://doi.org/10.3390/sym15081552 - 07 Aug 2023

Cited by 1 | Viewed by 859

Abstract

Symmetry and asymmetry play vital roles in prediction. Symmetrical data, which follows a predictable pattern, is easier to predict compared to asymmetrical data, which lacks a predictable pattern. Symmetry helps identify patterns within data that can be utilized in predictive models, while asymmetry [...] Read more.

Symmetry and asymmetry play vital roles in prediction. Symmetrical data, which follows a predictable pattern, is easier to predict compared to asymmetrical data, which lacks a predictable pattern. Symmetry helps identify patterns within data that can be utilized in predictive models, while asymmetry aids in identifying outliers or anomalies that should be considered in the predictive model. Among the various factors associated with storms and their impact on surface temperatures, wind speed stands out as a significant factor. This paper focuses on predicting wind speed by utilizing unified hybrid censoring data from the three-parameter Burr-XII distribution. Bayesian prediction bounds for future observations are obtained using both one-sample and two-sample prediction techniques. As explicit expressions for Bayesian predictions of one and two samples are unavailable, we propose the use of the Gibbs sampling process in the Markov chain Monte Carlo framework to obtain estimated predictive distributions. Furthermore, we present a climatic data application to demonstrate the developed uncertainty procedures. Additionally, a simulation research is carried out to examine and contrast the effectiveness of the suggested methods. The results reveal that the Bayes estimates for the parameters outperformed the Maximum likelihood estimators. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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24 pages, 358 KiB

Open AccessArticle

Confidence Intervals for Mean and Difference between Means of Delta-Lognormal Distributions Based on Left-Censored Data

by Warisa Thangjai and Sa-Aat Niwitpong

Symmetry 2023, 15(6), 1216; https://doi.org/10.3390/sym15061216 - 07 Jun 2023

Cited by 1 | Viewed by 1111

Abstract

A delta-lognormal distribution consists of zero and positive values. The positive values follow a lognormal distribution, which is an asymmetric distribution. It is well known that the logarithm of these values follows a normal distribution, which is a symmetric distribution. The delta-lognormal distribution [...] Read more.

A delta-lognormal distribution consists of zero and positive values. The positive values follow a lognormal distribution, which is an asymmetric distribution. It is well known that the logarithm of these values follows a normal distribution, which is a symmetric distribution. The delta-lognormal distribution is used in medical and environmental sciences. This study considers the challenges of constructing confidence intervals for the mean and difference between means of delta-lognormal distributions containing left-censored data and applies them to compare two daily rainfall average areas in Thailand. Three different approaches for constructing confidence intervals for the mean of the delta-lognormal distribution containing left-censored data, based on the generalized confidence interval approach, the Bayesian approach, and the parametric bootstrap approach, are developed. Moreover, four different approaches for constructing confidence intervals for the difference between means of delta-lognormal distributions containing left-censored data, based on the generalized confidence interval approach, the Bayesian approach, the parametric bootstrap approach, and the method of variance estimates recovery approach, are considered. The performance of the proposed confidence intervals is evaluated by Monte Carlo simulation. The simulation studies indicate that the Bayesian approach can be considered as an alternative to construct a credible interval for the mean of the delta-lognormal distribution containing left-censored data. Additionally, the generalized confidence interval and Bayesian approaches can be recommended as alternatives to estimate the confidence interval for the difference between means of delta-lognormal distributions containing left-censored data. All approaches are illustrated using the daily rainfall data from Chiang Mai and Lampang provinces in Thailand. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

22 pages, 659 KiB

Open AccessArticle

Simulation Techniques for Strength Component Partially Accelerated to Analyze Stress–Strength Model

by Manal M. Yousef, Aisha Fayomi and Ehab M. Almetwally

Symmetry 2023, 15(6), 1183; https://doi.org/10.3390/sym15061183 - 01 Jun 2023

Cited by 3 | Viewed by 1250

Abstract

Based on independent progressive type-II censored samples from two-parameter Burr-type XII distributions, various point and interval estimators of

δ = P (Y < X)

were proposed when the strength variable was subjected to the step–stress partially accelerated life test. The point [...] Read more.

