Abstract
There are a wide range of industries that make use of the Weibull distribution, including industrial engineering, general insurance, survival analysis, electrical engineering, and reliability engineering, to name just a few. The Weibull distribution is expanded to become the Weibull-Pareto distribution, also known simply as the Weibull distribution. A notable use of the Weibull-Pareto distribution is in the modeling of asymmetrical data, which is also one of its most important applications. During the course of this inquiry, a hierarchical Bayesian model will be created with the use of a Weibull-Pareto distribution as a reference.
Words to note: MCMC, survival model, right censoring, WPD, Heavy-tailed Dis tribution, hierarchical Bayesian model,
Table of Contents
Chapter One
Introduction
1.1 Study Background 6
1.2 Problem Statement 7
1.3 Study Objectives 8
CHAPTER TWO
LITERATURE REVIEW
2.1 WP Distribution 9
2.1.1 Estimation 9
2.2 Introduction to Bayesian Statistics 12
2.2.1 Bayesian WPD 18
CHAPTER THREE
RESEARCH METHODOLOGY
3.1 Weibull-Pareto Parameters 23
3.2 The Simulation Technique 28
3.2.1 The Generation of Data 28
3.3 Performance of Bayesian WPD mode: Simulation Study 1 29
3.4 Bayesian WPD under Censoring: Simulation Study 2 32
CHAPTER THREE
APPLICATION AND DISCUSSIONS
4.1 Tribolium Confusum and Tribolium Casteneum 36
4.2 Censoring with Melanoma data 42
5 CONCLUSION 45
6 REFERENCES 47
Chapter 1
Introduction
1.1 Study Background
In the field of modeling survival analysis and reliability, the Weibull distribution is applied rather frequently. Because of the value of the shape parameter, it is adaptable and can take on the characteristics of a variety of other distributions. Because of this, it is ex- tremely well-liked among quality control engineers and analysts, particularly those who have experience working with data modeling and reliability. A huge number of academics have come up with many versions of the Weibull distribution; one of these variants is called the EPD(Exponential-Weibull distribution). (Mudholkar et al.,1995).
Both the generalized Weibull distribution that Kollia and Mudholder (1994) produced, as well as the Beta-Weibull distribution that Famoye et al.(2005) constructed, have cer- tain similarities. A prior discussion introduced the idea of the New generalized Weibull distribution. This distribution can also be referred to as the Weibull Pareto distribution. It was generated from a family of probability density models known as ”Transformed- Transformer” (Alzaatreh et al., 2013). This distribution has a severely skewed nature when compared to the Weibull distribution, which is the standard. As a consequence of this, the tactic that is suggested is modeling extremely skewed data, which is something that frequently occurs in survival analysis and dependability.
According to Alzaatreh et al. (2013), the value of the shape parameter of the WPD is smaller than one. This conclusion was reached in 2013. There is no such thing as a Maximum Likelihood Estimator (MLE), and this applies to both the scale and the shape parameter. After that, they mandated two different parameter estimation methodologies, which were the modified maximum likelihood estimation (MMLE) and the alternative maximum likelihood estimation (AMLE). Despite the fact that this was the case, the AMLE resulted in a significant bias, and using the MMLE is costly. In the following part, 2.1.1, we will talk about the two different approaches.
The Bayesian Weibull Pareto Model will be explained in detail during the course of this article’s discussion. Bayes’ methodology differs from the frequentist approach in that it operates under the assumption that the parameters are subject to random variation and adhere to the probability that was previously stated. The researcher’s prior credence regarding the distribution parameters is defined by the random distribution, which is captured as the prior distribution.
Estimating Bayesian parameters has been the subject of multiple publications authored by a number of researchers over the past few years. Estimating the generalized lognormal distribution was the topic of an article that Perez and Martin (2009) wrote. With the use of Type-I censoring, Aslam and Noor (2013) were able to estimate the parameters of the inverse Weibull distribution. When conducting data analysis, the Weibull distribution has proven to be extremely helpful, particularly when used to censored data, which is the kind of study that is typically performed in survival analysis. For instance, survival time is the end result that is of interest in cancer research since it is the event of interest. In spite of this, it is sometimes known as the time between complete remission to the beginning of a recurrence. In addition, there is the possibility that some people will not experience the event of interest by the time the actual time to occurrence has passed. As a result, a censored observation takes place. As a consequence of this, in the event that it takes place, we will truncate the final findings due to random factors.
1.2 Problem Statement
Because of the extreme skewness of many real-world data sets, the most recent iteration of the Weibull Distribution is unable to effectively simulate the distributions of a number of these collections of data. Because of this, it is essential to broaden the application of the Weibull-Pareto distribution by incorporating a fresh component into the model. Be- cause of this, the current Weibull-Pareto distribution will have greater flexibility, and the resulting distribution will provide a better fit than the Weibull-Pareto Distribution. The Weibull-Pareto distribution needs to be extended in order to account for this necessity.
1.3 Study Objectives
The purpose of this study is to propose a novel distribution that will be referred to as the generalized Weibull-Pareto Distribution (Generalised WPD), and to deduce its features using the transformation.
The following is a list of the specific goals:
i. Construct the Generalized Weibull-Pareto Distribution, analyze the reliability of the data, and derive a variety of structural features.
While the General Objectives are
i. Make an educated guess as to the values of the modified distribution’s parameters.
ii. Evaluate how well the WPD is working.
1.4 Importance of the Study
The incorporation of a generalized parameter into the Weibull-Pareto Distribution will result in a significant increase in both the sensitivity and the efficacy of the statistical tests associated with the distribution. This will be the case because the sensitivity of the tests will increase while the efficacy of the tests will remain the same. This is going to be the case due to the fact that the test’s level of sensitivity is going to improve, although the test’s level of accuracy will remain unchanged. As a result of this, it will be possible to model and carry out flexible analyses of skewed data sets based on real-world examples in a wide variety of application domains. These studies can be conducted in a wide variety of application domains. There are some real-world data sets that do not follow the Weibull or Pareto distributions as expected. These data sets can be found all around the world. As a direct consequence of this, the Weibull-Pareto distribution needs to be extended by adding a new parameter to it in order to boost its adaptability and make it applicable to a wider number of scenarios. This can be accomplished by making the distribution more general.
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