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Symmetry
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Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines
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Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 July 2021) | Viewed by 29096

Special Issue Editors

Dr. Juan Carlos Castro-Palacio

E-Mail Website
Guest Editor
Centro de Tecnologías Físicas, Universitat Politècnica de València, 46022 Valencia, Spain
Interests: physics teaching; atomistic dynamics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Pedro José Fernández de Córdoba Castellá

E-Mail Website
Guest Editor
Grupo de Modelización Interdisciplinar, InterTech, Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera, 46022 Valencia, Spain
Interests: mathematical modeling; numerical simulation
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Skewed distributions are transversal and ubiquitous to all scientific disciplines. They have captured the attention of many researchers, as a deep understanding of their underlying probabilistic mechanisms is crucial in many fields. The right choice of the probability distribution for a non-normal stochastic process and the proper interpretation of its parameters can be very challenging and of enormous importance in fields like physics, chemistry, biology, and social sciences.

The guidelines for contributions to this Special Issue include (but are not limited to) the following topics, which are divided into two broad groups:

I. Methods and applications of skew distributions
- New applications and parameter interpretations of the main skewed distributions;
- Parameter estimation and statistical developments;
- Advances in modelling and simulations (i.e., Monte Carlo sampling) of processes in mathematics, physics, chemistry, biology, and social sciences;
- Efficient numerical methods to handle skewed distributions;
- Skewed distributions and the modelling of infectious diseases, which include COVID-19.

II. Skewed distributions in describing nature processes
- True meaning of skewed distributions in nature;
- Skewed distributions in psychological and neurological sciences;
- Non-normal distributions in biological and medical sciences;
- Skewed distributions in describing social processes;
- Origin and fundamental interpretations of skewed distributions in mathematics, physics, chemistry, biology, and social sciences.

Dr. Juan Carlos Castro-Palacio
Prof. Dr. Pedro José Fernández de Córdoba Castellá
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (15 papers)

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Editorial

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5 pages, 249 KiB  
Editorial
Editorial of the Special Issue “Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines”
by Juan Carlos Castro-Palacio and Pedro Fernández-de-Córdoba
Symmetry 2023, 15(3), 600; https://doi.org/10.3390/sym15030600 - 27 Feb 2023
Viewed by 888
Abstract
This Special Issue includes a total of 14 articles on skewed probability distributions and applications across disciplines [...] Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)