Based on independent progressive type-II censored samples from two-parameter Burr-type XII distributions, various point and interval estimators of

δ = P (Y < X)

were proposed when the strength variable was subjected to the step–stress partially accelerated life test. The point estimators computed were maximum likelihood and Bayesian under various symmetric and asymmetric loss functions. The interval estimations constructed were approximate, bootstrap-P, and bootstrap-T confidence intervals, and a Bayesian credible interval. A Markov Chain Monte Carlo approach using Gibbs sampling was designed to derive the Bayesian estimate of

δ

. Based on the mean square error, bias, confidence interval length, and coverage probability, the results of the numerical analysis of the performance of the maximum likelihood and Bayesian estimates using Monte Carlo simulations were quite satisfactory. To support the theoretical component, an empirical investigation based on two actual data sets was carried out. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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13 pages, 866 KiB

Open AccessArticle

Confidence Interval Estimation for the Ratio of the Percentiles of Two Delta-Lognormal Distributions with Application to Rainfall Data

by Warisa Thangjai, Sa-Aat Niwitpong, Suparat Niwitpong and Narudee Smithpreecha

Symmetry 2023, 15(4), 794; https://doi.org/10.3390/sym15040794 - 24 Mar 2023

Cited by 4 | Viewed by 1015

Abstract

The log-normal distribution (skewed distribution or asymmetry distribution) is used to describe random variables comprising positive real values. It is well known that the logarithm values of these are normally distributed (symmetry distribution). Positively right-skewed data applicable to the log-normal distribution are frequently [...] Read more.

The log-normal distribution (skewed distribution or asymmetry distribution) is used to describe random variables comprising positive real values. It is well known that the logarithm values of these are normally distributed (symmetry distribution). Positively right-skewed data applicable to the log-normal distribution are frequently observed in the fields of environmental studies, biology, and medicine. The number of zero observations follows a binomial distribution. However, problems can arise in the analysis of data containing zero observations along with log-normally distributed data, for which the delta-lognormal distribution is often referred to for using the analysis of the data. In statistics, the percentile provides the relative standing of a numerical data point when compared to all of the others in a distribution with reference to the observations at or below it. In this study, estimates for the confidence interval for the ratio of the percentiles of two delta-lognormal distributions are constructed using fiducial generalized confidence interval approaches based on the fiducial quantity and the optimal generalized fiducial quantity, the Bayesian approach, and the parametric bootstrap method. As assessed by Monte Carlo simulations using the RStudio programming in terms of the coverage probability and the average length, the Bayesian approach performed quite well by providing adequate coverage probabilities along with the shortest average lengths in all of the scenarios tested. Daily rainfall data contain both zero and positive values. The daily rainfall data can usually be fitted to the delta-lognormal distribution. Their application to rainfall data is also provided to illustrate their efficacies with real data. The efficacy of the approach is used to compare two rainfall dispersion populations. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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28 pages, 1358 KiB

Open AccessArticle

Analysis of Milk Production and Failure Data: Using Unit Exponentiated Half Logistic Power Series Class of Distributions

by Safar M. Alghamdi, Mansour Shrahili, Amal S. Hassan, Rokaya Elmorsy Mohamed, Ibrahim Elbatal and Mohammed Elgarhy

Symmetry 2023, 15(3), 714; https://doi.org/10.3390/sym15030714 - 13 Mar 2023

Cited by 5 | Viewed by 1494

Abstract

The unit exponentiated half logistic power series (UEHLPS), a family of compound distributions with bounded support, is introduced in this study. This family is produced by compounding the unit exponentiated half logistic and power series distributions. In the UEHLPS class, some interesting compound [...] Read more.

The unit exponentiated half logistic power series (UEHLPS), a family of compound distributions with bounded support, is introduced in this study. This family is produced by compounding the unit exponentiated half logistic and power series distributions. In the UEHLPS class, some interesting compound distributions can be found. We find formulas for the moments, density and distribution functions, limiting behavior, and other UEHLPS properties. Five well-known estimating approaches are used to estimate the parameters of one sub-model, and a simulation study is created. The simulated results show that the maximum product of spacing estimates had lower accuracy measure values than the other estimates. Ultimately, three real data sets from various scientific areas are used to analyze the performance of the new class. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)

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