scholarly journals Kumaraswamy-Janardan Distribution: A Generalized Janardan Distribution with Application to Real Data

2021 ◽  
pp. 111-122
Author(s):  
Nelson Doe Dzivor ◽  
Henry Otoo ◽  
Eric Neebo Wiah
Keyword(s):  
Density Function ◽  
Hazard Rate ◽  
Probability Distributions ◽  
Real Data ◽  
Moment Generating Function ◽  
Rate Function ◽  
Information Criteria ◽  
Cumulative Density Function ◽  
Baseline Model ◽  
Generalized Distribution

The quest to improve on flexibility of probability distributions motivated this research. Four-parameter Janardan generalized distribution known as Kumaraswamy-Janardan distribution is proposed through method of parameterization and studied. The probability density function, cumulative density function, survival rate function as well as hazard rate function of the distribution are established. Statistical properties such as moments, moment generating function as well as maximum likelihood of the model are discussed. The parameters are estimated using the simulated annealing optimization algorithm.   Flexibility of the model in comparison with the baseline model as well as other competing sub-models is verified using Akaike Information Criteria (AIC). The model is tested with real data and is proven to be more flexible in fitting real data than any of its sub-models considered. 

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 Kumaraswamy Generalized Power Lomax Distributionand Its Applications

Stats ◽  
2021 ◽  
Vol 4 (1) ◽  
pp. 28-45
Author(s):  
Vasili B.V. Nagarjuna ◽  
R. Vishnu Vardhan ◽  
Christophe Chesneau
Keyword(s):  
Hazard Rate ◽  
Real Data ◽  
Rate Function ◽  
Maximum Likelihood Estimates ◽  
Parameter Estimates ◽  
Parameter Distribution ◽  
Data Sets ◽  
Lomax Distribution ◽  
Entropy Measures ◽  
Modeling Behavior

In this paper, a new five-parameter distribution is proposed using the functionalities of the Kumaraswamy generalized family of distributions and the features of the power Lomax distribution. It is named as Kumaraswamy generalized power Lomax distribution. In a first approach, we derive its main probability and reliability functions, with a visualization of its modeling behavior by considering different parameter combinations. As prime quality, the corresponding hazard rate function is very flexible; it possesses decreasing, increasing and inverted (upside-down) bathtub shapes. Also, decreasing-increasing-decreasing shapes are nicely observed. Some important characteristics of the Kumaraswamy generalized power Lomax distribution are derived, including moments, entropy measures and order statistics. The second approach is statistical. The maximum likelihood estimates of the parameters are described and a brief simulation study shows their effectiveness. Two real data sets are taken to show how the proposed distribution can be applied concretely; parameter estimates are obtained and fitting comparisons are performed with other well-established Lomax based distributions. The Kumaraswamy generalized power Lomax distribution turns out to be best by capturing fine details in the structure of the data considered.

scholarly journals On a new distribution based on the arccosine function

2021 ◽  
Author(s):  
Christophe Chesneau ◽  
Lishamol Tomy ◽  
Jiju Gillariose
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Hazard Rate ◽  
Survival Function ◽  
Real Data ◽  
Cumulative Distribution ◽  
Data Sets ◽  
Power Functions ◽  
New Distribution

AbstractThis note focuses on a new one-parameter unit probability distribution centered around the inverse cosine and power functions. A special case of this distribution has the exact inverse cosine function as a probability density function. To our knowledge, despite obvious mathematical interest, such a probability density function has never been considered in Probability and Statistics. Here, we fill this gap by pointing out the main properties of the proposed distribution, from both the theoretical and practical aspects. Specifically, we provide the analytical form expressions for its cumulative distribution function, survival function, hazard rate function, raw moments and incomplete moments. The asymptotes and shape properties of the probability density and hazard rate functions are described, as well as the skewness and kurtosis properties, revealing the flexible nature of the new distribution. In particular, it appears to be “round mesokurtic” and “left skewed”. With these features in mind, special attention is given to find empirical applications of the new distribution to real data sets. Accordingly, the proposed distribution is compared with the well-known power distribution by means of two real data sets.

scholarly journals A Probability Density Function Generator Based on Deep Learning