Research

Jump to: Editorial

16 pages, 433 KiB  
Article
A Probability Mass Function for Various Shapes of the Failure Rates, Asymmetric and Dispersed Data with Applications to Coronavirus and Kidney Dysmorphogenesis
by Mahmoud El-Morshedy, Morad Alizadeh, Afrah Al-Bossly and Mohamed S. Eliwa
Symmetry 2021, 13(10), 1790; https://doi.org/10.3390/sym13101790 - 26 Sep 2021
Cited by 3 | Viewed by 1608
Abstract
In this article, a discrete analogue of an extension to a two-parameter half-logistic model is proposed for modeling count data. The probability mass function of the new model can be expressed as a mixture representation of a geometric model. Some of its statistical [...] Read more.
In this article, a discrete analogue of an extension to a two-parameter half-logistic model is proposed for modeling count data. The probability mass function of the new model can be expressed as a mixture representation of a geometric model. Some of its statistical properties, including hazard rate function, moments, moment generating function, conditional moments, stress-strength analysis, residual entropy, cumulative residual entropy and order statistics with its moments, are derived. It is found that the new distribution can be utilized to model positive skewed data, and it can be used for analyzing equi- and over-dispersed data. Furthermore, the hazard rate function can be either decreasing, increasing or bathtub. The parameter estimation through the classical point of view has been performed using the method of maximum likelihood. A detailed simulation study is carried out to examine the outcomes of the estimators. Finally, two distinctive real data sets are analyzed to prove the flexibility of the proposed discrete distribution. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
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19 pages, 2256 KiB  
Article
Experimental Design for the Lifetime Performance Index of Weibull Products Based on the Progressive Type I Interval Censored Sample
by Shu-Fei Wu, Yu-Cheng Wu, Chi-Han Wu and Wei-Tsung Chang
Symmetry 2021, 13(9), 1691; https://doi.org/10.3390/sym13091691 - 14 Sep 2021
Cited by 8 | Viewed by 1369
Abstract
In this study, the experimental design is developed based on the testing procedure for the lifetime performance index of products following Weibull lifetime distribution under progressive type I interval censoring. This research topic is related to asymmetrical probability distributions and applications across disciplines. [...] Read more.
In this study, the experimental design is developed based on the testing procedure for the lifetime performance index of products following Weibull lifetime distribution under progressive type I interval censoring. This research topic is related to asymmetrical probability distributions and applications across disciplines. The asymptotic distribution of the maximum likelihood estimator of the lifetime performance index is utilized to develop the testing procedure. In order to reach the given power level, the minimum sample size is determined and tabulated. In order to minimize the total cost that occurred under progressive type I interval censoring, the sampling design is investigated to determine the minimum number of inspection intervals and equal interval lengths when the termination time of experiment is fixed or not fixed. For illustrative aims, one practical example is given for the implementation of our proposed sampling design to collect the progressive type I interval censored sample so that the users can use this sample to test if the lifetime performance index exceeds the desired target level. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
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18 pages, 1290 KiB  
Article
Hedging and Evaluating Tail Risks via Two Novel Options Based on Type II Extreme Value Distribution
by Hang Lin, Lixin Liu and Zhengjun Zhang
Symmetry 2021, 13(9), 1630; https://doi.org/10.3390/sym13091630 - 5 Sep 2021
Cited by 6 | Viewed by 3198
Abstract
Tail risk is an important financial issue today, but directly hedging tail risks with an ad hoc option is still an unresolved problem since it is not easy to specify a suitable and asymmetric pricing kernel. By defining two ad hoc underlying “assets”, [...] Read more.
Tail risk is an important financial issue today, but directly hedging tail risks with an ad hoc option is still an unresolved problem since it is not easy to specify a suitable and asymmetric pricing kernel. By defining two ad hoc underlying “assets”, this paper designs two novel tail risk options (TROs) for hedging and evaluating short-term tail risks. Under the Fréchet distribution assumption for maximum losses, the closed-form TRO pricing formulas are obtained. Simulation examples demonstrate the accuracy of the pricing formulas. Furthermore, they show that, no matter whether at scale level (symmetric “normal” risk, with greater volatility) or shape level (asymmetric tail risk, with a smaller value in tail index), the greater the risk, the more expensive the TRO calls, and the cheaper the TRO puts. Using calibration, one can obtain the TRO-implied volatility and the TRO-implied tail index. The former is analogous to the Black-Scholes implied volatility, which can measure the overall symmetric market volatility. The latter measures the asymmetry in underlying losses, mirrors market sentiment, and provides financial crisis warnings. Regarding the newly proposed TRO and its implied tail index, economic implications can be offered to investors, portfolio managers, and policy-makers. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
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12 pages, 4239 KiB  
