Asymmetry approach to study for chemotherapy treatment and devices failure times data using modified Power function distribution with some modified estimators

Abstract

In order to improve the already existing models that are used extensively in bio sciences and applied sciences research, a new class of Weighted Power function distribution (WPFD) has been proposed with its various properties and different modifications to be more applicable in real life. We have provided the mathematical derivations for the new distribution including moments, incomplete moments, conditional moments, inverse moments, mean residual function, vitality function, order statistics, mills ratio, information function, Shannon entropy, Bonferroni and Lorenz curves and quantile function. We have also characterized the WPFD, based on doubly truncated mean. The aim of the study is to increase the application of the Power function distribution. The main feature of the proposed distribution is that there is no induction of parameters as compare to the other generalization of the distributions, which are complexed having many parameters. We have used R programming to estimate the parameters of the new class of WPFD using Maximum Likelihood Method (MLM), Percentile Estimators (P.E) and their modified estimators. After analyzing the data, we conclude that the proposed model WPFD performs better in the data sets while compared to different competitor models.

Full text

11 Asymmetry approach to study for chemotherapy treatment and devices failure time’s data using modified Power function distribution with some modified estimators

Azam Zaka , Ahmad Saeed Akhter and Riffat Jabeen College of Statistical and Actuarial Sciences, University of the Punjab, Lahore, Pakistan COMSATS University Islamabad Lahore Campus, Lahore, Pakistan *Corresponding author: Tel: 92 300 4364368 E-mail: azamzka@gmail.com (Azam Zaka); asakhter@yahoo.com (Ahmad S. Akhter); drriffatjabeen@cuilahore.edu.pk (Riffat Jabeen) Abstract

In order to improve the already existing models that are used extensively in bio sciences and applied sciences research, a new class of Weighted Power function distribution (WPFD) has been proposed with its various properties and different modifications to be more applicable in real life. We have provided the mathematical derivations for the new distribution including moments, incomplete moments, conditional moments, inverse moments, mean residual function, vitality function, order statistics, mills ratio, information function, Shannon entropy, Bonferroni and Lorenz curves and quantile function. We have also characterized the WPFD, based on doubly truncated mean. The aim of the study is to increase the application of the Power function distribution. The main feature of the proposed distribution is that there is no induction of parameters as compare to the other generalization of the distributions, which are complexed having many parameters. We have used R programming to estimate the parameters of the new class of WPFD using Maximum Likelihood Method (MLM), Percentile Estimators (P.E) and their modified estimators. After analyzing the data, we conclude that the proposed model WPFD performs better in the data sets while compared to different competitor models.

Key words:

Power function distribution; weighted distribution; characterization; adequacy model.

Biographical notes:

Azam Zaka is a PhD student in the College of Statistical and Actuarial Sciences, University of the Punjab, Lahore, Pakistan. He gained his MPhil and MSc in Statistics from the College of Statistical and Actuarial Sciences, University of the Punjab, Lahore, Pakistan. He is presently working as Assistant Professor of Statistics at the Govt. College of Science, Wahdat Road, Lahore, Pakistan. His research interests are distribution theory, statistical inference and quality control charts.

Dr. Ahmad Saeed Akhter is working as a Professor of Statistics at College of Statistical and Actuarial Sciences, University of the Punjab, Lahore. He has 37 years of teaching experience at university level. He has several research publications to his credit in different reputed national and international conferences and journals. His areas of interests are distribution theory and statistical inference.

