Type II Topp-Leone Generalized Exponentiated Weibull Distribution: Properties and Application to Cancer Stem Cell Data

Type II Topp-Leone Generalized Exponentiated Weibull Distribution: Properties and Application to Cancer Stem Cell Data

¹Oseghale innocent Osezuwa, ²Oyebimpe Emmanuel Adeniji, ³Solanke Gbenga

¹Department of Mathematics and Statistics, Federal University Otuoke, Bayelsa State, Nigeria.

^2,3Department of Statistics, University of Ibadan, Oyo State, Nigeria.

DOI: https://doi.org/10.51584/IJRIAS.2024.909005

Received: 08 August 2024; Accepted: 13 August 2024; Published: 28 September 2024

ABSTRACT

This work presents a flexible lifetime distribution with increasing, increasing and decreasing and non-monotonic hazard rate called Type II Top-Leone Exponentiated Weibull ( ) distribution. The density function of  model has right-skewed and symmetrical shapes. Descriptive properties such as quantile function, moments, incomplete moments, probability weighted moments, moment generating functions, Renyl and Shannon entropies are theoretically established. Parameters of  distribution are estimated using maximum likelihood method. The potentiality/tractability of  distribution is demonstrated by its to cancer stem cell data.

Keywords: Quantile, Renyl and Shannon entropies, Probability Weighted Moments, Incomplete Moments.

INTRODUCTION

The Weibull (W) and exponentiated Weibull (EW) distributions give a close form solution to several problems in reliability studies. However, they do not possess a good/ reasonable parametric fit for real life applications; for example, when modeling phenomenon with non-monotonic failure rates, the Weibull distribution should not be considered because it does not provide a reasonable parametric fit. The unimodal and bathtub failure rate which are commonly observed in biological and reliability studies which cannot be modeled using the Weibull distribution. In recent decade, several attempts have been made to develop new families of distribution that extent the well-known families of distribution and also inducing flexibility which improves its modeling potentials of the baseline distribution in modeling real life data. Such work includes: the exponentiated Weibull distribution by Mudholkar and Srivastava (1993), Weibull-geometric (WG) distribution by Barreto-Souza et al. (2011), Exponentiated Weibull-geometric (EWG) distribution by Mahmoudi and Shiran (2012). Further, complementary versions of the Exponential Geometric and Weibull Geometric distributions, so-called Complimentary Exponential Geometric and Complimentary Weibull Geometric distribution, respectively, have been introduced by Louzada et a (2011) and Tojeiro et al. (2014). Marshall-Olkin Exponentiated Weibull distribution by Bidram et al. (2015), Transmuted Exponentiated Weibull distribution by Khan et al. (2019).

Motivation of study

The main purpose of the modification and extension forms of the Weibull distribution is to describe and fit the data sets with non-monotonic hazard rate, such as the bathtub, unimodal and modified unimodal hazard rate. Many modifications of the Weibull distribution have achieved the above purpose. On the other hand, unfortunately, the number of parameters has increased, the forms of the survival and hazard functions have been complicated and estimation problems have risen.

EW Distribution: A Brief Review

The EW distribution is an extension of the Weibull family and was developed by Mudholkar and Srivastava (1993). The EW distribution exhibits a non-monotone failure rate, making it a reliable model in lifetime data modeling. Mudholkar et al. (1993), Mudholkar and Huston (1996), Gupta and Kundu (2001), Nassar and Eissa (2003), and Choudhury (2005) applied the EW model in reliability and survival data modeling.

Cumulative Distribution Function (CDF)

The random variable \( X \) follows an EW distribution if its cumulative density function (CDF) is given by:
\[
F(x; \alpha, \beta, \theta) = 1 – e^{-\beta x^\theta \alpha}, \quad x > 0
\]
where \( \alpha \) and \( \theta \) are positive shape parameters and \( \beta \) is a positive scale parameter.