Article
Bayesian Testing Procedure on the Lifetime Performance Index of Products Following Chen Lifetime Distribution Based on the Progressive Type-II Censored Sample
by Shu-Fei Wu and Wei-Tsung Chang
Symmetry 2021, 13(8), 1322; https://doi.org/10.3390/sym13081322 - 22 Jul 2021
Cited by 5 | Viewed by 1389
Abstract
With the high demands on the quality of high-tech products for consumers, assuring the lifetime performance is a very important task for competitive manufacturing industries. The lifetime performance index CL is frequently used to monitor the larger-the-better lifetime performance of products. This [...] Read more.
With the high demands on the quality of high-tech products for consumers, assuring the lifetime performance is a very important task for competitive manufacturing industries. The lifetime performance index CL is frequently used to monitor the larger-the-better lifetime performance of products. This research is related to the topic of asymmetrical probability distributions and applications across disciplines. Chen lifetime distribution with a bathtub shape or increasing failure rate function has many applications in the lifetime data analysis. We derived the uniformly minimum variance unbiased estimator (UMVUE) for CL, and we used this estimator to develop a hypothesis testing procedure of CL under a lower specification limit based on the progressive type-II censored sample. The Bayesian estimator for CL is also derived, and it is used to develop another hypothesis testing procedure. A simulation study is conducted to compare the average confidence levels for two procedures. Finally, one practical example is given to illustrate the implementation of our proposed non-Bayesian and Bayesian testing procedure. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
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12 pages, 299 KiB  
Article
Temporal Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient
by Wensheng Wang and Changkai Zhou
Symmetry 2021, 13(7), 1306; https://doi.org/10.3390/sym13071306 - 20 Jul 2021
Cited by 2 | Viewed by 1274
Abstract
Let U=U(t,x) for (t,x)∈R+×Rd and ∂xU=∂xU(t,x) for [...] Read more.
Let U=U(t,x) for (t,x)∈R+×Rd and ∂xU=∂xU(t,x) for (t,x)∈R+×R be the solution and gradient solution of the fourth order linearized Kuramoto–Sivashinsky (L-KS) SPDE driven by the space-time white noise in one-to-three dimensional spaces, respectively. We use the underlying explicit kernels and symmetry analysis, yielding exact, dimension-dependent, and temporal moduli of non-differentiability for U(·,x) and ∂xU(·,x). It has been confirmed that almost all sample paths of U(·,x) and ∂xU(·,x), in time, are nowhere differentiable. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
11 pages, 278 KiB  
Article
Spatial Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient
by Wensheng Wang
Symmetry 2021, 13(7), 1251; https://doi.org/10.3390/sym13071251 - 12 Jul 2021
Cited by 2 | Viewed by 1105
Abstract
We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs [...] Read more.
We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
15 pages, 2280 KiB  
Article
The Evaluation on the Process Capability Index CL for Exponentiated Frech’et Lifetime Product under Progressive Type I Interval Censoring
by Shu-Fei Wu and Wei-Tsung Chang
Symmetry 2021, 13(6), 1032; https://doi.org/10.3390/sym13061032 - 8 Jun 2021
Cited by 3 | Viewed by 1598
Abstract
We present the likelihood inferences on the lifetime performance index CL to evaluate the performance of lifetimes of products following the skewed Exponentiated Frech’et distribution in many manufacturing industries. This research is related to the topic of skewed Probability Distributions and Applications [...] Read more.
We present the likelihood inferences on the lifetime performance index CL to evaluate the performance of lifetimes of products following the skewed Exponentiated Frech’et distribution in many manufacturing industries. This research is related to the topic of skewed Probability Distributions and Applications across Disciplines. Exponentiated Frech’et distribution is a generalization of some lifetime distributions. The maximum likelihood estimator for CL for lifetimes with exponentiated Frech’et distribution is derived to develop a computational testing procedure so that experimenters can implement it to test whether the lifetime performance reached the pre-assigned level of significance with a given lower specification limit under progressive type I interval censoring. At the end, two examples are provided to demonstrate the implementation on the algorithm for our proposed computational testing procedure. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
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22 pages, 502 KiB  
Article
A New Generalization of the Generalized Inverse Rayleigh Distribution with Applications
by Rana Ali Bakoban and Ashwaq Mohammad Al-Shehri
Symmetry 2021, 13(4), 711; https://doi.org/10.3390/sym13040711 - 18 Apr 2021
Cited by 3 | Viewed by 2026
Abstract
In this article, a new four-parameter lifetime model called the beta generalized inverse Rayleigh distribution (BGIRD) is defined and studied. Mixture representation of this model is derived. Curve’s behavior of probability density function, reliability function, and hazard function are studied. Next, we derived [...] Read more.