Dr. Riffat Jabeen has earned a doctoral degree in Statistics from National College of Business Administration and Economics Lahore Pakistan. She has an M.Phil degree in Statistics from College of Statistical and Actuarial Sciences, University of the Punjab, Lahore, Pakistan, and Master in Statistics from LCWU, Lahore, Pakistan. Dr. Riffat Jabeen is currently serving for COMSATS University Islamabad, Lahore Campus, as an Assistant Professor at Department of Statistics. Her research interests are distribution theory, quality control charts and their applications, and survey sampling. Introduction

Weighted distributions have been extensively applied in the field of sampling which deals with unequal weighting of the units for example actuarial sciences, biomedicine, ecology and survival data analysis. Fisher (1934) has used firstly weighted distributions in order to estimate the frequencies by using methods of ascertainment. Let we have a random variable x with the following probability density function, 𝑓(𝑥; 𝛼, 𝛽) = 𝑤(𝑥;𝛽)𝑓(𝑥;𝛼)𝐸[𝑤(𝑥;𝛽)] (1.1) We take 𝑤(𝑥; 𝛽) as the non-negative weight function. Patil and Ord (1976) utilized the concept of weighted distribution and presented the idea of 𝛽𝑡ℎ order size biased distribution utilizing the weight function as 𝑤(𝑥) = 𝑥 𝛽 , and that was called moment distribution. It is is called as size biased when 𝛽 =1, whereas it is called the area biased distribution for 𝛽 =2. Afterwards many statisticians worked on weighted distribution such as Patil and Rao (1978), Arnold and Nagaraja (1991), Gove (2003), Mir and Ahmed (2009), Das and Roy (2011) applied this concept on different probability distributions. Ramos and Louszada (2016) discussed generalized weighted Lindley distributions with its different properties. Dar et al. (2017) introduced transmuted weighted exponential distribution and discussed its application. Balakrishnan et al. (2017) introduced the weighted Poisson distribution and its application to cure rate models. Different works on the weighted distributions and its parameters estimations are discussed in (Para and Jan (2018), Perveen and Ahmad (2018), Acitas (2019)) Dallas (1976) introduced the power function as the inverse of Pareto distribution. Meniconi and Barry (1996) showed that Power function distribution (PFD) is better to fit for failure data over exponential, lognormal and Weibull because it provides a better fit. Zaka and Akhter (2013) worked on parameters estimation for Power function distribution. Afterwards Zaka and Akhter (2014) provided the different modifications and Bayes inference of the parameters from power function distribution. Zaka et al. (2020) proposed the exponentiated class of Power function distribution. In this research paper, the effort is to introduce a new model called the Weighted Power function distribution (WPFD) which may be more suitable to the applied bio sciences and applied sciences data. We have studied the various properties of the under discussion distribution as moments, inverse moments, conditional moments, moments generating function, quantile function, mean residual function, vitality function, information function, mills ratio, bonferroni curve, lorenz curve, some entropies and order statistics. We have also produced some modifications of the WPFD. We have demonstrated the performance of the new models over already existing distributions by using a real life example from medical and applied sciences. The main feature of the proposed distributions is that there is no induction of parameters as compare to the other generalization of the distributions, which are complexed having many parameters. Weighted Power Function Distribution (WPFD)

Power function distribution (PFD) may model life time data as a good fit. The pdf (probability distribution function) may be written as: 𝑓(𝑥) = 𝛾𝑥 𝛾−1 𝛽 𝛾 ; 0 < 𝑥 < 𝛽 (2.1) and 𝐹(𝑥) = ( 𝑥𝛽 ) 𝛾 ; where β and γ are the scale and shape parameters. We may consider the following weight function as: w(x; α) =F (αx) (2.2) Hence using (2.1) and (2.2) in (1.1), the pdf of the Weighted Power function distribution (WPFD) is 𝑔(𝑥) =

2𝛾 𝑥 𝛽 ; 0 < 𝑥 < 𝛽 (2.3) The cumulative distribution function (cdf), survival, and hazard functions of WPFD are 𝐺(𝑥) = ( 𝑥𝛽 ) (2.4) 𝑠(𝑥) = 1 − ( 𝑥𝛽 ) (2.5) h(x) = (2γ)(X) (β) −(x) (2.6) Asymptotic Behavior