Probability Density Function (PDF)

The associated probability density function (PDF) is given as:
\[
f(x; \alpha, \beta, \theta) = \alpha \beta \theta x^{\theta-1} e^{-\beta x^\theta} \left(1 – e^{-\beta x^\theta \alpha}\right), \quad x > 0
\]

Reliability and Hazard Rate Functions

The reliability \( R(x; \alpha, \beta, \theta) \) and hazard rate \( h(x; \alpha, \beta, \theta) \) functions of the EW distribution are respectively given as:
\[
R(x; \alpha, \beta, \theta) = 1 – \left(1 – e^{-\beta x^\theta \alpha}\right)
\]
and
\[
h(x; \alpha, \beta, \theta) = \frac{\alpha \beta \theta x^{\theta-1} e^{-\beta x^\theta}}{1 – \left(1 – e^{-\beta x^\theta \alpha}\right)}
\]

Moments of the EW Distribution

The \( p \)-th moment about the origin of the EW distribution is given by:
\[
E(X^p) = \alpha \beta^{-p} \Gamma(p\theta + 1) N_p(\theta)
\]
where:
\[
N_p(\theta) = 1 + \sum_{l=1}^{\alpha-1} (-1)^l \binom{\alpha-1}{l} (l+1)^{-p\theta – 1}
\]
\( \Gamma \) represents the incomplete gamma function.

Type II Topp-Leone Exponentiated Weibull Distribution

Using the generalization by Elgarhy et al. (2018), the cumulative distribution function (CDF) of the Type II Topp-Leone Exponentiated Weibull (TIITLEW) distribution is given by:
\[
F(x; \alpha, \beta, \theta, v) = 1 – \left(1 – \left(1 – e^{-\beta x^\theta}\right)^{2\alpha}\right)^v
\]

Figure 1. Graph of the density and distribution functions of  distribution

Probability Density Function (PDF)

The PDF corresponding to the above CDF is given by:
\[
f(x; \alpha, \beta, \theta, v) = \alpha \beta \theta x^{\theta-1} e^{-\beta x^\theta} \left(1 – e^{-\beta x^\theta}\right)^{2\alpha-1} \left(1 – \left(1 – e^{-\beta x^\theta}\right)^{2\alpha}\right)^{v-1}
\]

Figure 2. Graph of the hazard functions of  distribution

Reliability and Hazard Rate Functions

The reliability function is:
\[
R(x; \alpha, \beta, \theta, v) = 1 – \left(1 – \left(1 – e^{-\beta x^\theta}\right)^{2\alpha}\right)^v
\]
The hazard rate function is:
\[
h(x; \alpha, \beta, \theta, v) = \frac{\alpha \beta \theta x^{\theta-1} e^{-\beta x^\theta} \left(1 – e^{-\beta x^\theta}\right)^{2\alpha-1}}{1 – \left(1 – \left(1 – e^{-\beta x^\theta}\right)^{2\alpha}\right)^v}
\]

Figure 3. Graph of the hazard functions of  distribution

The cumulative hazard function is:
\[
H(x; \alpha, \beta, \theta, v) = \log\left(1 – \left(1 – \left(1 – e^{-\beta x^\theta}\right)^{2\alpha}\right)^v\right)
\]

Quantiles, Median, and Upper Quartile

The quantile \( x_u \) of the TIITLEW distribution is given by:
\[
x_u = \frac{1}{\beta} \left[-\log\left(1 – \left(1 – u^{\frac{1}{v}}\right)^{\frac{1}{2\alpha}}\right)\right]^{\frac{1}{\theta}}
\]

The median and upper quartile are found by substituting \( u = 0.5 \) and \( u = 0.75 \) in the above expression, respectively:
\[
x_{0.5} = \frac{1}{\beta} \left[-\log\left(1 – \left(1 – 0.5^{\frac{1}{v}}\right)^{\frac{1}{2\alpha}}\right)\right]^{\frac{1}{\theta}}
\]
and
\[
x_{0.75} = \frac{1}{\beta} \left[-\log\left(1 – \left(1 – 0.25^{\frac{1}{v}}\right)^{\frac{1}{2\alpha}}\right)\right]^{\frac{1}{\theta}}
\]

Moments and Incomplete Moments

The \( r \)-th ordinary moment of \( X \) is given by:
\[
\mu_r’ = 2v\alpha \sum_{i=0}^{\infty} (-1)^i \binom{v-1}{i} \frac{(2\alpha i + 1) \Gamma(1 + r\theta)}{(i+1)^{r\theta} \beta^r}
\]

The \( r \)-th incomplete moment of \( X \) is given by:
\[
\varsigma_r(t) = 2v\alpha \sum_{i=0}^{\infty} (-1)^i \binom{v-1}{i} \frac{(2\alpha i + 1) \Gamma(1 + r\theta, (i+1)(\beta t)^\theta)}{(i+1)^{r\theta} \beta^r}
\]

Moment Generating Function (MGF)

The MGF of the TIITLEW distribution is given by:
\[
M_X(t) = E[e^{tX}] = \sum_{r=0}^{\infty} \frac{t^r}{r!} E[X^r]
\]
Substituting the moments provides the complete MGF.