In this article, a new four-parameter lifetime model called the beta generalized inverse Rayleigh distribution (BGIRD) is defined and studied. Mixture representation of this model is derived. Curve’s behavior of probability density function, reliability function, and hazard function are studied. Next, we derived the quantile function, median, mode, moments, harmonic mean, skewness, and kurtosis. In addition, the order statistics and the mean deviations about the mean and median are found. Other important properties including entropy (Rényi and Shannon), which is a measure of the uncertainty for this distribution, are also investigated. Maximum likelihood estimation is adopted to the model. A simulation study is conducted to estimate the parameters. Four real-life data sets from difference fields were applied on this model. In addition, a comparison between the new model and some competitive models is done via information criteria. Our model shows the best fitting for the real data. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
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27 pages, 1565 KiB  
Article
The Flexible Burr X-G Family: Properties, Inference, and Applications in Engineering Science
by Abdulhakim A. Al-Babtain, Ibrahim Elbatal, Hazem Al-Mofleh, Ahmed M. Gemeay, Ahmed Z. Afify and Abdullah M. Sarg
Symmetry 2021, 13(3), 474; https://doi.org/10.3390/sym13030474 - 14 Mar 2021
Cited by 37 | Viewed by 1986
Abstract
In this paper, we introduce a new flexible generator of continuous distributions called the transmuted Burr X-G (TBX-G) family to extend and increase the flexibility of the Burr X generator. The general statistical properties of the TBX-G family are calculated. One special sub-model, [...] Read more.
In this paper, we introduce a new flexible generator of continuous distributions called the transmuted Burr X-G (TBX-G) family to extend and increase the flexibility of the Burr X generator. The general statistical properties of the TBX-G family are calculated. One special sub-model, TBX-exponential distribution, is studied in detail. We discuss eight estimation approaches to estimating the TBX-exponential parameters, and numerical simulations are conducted to compare the suggested approaches based on partial and overall ranks. Based on our study, the Anderson–Darling estimators are recommended to estimate the TBX-exponential parameters. Using two skewed real data sets from the engineering sciences, we illustrate the importance and flexibility of the TBX-exponential model compared with other existing competing distributions. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
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16 pages, 1566 KiB  
Article
Development of Perceptual Inhibition in Adolescents—A Critical Period?
by Isabel María Introzzi, María Marta Richard’s, Yesica Aydmune, Eliana Vanesa Zamora, Florencia Stelzer, Ana García Coni, María Fernanda Lopez-Ramon and Esperanza Navarro-Pardo
Symmetry 2021, 13(3), 457; https://doi.org/10.3390/sym13030457 - 11 Mar 2021
Cited by 3 | Viewed by 2165
Abstract
Recent studies suggest that the developmental curves in adolescence, related to the development of executive functions, could be fitted to a non-linear trajectory of development with progressions and retrogressions. Therefore, the present study proposes to analyze the pattern of development in Perceptual Inhibition [...] Read more.
Recent studies suggest that the developmental curves in adolescence, related to the development of executive functions, could be fitted to a non-linear trajectory of development with progressions and retrogressions. Therefore, the present study proposes to analyze the pattern of development in Perceptual Inhibition (PI), considering all stages of adolescence (early, middle, and late) in intervals of one year. To this aim, we worked with a sample of 275 participants between 10 and 25 years, who performed a joint visual and search task (to measure PI). We have fitted ex-Gaussian functions to the probability distributions of the mean response time across the sample and performed a covariance analysis (ANCOVA). The results showed that the 10- to 13-year-old groups performed similarly in the task and differ from the 14- to 19-year-old participants. We found significant differences between the older group and all the rest of the groups. We discuss the important changes that can be observed in relation to the nonlinear trajectory of development that would show the PI during adolescence. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
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9 pages, 4728 KiB  
Article
Human Reaction Times: Linking Individual and Collective Behaviour Through Physics Modeling
by Juan Carlos Castro-Palacio, Pedro Fernández-de-Córdoba, J. M. Isidro, Sarira Sahu and Esperanza Navarro-Pardo
Symmetry 2021, 13(3), 451; https://doi.org/10.3390/sym13030451 - 10 Mar 2021
Cited by 5 | Viewed by 3564
Abstract
An individual’s reaction time data to visual stimuli have usually been represented in Experimental Psychology by means of an ex-Gaussian function. In most previous works, researchers have mainly aimed at finding a meaning for the parameters of the ex-Gaussian function which are known [...] Read more.
An individual’s reaction time data to visual stimuli have usually been represented in Experimental Psychology by means of an ex-Gaussian function. In most previous works, researchers have mainly aimed at finding a meaning for the parameters of the ex-Gaussian function which are known to correlate with cognitive disorders. Based on the recent evidence of correlations between the reaction time series to visual stimuli produced by different individuals within a group, we go beyond and propose a Physics-inspired model to represent the reaction time data of a coetaneous group of individuals. In doing so, a Maxwell–Boltzmann-like distribution appeared, the same distribution as for the velocities of the molecules in an Ideal Gas model. We describe step by step the methodology we use to go from the individual reaction times to the distribution of the individuals response within the coetaneous group. In practical terms, by means of this model we also provide a simple entropy-based methodology for the classification of the individuals within the collective they belong to with no need for an external reference which can be applicable in diverse areas of social sciences. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