We may see the asymptotic behavior of the pdf, cdf, hazard and survival functions of WPFD as x → 0 and x → ∞. i. 𝑙𝑖𝑚 𝑥 → 0 𝑔(𝑥) = 0 ; 𝑖𝑓 𝛾 = 0 𝑎𝑛𝑑 𝛽 > 0. ii. 𝑙𝑖𝑚 𝑥 → ∞ 𝑔(𝑥) = ∞ ; ∀ 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝛾 𝑎𝑛𝑑 𝛽. iii. 𝑙𝑖𝑚 𝑥 → 0 𝐺(𝑥) = 0 ; ∀ 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝛾 𝑎𝑛𝑑 𝛽. iv. 𝑙𝑖𝑚 𝑥 → 0

𝐺(𝑥) = 1 ; 𝑖𝑓 𝛾 = 0 𝑎𝑛𝑑 𝛽 > 0. v. 𝑙𝑖𝑚 𝑥 → 0 𝑆(𝑥) = 0 ; 𝑖𝑓 𝛾 > 0 𝑎𝑛𝑑 𝛽 > 0. vi. 𝑙𝑖𝑚 𝑥 → 0

𝑆(𝑥) = ∞ ; 𝑖𝑓 𝛾 > 0 𝑎𝑛𝑑 𝛽 = 0. vii. 𝑙𝑖𝑚 𝑥 → 0

𝑆(𝑥) = 1 ; 𝑖𝑓 𝛾 = 0 𝑎𝑛𝑑 𝛽 ≥ 1. viii. 𝑙𝑖𝑚 𝑥 → ∞

𝑆(𝑥) = ∞ ; 𝑖𝑓 𝛾 > 0 𝑎𝑛𝑑 𝛽 ≥ 1. ix. 𝑙𝑖𝑚 𝑥 → 0 ℎ(𝑥) = 0 ; 𝑖𝑓 𝛾 = 0 𝑎𝑛𝑑 𝛽 > 0. x. 𝑙𝑖𝑚 𝑥 → ∞ ℎ(𝑥) = 0 ; 𝑖𝑓 𝛽 > 0 𝑎𝑛𝑑 𝛾 ≥ 1. Characteristics of Hazard function using Glaser method

We may use the conditions defined by Glaser (1980) as 𝜂(𝑥) = − 𝑔′(𝑥)𝑔(𝑥) 𝜂(𝑥) = − (2𝛾 − 1)𝑥 𝜂́ (𝑥) = (2𝛾 − 1)𝑥 If x > 0 , then 𝜂́ (𝑥) > 0 under the following conditions i. If 𝛾 ≥ 1 , then 𝜂́ (𝑥) > 0 . ii. If 𝛾 = 0 , then 𝜂́ (𝑥) = 0 . iii. If 𝛾 < 1 𝑜𝑟 𝛾 = 0 , then 𝜂́ (𝑥) < 0 The above conditions shows that the hazard function of WPFD is increasing but if 𝛾 < 1 𝑜𝑟 𝛾 = 0 , then it will be decreasing function.

Shapes

Figures.1-3 (See Appendix) shows some plots of the pdf, cdf and hrf for some parameter values of WPFD. WPFD have different shapes like increasing, right and left skewed and J shapes. Mathematical Properties of the WPFD

We may discuss some general properties of WPFD under this current section;

Quantile Function

By inverting (2.4), we get the quantile function as:

𝑄(𝑢) = 𝛽𝑈 (1/2𝛾)

Moments and Inverse Moments

The r th moment and the r th inverse moment of the random variable “ X” , say 𝜇 𝑟′ may be expressed as 𝜇 𝑟′ = 𝑟 (𝑟+2𝛾) and 𝜇 (−𝑟)′ =

2𝛾 𝛽 −𝑟 (−𝑟+2𝛾) Incomplete Moments and Conditional Moments

The Incomplete and Conditional moments may be expressed as: 𝜇 𝑥|(𝛽,𝛾);𝑟 (𝑝) = ∫ 𝑥 𝑟𝑝0 2𝛾 𝑥 𝛽 𝑑𝑥 = ( (2𝛾)𝛽 ) (𝑝) 𝑟+2𝛾 𝑟+2𝛾 And Conditional moments of “X”