2018 ◽  
Author(s):  
Chi-Hua Chen ◽  
Fangying Song ◽  
Feng-Jang Hwang ◽  
Ling Wu
Keyword(s):  
Deep Learning ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Distributions ◽  
Real Data ◽  
Learning Model ◽  
Cumulative Distribution ◽  
Function Generator ◽  
Deep Learning Model

To generate a probability density function (PDF) for fitting probability distributions of real data, this study proposes a deep learning method which consists of two stages: (1) a training stage for estimating the cumulative distribution function (CDF) and (2) a performing stage for predicting the corresponding PDF. The CDFs of common probability distributions can be adopted as activation functions in the hidden layers of the proposed deep learning model for learning actual cumulative probabilities, and the differential equation of trained deep learning model can be used to estimate the PDF. To evaluate the proposed method, numerical experiments with single and mixed distributions are performed. The experimental results show that the values of both CDF and PDF can be precisely estimated by the proposed method.

scholarly journals The Exponentiated Burr–Hatke Distribution and Its Discrete Version: Reliability Properties with CSALT Model, Inference and Applications

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2277
Author(s):  
Mahmoud El-Morshedy ◽  
Hassan M. Aljohani ◽  
Mohamed S. Eliwa ◽  
Mazen Nassar ◽  
Mohammed K. Shakhatreh ◽  
...  
Keyword(s):  
Hazard Rate ◽  
Real Life ◽  
Real Data ◽  
Rate Function ◽  
Discrete Analog ◽  
Discrete Distributions ◽  
Discrete Version ◽  
Continuous Version ◽  
Data Set ◽  
Accelerated Life Tests

Continuous and discrete distributions are essential to model both continuous and discrete lifetime data in several applied sciences. This article introduces two extended versions of the Burr–Hatke model to improve its applicability. The first continuous version is called the exponentiated Burr–Hatke (EBuH) distribution. We also propose a new discrete analog, namely the discrete exponentiated Burr–Hatke (DEBuH) distribution. The probability density and the hazard rate functions exhibit decreasing or upside-down shapes, whereas the reversed hazard rate function. Some statistical and reliability properties of the EBuH distribution are calculated. The EBuH parameters are estimated using some classical estimation techniques. The simulation results are conducted to explore the behavior of the proposed estimators for small and large samples. The applicability of the EBuH and DEBuH models is studied using two real-life data sets. Moreover, the maximum likelihood approach is adopted to estimate the parameters of the EBuH distribution under constant-stress accelerated life-tests (CSALTs). Furthermore, a real data set is analyzed to validate our results under the CSALT model.

scholarly journals The partly parametric and partly nonparametric additive risk model

2021 ◽  
Author(s):  
Nils Lid Hjort ◽  
Emil Aas Stoltenberg
Keyword(s):  
Hazard Rate ◽  
Goodness Of Fit ◽  
Risk Model ◽  
Real Data ◽  
Rate Function ◽  
Nonparametric Model ◽  
Additive Risk Model ◽  
Data Application ◽  
Additive Risk ◽  
Hazard Factor

AbstractAalen’s linear hazard rate regression model is a useful and increasingly popular alternative to Cox’ multiplicative hazard rate model. It postulates that an individual has hazard rate function $$h(s)=z_1\alpha _1(s)+\cdots +z_r\alpha _r(s)$$ h ( s ) = z 1 α 1 ( s ) + ⋯ + z r α r ( s ) in terms of his covariate values $$z_1,\ldots ,z_r$$ z 1 , … , z r . These are typically levels of various hazard factors, and may also be time-dependent. The hazard factor functions $$\alpha _j(s)$$ α j ( s ) are the parameters of the model and are estimated from data. This is traditionally accomplished in a fully nonparametric way. This paper develops methodology for estimating the hazard factor functions when some of them are modelled parametrically while the others are left unspecified. Large-sample results are reached inside this partly parametric, partly nonparametric framework, which also enables us to assess the goodness of fit of the model’s parametric components. In addition, these results are used to pinpoint how much precision is gained, using the parametric-nonparametric model, over the standard nonparametric method. A real-data application is included, along with a brief simulation study.

