On the existence and uniqueness of the MLEs of the parameters of a general class of exponentiated distributions

To study the high quality processes, Time-between-events (TBE) control charts have several advantages over the ordinary control charts. However, the existing TBE charts are based on the exponential distribution, which limit the application of these charts to monitor rare events. Therefore, to generalize existing exponential TBE charts, recently two new TBE charts assuming Weibull and generalized exponential distributions for the inter-arrival times of the renewal process, have been proposed in the literature. The main aim of this study is to present a detailed comparison of these charts. We use the expected quadratic loss (EQL), relative average run length (RARL), continuous rank probability score (CRPS) and closeness measure to assess the performance of the Weibull and the generalized exponential charts. We also discuss four real data examples in this article. INDEX TERMS Average run length, continuous rank probability score, Expected quadratic loss, Generalized exponential distribution, High quality process, Relative average run length, Weibull distribution.

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“…Ghitany et al [17] proposed the following cumulative distribution function of a general class of exponentiated distributions:…”

Section: Design Structure Of the Weibull And Ge Chartsmentioning

confidence: 99%

A Comparison of Shewhart-Type Time-Between-Events Control Charts Based on the Renewal Process

et al. 2020

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“…Ghitany et al defined the following general class of exponentiated distributions with the cumulative distribution function (cdf):

F false(x; λ, β false) = {((), 1 - \exp false(- λ M false(x false) false))}^{β}, x, λ, β > 0,

where M ( x ) is an increasing function with M (0)=0 and

M false(\infty false) = \infty

. The corresponding probability density function (pdf) is given by

\begin{matrix} right f (x; λ, β) & left = β λ M^{'} (x) \exp (- λ M (x)) {(1 - \exp (- λ M (x)))}^{β - 1}, & right \\ right & left x, λ, β > 0, \end{matrix}

where

M^{'} false(x false)

is the derivative of M ( x ).…”

Section: Some Preliminaries About Gementioning

confidence: 99%

Time‐between‐events control charts for an exponentiated class of distributions of the renewal process

Ali

2017

Quality & Reliability Eng

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Time‐between‐events control charts are commonly used to monitor high‐quality processes and have several advantages over the ordinary control charts. In this article, we present some new control charts based on the renewal process, where a class of absolutely continuous exponentiated distributions is assumed for the time between events. This class includes the generalized exponential, generalized Rayleigh, and exponentiated Pareto distributions. Although we discuss the design structure for all the mentioned distributions, our main focus will be on the generalized exponential distribution due to its practical relevance and popularity. Since the generalized exponential distribution is a generalization of the traditional exponential distribution, the new control chart is more flexible than the existing exponential time‐between‐events charts. The control chart performance is evaluated in terms of some useful measures, including the average run length (ARL), the expected quadratic loss, continuous ranked probability, and the relative ARL. The effect of parameter estimation using the maximum likelihood and Bayesian methods on the ARL is also discussed in this article. The study also presents an illustrative example and 4 case studies to highlight the practical relevance of the proposal.

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“…Moreover, assume that at the first failure time (y 1 ), s 1 surviving experimental units are removed and at the second failure time For more detail about Type-II progressive censoring readers can refer to Balakrishnan and Aggarwala (2000), Balakrishnan (2007), , Balakrishnan et al (2008, Burkschat (2008), Burkschat et al (2006), Cohen (1963), Cramer (2014), Cramer and Kamps (2001), Herd (1956). Some valuable results can also be found in Ghitany et al (2013Ghitany et al ( , 2014, Kang and Seo (2011), Krishna and Kumar (2013), Pakyari and Balakrishnan (2013), Rezapour et al (2013aRezapour et al ( , 2013b, Seo and Kang (2014). Cramer and Iliopoulos (2009) proposed an adaptive Type-II progressive censoring such that the number of removals has been involved in previous failure times and the number of previous censored items.…”

Section: Introductionmentioning

confidence: 99%

Type-II stepwise progressive censoring

Bayat

Torabi

2016

CSAM

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Type-II progressive censoring is one of the censoring methods frequently used in clinical studies, reliability trials, quality control of products and industrial experiments. Sometimes in Type-II progressive censoring experiments, the failure rate is low so the waiting time to observe the m th failure will be very long; however, the experimenter may have to terminate the experiment before a predetermined time. In this article, if two generalized types of Type-II progressive censoring are reminded, we then make some changes in the removal method of Type-II progressive censoring such that without reducing the deduction quality, the termination time of the experiment decreases. This can be done with decreasing withdraws throughout the steps of the experiment with a special reasonable method. A simulation study is done and the results are tabulated at the end of this article for a comparison between introduced method and Type-II progressive censoring.

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On the existence and uniqueness of the MLEs of the parameters of a general class of exponentiated distributions

Cited by 31 publications

References 11 publications

A Comparison of Shewhart-Type Time-Between-Events Control Charts Based on the Renewal Process

A Comparison of Shewhart-Type Time-Between-Events Control Charts Based on the Renewal Process

Time‐between‐events control charts for an exponentiated class of distributions of the renewal process

Type-II stepwise progressive censoring

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