Probability-Weighted Moments (PWM)

The probability-weighted moment \( \zeta_{rs} \) is given by:
\[
\zeta_{rs} = 2v\alpha \sum_{i=0}^{\infty} (-1)^i \binom{v-1}{i} \frac{(2\alpha i + 1) \Gamma(r+1)}{(i+1-r\theta)\Gamma(1+r\theta)}
\]

Rényi Entropy Function and \( \rho \)-Entropy

The Rényi entropy of the TIITLEW distribution is given by:
\[
I_\rho(X) = \frac{1}{1-\rho} \log\left(\int_{-\infty}^{\infty} f^\rho(x) dx\right)
\]
The \( \rho \)-entropy is given by:
\[
H_\rho(X) = \frac{1}{1-\rho} \log\left(1 – \frac{1}{\rho} I_\rho(X)\right)
\]

Stress-Strength Reliability

The stress-strength reliability parameter \( K \) is given by:
\[
K = P(X_2 < X_1) = \int_0^{\infty} g_1(x; \alpha_1, \beta_1, v_1, \theta) G_2(x; \alpha_2, \beta_2, v_2, \theta) dx
\]

Maximum Likelihood Estimation (MLE)

Let \( X_1, X_2, \dots, X_n \) be a random sample drawn from the TIITLEW distribution. The log-likelihood function is:
\[
l = n \log(2\alpha \theta v \beta^\theta) + (\theta-1)\sum_{i=1}^{n} \log(x_i) – \sum_{i=1}^{n} \beta x_i^\theta + (2\alpha-1)\sum_{i=1}^{n} \log\left(1 – e^{-\beta x_i^\theta}\right)
\]
Differentiating the log-likelihood function with respect to \( \alpha, \beta, \theta, v \), the score vector is obtained.

Application of  model

In this section, the  model is compared with Type II Topp-Leone Exponentiated Exponential ( ), Type II Top-Leone Weibull ( ), Weibull (W) and Exponential € distributions. Different goodness of fit measures like Cramer-von Mises (W), Anderson Darling (A), Kolmogorov- Smirnov (KS) statistics with Probability values (P-v), Akaike Information Criterion (AIC), consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), and Hannan-Quinn Information Criterion (HQIC). The data set represents the remission times (in months) of a random sample of 128 bladder cancer patients. For previous study see Lee and Wang (2003). That data are: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69. Table 1.0 gives the exploratory data analysis of the cancer data which shows that the data is over-dispersed and leptokurtic. Figure 4.0 represents the boxplot for the cancer data which shows that the data is positively skewed. Total time on test plot is given in Figure 5.0 which shows that the cancer data exhibits bathtub failure rate. The better fit corresponds to smaller W, A, KS, AIC, CAIC, BIC, HQIC and the larger the  The Maximum Likelihood Estimates (MLEs) of the unknown parameters and values of goodness of fit measures are computed for  distribution and its sub-models.

Table 4.4: Exploratory data Analysis of Bladder cancer patients

Figure 4.0 Boxplot for cancer data

Figure 5. Total Time on Test (TTT) plot

Table 2:  Result of the MLEs and standard error for cancer

Table 3: Goodness-of-fit statistics for bladder cancer data set

From Tables, we observe that  model has a better fit than its existing sub-model models which includes and model because it possesses the smallest and also possesses the highest P-value.

CONCLUSION

A new four-parameter distribution called the  distribution is developed. This distribution is a generalization of the EW distribution and contains several lifetime sub-models such as:   , ,  and . A characteristic of the  distribution is that its failure rate function can be decreasing, increasing, bathtub-shaped and unimodal depending on its parameter values. Several statistical properties of the new distribution such as its probability density function, its cumulative density function, quantiles, moments, incomplete moments, moments generating functions, probability weighted moments, stress-strength reliability function, Renyi and ρ-entropies are obtained. Fitting the model to areal data sets indicates the flexibility and usefulness of the proposed distribution in modeling cancer remission times data because it provides a good fit when compared with other competing models considered in this study.

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