scholarly journals A generalized beta function and associated probability density

2002 ◽  
Vol 30 (8) ◽  
pp. 467-478 ◽  
Author(s):  
Y. Ben Nakhi ◽  
S. L. Kalla
Keyword(s):  
Probability Density ◽  
Hazard Rate ◽  
Beta Function ◽  
Moment Generating Function ◽  
Residue Function ◽  
Rate Function ◽  
Hazard Rate Function ◽  
Scale Parameters ◽  
Special Cases ◽  
Basic Functions

We introduce and establish some properties of a generalized form of the beta function. Corresponding generalized incomplete beta functions are also defined. Moreover, we define a new probability density function (pdf) involving this new generalized beta function. Some basic functions associated with the pdf, such as moment generating function, mean residue function, and hazard rate function are derived. Some special cases are mentioned. Some figures for pdf, hazard rate function, and mean residue life function are given. These figures reflect the role of shape and scale parameters.

scholarly journals A Four-Parameter Extension of Burr III Distribution with Applications

2021 ◽  
Vol 3 (1) ◽  
pp. 7-19
Author(s):  
Emmanuel W. Okereke ◽  
Johnson Ohakwe
Keyword(s):  
Maximum Likelihood ◽  
Hazard Rate ◽  
Linear Representation ◽  
Quantile Function ◽  
Moment Generating Function ◽  
Rate Function ◽  
Hazard Rate Function ◽  
Maximum Likelihood Procedure ◽  
Burr Iii Distribution ◽  
New Distribution

AbstractIn this paper, we defined and studied a new distribution called the odd exponentiated half-logistic Burr III distribution. Properties such as the linear representation of the probability density function (PDF) of the distribution, quantile function, ordinary and incomplete moments, moment generating function and distribution of the order statistic were derived. The PDF and hazard rate function were found to be capable of having various shapes, making the new distribution highly flexible. In particular, the hazard rate function can be nonincreasing, unimodal and nondecreasing. It can also have the bathtub shape among other non- monotone shapes. The maximum likelihood procedure was used to estimate the parameters of the new model. We gave two numerical examples to illustrate the usefulness and the ability of the distribution to provide better fits to a number of data sets than several distributions in existence.Keywords: Burr III distribution; maximum likelihood procedure; moments; odd exponentiated half-logistic-G family; order statistics. AbstrakPada artikel ini akan didefinisikan dan dipelajari mengenai distribusi baru yang disebut distribusi Burr III setengah logistik tereksponen ganjil. Kami menurunkan beberapa sifat dari distribusi tersebut yaitu representasi linier dari fungsi kepadatan peluang (FKP), fungsi kuantil, momen biasa dan momen tidak lengkap, fungsi pembangkit momen dan distribusi statistik terurut. Fungsi FKP dan fungsi tingkat hazard diperoleh memiliki bermacam-macam bentuk, membuat distribusi baru ini sangat fleksibel. Secara khusus, fungsi tingkat hazard dapat berupa fungsi taknaik, bermodus tunggal, bisa juga tidak turun. Selain itu, fungsi ini juga dapat berbentuk seperti bak mandi di antara bentuk-bentuk tak monoton lainnya. Prosedur kemungkinan maksimum digunakan untuk mengestimasi parameter model yang baru. Kami memberikan dua contoh numerik untuk mengilustrasikan kegunaan dan kemampuan distribusi untuk menghasilkan kesesuaian yang lebih baik pada sejumlah kumpulan data dibandingkan beberapa distribusi yang ada.Kata kunci: distribusi Burr III; prosedur kemungkinan maksimum; momen; keluarga setengah logistik-G teresponen ganjil; statistic terurut.

scholarly journals Modeling Extreme Values Utilizing an Asymmetric Probability Function

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1730
Author(s):  
Mohammed M. A. Almazah ◽  
Muqrin A. Almuqrin ◽  
Mohamed. S. Eliwa ◽  
Mahmoud El-Morshedy ◽  
Haitham M. Yousof
Keyword(s):  
Hazard Rate ◽  
Extreme Values ◽  
Residual Life ◽  
Real Data ◽  
Moment Generating Function ◽  
Rate Function ◽  
Estimation Methods ◽  
Model Parameters ◽  
Hazard Rate Function ◽  
Data Application