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21 pages, 356 KiB  
Article
Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions
by Wensheng Wang and Dazhong Wang
Symmetry 2021, 13(1), 73; https://doi.org/10.3390/sym13010073 - 3 Jan 2021
Cited by 5 | Viewed by 1654
Abstract
We study the realized power variations for the fourth order linearized Kuramoto–Sivashinsky (LKS) SPDEs and their gradient, driven by the space–time white noise in one-to-three dimensional spaces, in time, have infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. This class was introduced-with Brownian-time-type [...] Read more.
We study the realized power variations for the fourth order linearized Kuramoto–Sivashinsky (LKS) SPDEs and their gradient, driven by the space–time white noise in one-to-three dimensional spaces, in time, have infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. This class was introduced-with Brownian-time-type kernel formulations by Allouba in a series of articles starting in 2006. He proved the existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above class of equations in d=1,2,3. We use the relationship between LKS-SPDEs and the Houdré–Villaa bifractional Brownian motion (BBM), yielding temporal central limit theorems for LKS-SPDEs and their gradient. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on the delicate analysis of variations of general Gaussian processes and stochastic heat equation driven by the space–time white noise. On the other hand, it builds on and complements Allouba’s earlier works on the LKS-SPDEs and their gradient. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
11 pages, 277 KiB  
Article
Multistage Estimation of the Rayleigh Distribution Variance
by Ali Yousef, Ayman A. Amin, Emad E. Hassan and Hosny I. Hamdy
Symmetry 2020, 12(12), 2084; https://doi.org/10.3390/sym12122084 - 15 Dec 2020
Cited by 2 | Viewed by 1579
Abstract
In this paper we discuss the multistage sequential estimation of the variance of the Rayleigh distribution using the three-stage procedure that was presented by Hall (Ann. Stat. 9(6):1229–1238, 1981). Since the Rayleigh distribution variance is a linear function of the distribution scale parameter’s [...] Read more.
In this paper we discuss the multistage sequential estimation of the variance of the Rayleigh distribution using the three-stage procedure that was presented by Hall (Ann. Stat. 9(6):1229–1238, 1981). Since the Rayleigh distribution variance is a linear function of the distribution scale parameter’s square, it suffices to estimate the Rayleigh distribution’s scale parameter’s square. We tackle two estimation problems: first, the minimum risk point estimation problem under a squared-error loss function plus linear sampling cost, and the second is a fixed-width confidence interval estimation, using a unified optimal stopping rule. Such an estimation cannot be performed using fixed-width classical procedures due to the non-existence of a fixed sample size that simultaneously achieves both estimation problems. We find all the asymptotic results that enhanced finding the three-stage regret as well as the three-stage fixed-width confidence interval for the desired parameter. The procedure attains asymptotic second-order efficiency and asymptotic consistency. A series of Monte Carlo simulations were conducted to study the procedure’s performance as the optimal sample size increases. We found that the simulation results agree with the asymptotic results. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
14 pages, 303 KiB  
Article
Multistage Estimation of the Scale Parameter of Rayleigh Distribution with Simulation
by Ali Yousef, Emad E. H. Hassan, Ayman A. Amin and Hosny I. Hamdy
Symmetry 2020, 12(11), 1925; https://doi.org/10.3390/sym12111925 - 22 Nov 2020
Cited by 5 | Viewed by 1823
Abstract
This paper discusses the sequential estimation of the scale parameter of the Rayleigh distribution using the three-stage sequential sampling procedure proposed by Hall (Ann. Stat.1981, 9, 1229–1238). Both point and confidence interval estimation are considered via a unified optimal [...] Read more.
This paper discusses the sequential estimation of the scale parameter of the Rayleigh distribution using the three-stage sequential sampling procedure proposed by Hall (Ann. Stat.1981, 9, 1229–1238). Both point and confidence interval estimation are considered via a unified optimal decision framework, which enables one to make the maximum use of the available data and, at the same time, reduces the number of sampling operations by using bulk samples. The asymptotic characteristics of the proposed sampling procedure are fully discussed for both point and confidence interval estimation. Since the results are asymptotic, Monte Carlo simulation studies are conducted to provide the feel of small, moderate, and large sample size performance in typical situations using the Microsoft Developer Studio software. The procedure enjoys several interesting asymptotic characteristics illustrated by the asymptotic results and supported by simulation. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines)
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