Alpha Power-Kumaraswamy Distribution with An Application on Survival Times of Cancer Patients

2020 ◽  
Vol 2 (2) ◽  
Author(s):  
Fatima Ulubekova ◽  
Gamze Ozel
Keyword(s):  
Hazard Rate ◽  
Real Data ◽  
Rate Function ◽  
Data Sets ◽  
Alpha Power ◽  
Survival Times ◽  
Kumaraswamy Distribution ◽  
Data Application ◽  
The Right ◽  
Better Than

The aim of the study is to obtain the alpha power Kumaraswamy (APK) distribution. Some main statistical properties of the APK distribution are investigated including survival, hazard rate and quantile functions, skewness, kurtosis, order statistics. The hazard rate function of the proposed distribution could be useful to model data sets with bathtub hazard rates. We provide a real data application and show that the APK distribution is better than the other compared distributions fort the right-skewed data sets.

The Lindley negative-binomial distribution: Properties, estimation and applications to lifetime data

2020 ◽  
Vol 70 (4) ◽  
pp. 917-934
Author(s):  
Muhammad Mansoor ◽  
Muhammad Hussain Tahir ◽  
Gauss M. Cordeiro ◽  
Sajid Ali ◽  
Ayman Alzaatreh
Keyword(s):  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Hazard Rate ◽  
Negative Binomial ◽  
Real Data ◽  
Moment Generating Function ◽  
Estimation Methods ◽  
Model Parameters ◽  
Proposed Model ◽  
Rate Functions

AbstractA generalization of the Lindley distribution namely, Lindley negative-binomial distribution, is introduced. The Lindley and the exponentiated Lindley distributions are considered as sub-models of the proposed distribution. The proposed model has flexible density and hazard rate functions. The density function can be decreasing, right-skewed, left-skewed and approximately symmetric. The hazard rate function possesses various shapes including increasing, decreasing and bathtub. Furthermore, the survival and hazard rate functions have closed form representations which make this model tractable for censored data analysis. Some general properties of the proposed model are studied such as ordinary and incomplete moments, moment generating function, mean deviations, Lorenz and Bonferroni curve. The maximum likelihood and the Bayesian estimation methods are utilized to estimate the model parameters. In addition, a small simulation study is conducted in order to evaluate the performance of the estimation methods. Two real data sets are used to illustrate the applicability of the proposed model.

scholarly journals A Novel Generator of Continuous Probability Distributions for the Asymmetric Left-skewed Bimodal Real-life Data with Properties and Copulas

2021 ◽  
pp. 943-961 ◽  
Author(s):  
Wahid A. M. Shehata ◽  
Haitham Yousof ◽  
Mohamed Aboraya
Keyword(s):  
Probability Distributions ◽  
Real Life ◽  
Real Data ◽  
Moment Generating Function ◽  
Data Sets ◽  
Base Line ◽  
Survival Times ◽  
New Family ◽  
Real Life Data ◽  
Two Parameter

This paper presents a novel two-parameter G family of distributions. Relevant statistical properties such as the ordinary moments, incomplete moments and moment generating function are derived.  Using common copulas, some new bivariate type G families are derived. Special attention is devoted to the standard exponential base line model. The density of the new exponential extension can be “asymmetric and right skewed shape” with no peak, “asymmetric right skewed shape” with one peak, “symmetric shape” and “asymmetric left skewed shape” with one peak. The hazard rate of the new exponential distribution can be “increasing”, “U-shape”, “decreasing” and “J-shape”. The usefulness and flexibility of the new family is illustrated by means of two applications to real data sets. The new family is compared with many common G families in modeling relief times and survival times data sets.

scholarly journals A New Extension of Lindley Geometric Distribution and its Applications

2019 ◽  
pp. 249-263
Author(s):  
I. Elbatal ◽  
Mohamed G. Khalil
Keyword(s):  
Hazard Rate ◽  
Maximum Likelihood Method ◽  
Geometric Distribution ◽  
Real Data ◽  
Rate Function ◽  
Likelihood Method ◽  
Parameter Distribution ◽  
Model Parameters ◽  
Data Set ◽  
New Distribution

A new four-parameter distribution called the beta Lindley-geometric distribution is proposed. The hazard rate function of the new model can be constant, decreasing, increasing, upside down bathtub or bathtub failure rate shapes. Various structural properties including of the new distribution are derived. The estimation of the model parameters is performed by maximum likelihood method. The usefulness of the new distribution is illustrated using a real data set.

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