TYPE II TOPPE LEONE EXPONENTIAL DISTRIBUTION WITH APPLICATION TO LIFE TIME DATA

CHAPTER ONE

INTRODUCTION

1.1 Background of the study

The rapid growth in data generation across diverse fields, combined with continuous advancements in computational science, has fueled the need for the development of robust and adaptable statistical models. Such models are often derived from a wide range of distributions that possess desirable properties, including those constructed through the use of generated distributions. This approach allows for the creation of models tailored to meet the specific demands of various applications.

“In disciplines such as engineering, medicine, and reliability analysis, examining the lifespans of systems, components, or living organisms is essential. Lifetime data, as it is often called, plays a significant role in predicting failures, optimizing maintenance strategies, and evaluating product performance” (Al-Shomrani, Arif, Shawky, Hanif, and Shahbaz 2016). “Exponential distributions are among the models commonly employed for analyzing lifetime data. This distribution operates under the assumption of a constant failure rate, implying that the likelihood of failure does not depend on how long the item or system has been in operation” (Cordeiro and Castro 2013). “However, in many practical situations, this assumption may not hold true. Real-world scenarios often exhibit varying failure rates, including higher failure rates at the beginning due to defects, a "burn-in" period where the failure rate stabilizes, and an increased failure rate over time caused by wear and tear” (Alzaatreh, Lee and Famoye 2013).

The Topp-Leone (TL) distribution offers a more suitable alternative to the standard exponential distribution by accounting for changing failure rates. Unlike the exponential model, the TL distribution introduces an additional parameter that adjusts the skewness of the distribution, allowing it to better represent data with early failures or aging-related failures. To enhance its applicability, researchers have combined the TL distribution with the exponential distribution, resulting in the Topp-Leone Extended Exponential (TL-EE) distribution. “The TL-EE model provides greater versatility for capturing diverse lifetime data patterns and can address a broader range of failure behaviors.

The Topp-Leone (TL) distribution was first introduced by Topp and Leone in 1955 as a viable alternative to the beta distribution. Its structure makes it particularly useful for analyzing failure data. This one-parameter distribution is defined by its cumulative distribution function (CDF) and probability density function (PDF), which allow for effective modeling in various reliability and survival analysis contexts” (Al-Shomrani et al., 2016).

F (x) = 1 - (1-G²(x) ^α α > 0

F (x) = 2 αg (x) G (x) 1- G2 (x) ^{α -1}

“Equations 1 and 2 represent the cumulative distribution function (CDF) and probability density function (PDF) of the Topp-Leone (TL) distribution. These functions belong to a family of statistical distributions widely recognized for their flexibility and applicability. Over the years, the TL distribution has gained significant attention in statistical research due to its ability to model diverse real-world data” (Kotz and Nadarajah 2006). Studies and advancements by researchers like Nadarajah and Kotz (2003), Kotz and Seier (2007) and Zghoul (2011) have further solidified its importance in the field.

According to Bayoud (2016), “statistical distributions play an essential role in describing and analyzing real-world data across various domains, including economics, engineering, and biological sciences. However, classical distributions often fall short when applied to complex datasets. To address these limitations, researchers have introduced generated classes of probability distributions by incorporating one or more additional parameters, resulting in new, more flexible distributions. These generated families extend and generalize traditional distributions, providing better fits for real data in numerous applications”.

1. Statement of Problem

The exponential distribution is widely recognized for its application in modeling lifetime data. However, its inherent property of a constant failure rate makes it unsuitable for modeling many real-life datasets. “In practical scenarios, failure rates often vary over time, exhibiting patterns such as increasing, decreasing, bathtub-shaped, or inverted bathtub-shaped failure rates. This limitation restricts the exponential distribution's ability to effectively model lifetime data with non-constant failure rates” (Elgarhy et al. 2018).

To address this issue, there is a growing need to modify the exponential distribution to overcome its rigid constant failure rate property. Many datasets arising from survival analysis and reliability studies require more flexible models that can accommodate varying failure rate patterns. Advances in statistical modeling have focused on enhancing the flexibility of classical distributions to capture these diverse failure rate shapes.

The exponential distribution has been generalized in several ways to improve its fit to real-world lifetime data. “This study seeks to enhance the flexibility of the exponential distribution by incorporating an additional shape parameter, which allows for more versatile modeling of failure rate patterns”. Specifically, this work builds on the “Type II Topp-Leone family” of distributions proposed by El-Bassiouny and Mohamed (2019). which provides a framework for achieving better alignment with the complex failure rate behaviors observed in lifetime data.

1.3 Aim and Objectives

The aim of this study is to derive a new flexible modified exponential distribution called Type II Topp-Leone exponential (TIITL-E) distribution.

The following are the specific objectives of the study:

To derive the probability density function (pdf) and cumulative density function (cdf) of the new distribution.
To study some of the properties of the new distribution
Estimate the parameters of the new distribution.
To demonstrate the potentials of TIITL-E distribution in modeling lifetime data and compare its performance with other distributions with the same baseline distribution.

1.4 Significance of the study

“The Type II Topp-Leone Exponential (TII-TLE) distribution plays a crucial role in improving the analysis of complex lifetime data” (El-Bassiouny et al, 2019). It offers a level of flexibility that allows for modeling non-monotonic hazard rates, which traditional distributions, such as the exponential or Weibull distributions, often fail to capture effectively. This flexibility is particularly important in fields like reliability engineering, where accurately predicting failure times can lead to more efficient maintenance strategies. Similarly, in medical survival analysis, it enhances insights into disease progression and patient survival patterns, ultimately improving healthcare planning and interventions.

Moreover, the TII-TLE distribution proves valuable in areas such as public health and insurance, where it supports more precise risk evaluations. By aligning better with real-world risk patterns, it enables informed decision-making and resource allocation. This study significantly contributes to distribution theory by broadening the range of lifetime models available for researchers and practitioners. Additionally, it refines parameter estimation methods, such as the “Maximum Likelihood Estimation (MLE)” and the “Bayesian techniques”, making the TII-TLE distribution more practical for widespread application.

The findings from this research also pave the way for future exploration and innovation. The TII-TLE distribution could be instrumental in areas like artificial intelligence-driven predictive maintenance and genomics, where understanding lifetime behavior is essential. By extending the applicability of the TII-TLE distribution, this work establishes a foundation for its use in various cutting-edge and interdisciplinary fields.

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