In this article, a new flexible probability density function with three parameters is proposed for modeling asymmetric data (positive and negative) with different types of kurtosis (mesokurtic, leptokurtic and platykurtic). Some of its statistical and reliability properties, including hazard rate function, moments, moment generating function, incomplete moments, mean deviations, moment of the residual life, moment of the reversed residual life, and order statistics are derived. Its hazard rate function can be either constant, increasing-constant, decreasing-constant, U shape, upside down shape or upside down-U shape. Seven classical estimation methods are considered to estimate the unknown model parameters. Monte Carlo simulation experiments are performed to compare the performance of the seven different estimation methods. Finally, a distinctive asymmetric real data application is analyzed for illustrating the flexibility of the new model.

scholarly journals PCM Transformation: Properties and Their Estimation

2021 ◽  
Author(s):  
Dinesh Kumar ◽  
Pawan Kumar ◽  
Pradip Kumar ◽  
Sanjay Kumar Singh ◽  
Umesh Singh
Keyword(s):  
Hazard Rate ◽  
Survival Function ◽  
Stochastic Ordering ◽  
Information Criterion ◽  
Moment Generating Function ◽  
Rate Function ◽  
Hazard Rate Function ◽  
Test Statistic ◽  
Bayesian Approaches

In the present piece of work, we are going to propose a new trigonometry based transformation called PCM transformation. We have been obtained its various statistical properties such as survival function, hazard rate function, reverse-hazard rate function, moment generating function, median, stochastic ordering etc. Maximum Likelihood Estimator (MLE) method under classical approach and Bayesian approaches are tackled to obtain the estimate of unknown parameter. A real dataset has been applied to check its fitness on the basis of fitting criterions Akaike Information criterion (AIC), Bayesian Information criterion (BIC), log-likelihood (-LL) and Kolmogrov-Smirnov (KS) test statistic values in real sense. A simulation study is also being conducted to assess the estimator’s long-term attitude and compared over some chosen distributions.

scholarly journals A New Five Parameter Lifetime Distribution: Properties and Application

2017 ◽  
Vol 13 (3) ◽  
pp. 7205-7218
Author(s):  
Shimaa A. Dessoky ◽  
Ahmed M. T. Abd El-Bar
Keyword(s):  
Hazard Rate ◽  
Quantile Function ◽  
Moment Generating Function ◽  
Rate Function ◽  
Lifetime Distribution ◽  
Maximum Likelihood Estimates ◽  
Hazard Rate Function ◽  
Unknown Parameters ◽  
Data Set ◽  
Present Distribution

This paper deals with a new generalization of the Weibull distribution. This distribution is called exponentiated exponentiated exponential-Weibull (EEE-W) distribution. Various structural properties of the new probabilistic model are considered, such as hazard rate function, moments, moment generating function, quantile function, skewness, kurtosis, Shannon entropy and Rényi entropy. The maximum likelihood estimates of its unknown parameters are obtained. Finally, areal data set is analyzed and it observed that the present distribution can provide a better fit than some other known distributions.

Reliability Concepts

2018 ◽  
pp. 1-22
Keyword(s):  
Hazard Rate ◽  
Failure Process ◽  
Reliability Function ◽  
Rate Function ◽  
Cumulative Distribution ◽  
Hazard Rate Function ◽  
Reliability Engineering ◽  
Probability Models ◽  
Engineering Design Process ◽  
Basic Concepts

The first chapter introduces basic concepts of Reliability and their relationships. Four probability functions—reliability function, cumulative distribution function, probability density function, and hazard rate function—that completely characterize the failure process are defined. Three failure rates—MTBF, MTTF, MTTR—that play important role in reliability engineering design process are explained here. The three patterns of failures, DFR, CFR, and IFR, are discussed with reference to the bathtub curve. Two probability models, Exponential and Weibull, are presented. Series and parallel systems and application areas of reliability are also presented.

Bounds for the hazard rate and the reversed hazard rate of the convolution of dependent random lifetimes

2019 ◽  
Vol 56 (4) ◽  
pp. 1033-1043 ◽  
Author(s):  
Félix Belzunce ◽  
Carolina Martínez-Riquelme
Keyword(s):  
Recent Result ◽  
Upper Bound ◽  
Hazard Rate ◽  
Rate Function ◽  
Semiparametric Models ◽  
Hazard Rate Function ◽  
Reversed Hazard Rate ◽  
Parametric And Semiparametric Models ◽  
Reversed Hazard Rate Function

AbstractAn upper bound for the hazard rate function of a convolution of not necessarily independent random lifetimes is provided, which generalizes a recent result established for independent random lifetimes. Similar results are considered for the reversed hazard rate function. Applications to parametric and semiparametric models are also given.

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