𝐸(𝑥 𝑟 | 𝑥 > 𝑡) = ( (2𝛾)𝐹̅(𝑡)𝛽 ) (𝛽) 𝑟+2𝛾 −(𝑡) 𝑟+2𝛾 𝑟+2𝛾 Moments Generating Function (MGF)

The MGF of WPFD is expressed as 𝑀 (𝑡) = 1 + ∑ (𝑡𝛽) 𝑟 𝑟!( 𝑟2𝛾 +1)∞𝑟=1 Mean Residual function(MRF) and Vitality function(VF)

The mean residual function is given by the relation: 𝑒(𝑥) = ∫ 𝑆(𝑡) ∞𝑥 𝑑𝑡𝑆(𝑥) = (𝛽−𝑥)− ( 𝛽2𝛾+1−𝑥2𝛾+12𝛾+1 )1−( 𝑥𝛽 ) and 𝑉(𝑥) = ∫ 𝑥 𝑓(𝑥) ∞𝑥 𝑑𝑥𝑆(𝑥) = ( 𝛽2𝛾+1−𝑥2𝛾+12𝛾+1 )1−( 𝑥𝛽 ) Some Entropies and Information function

The

Rený i entropy of a random variable “X” is defined as; 𝐼 𝑅 (𝑠) = 𝑙𝑜𝑔 [∫ 𝑓 𝑠 (𝑥) ∞0 𝑑𝑥] = 𝑙𝑜𝑔 {( ) 𝑠 ( 𝛽 𝑠(2𝛾−1)+1 𝑠(2𝛾−1)+1 )} And Shannon entropy of “X” is defined as

𝐸{− 𝑙𝑜𝑔[𝑓(𝑥)]} = − [𝑙𝑜𝑔 + (2𝛾 − 1) {𝑙𝑜𝑔𝛽 − }] Also Information function provides the moments of self-information of the probability density function by taking the derivatives at certain at certain place

𝐸{𝑓(𝑥)} 𝑠 = ( ) 𝑠 ( 𝛽 𝑠(2𝛾−1)+1 𝑠(2𝛾−1)+1 ) Order Statistics

The pdf of jth order statistics may be written as following, 𝑓 𝑗,𝑛 (𝑥) = 𝑛!(𝑗−1)!(𝑛−𝑖)! 𝑓(𝑥)𝐹 𝑗−1 (𝑥){1 − 𝐹(𝑥)} 𝑛−𝑗 ; 𝑗 = 1, … , 𝑛 Therefore the pdf of lower order statistics 𝑓 (𝑥) = { 𝑛(2𝛾)(𝑋) 𝛽 } (1 − (𝑥) 𝛽 ) 𝑛−1 And the pdf of highest order statistic 𝑓 𝑛,𝑛 (𝑥) = { 𝑛(2𝛾)(𝑋) 𝛽 } ( (𝑥) 𝛽 ) 𝑛−1 The Mills Ratio

This is defined as the inverse of hazard rate function and mathematically expressed as m(x) = s(x)g(x) = (β −x )(2γ)x Bonferroni and Lorenz curves

𝐿(𝑝) = ∫ 𝑥 𝑓(𝑥)𝑑𝑥 = 𝑞0 (2𝛾)(𝑞) 𝜇 𝛽 (2𝛾+1) and 𝐵(𝑝) = 𝐿(𝑝)/𝑝 Some Modified Weighted Power Function Distribution (MWPFD) st Modified Weighted Power Function Distribution (MWPFD-1)

In this modification we replace the weight function in (2.2) by

𝐹 (𝛼 𝜃 𝑥 𝜃 ) i.e. 𝑤(𝑥; 𝛼, 𝜃) = 𝐹 (𝛼 𝜃 𝑥 𝜃 ) (See Table 1 in Appendix) nd Modified Weighted Power Function Distribution (MWPFD-2)

In this modification we replace the weight function in (2.2) by

𝐹 (𝛼 𝑥 ) i.e. 𝑤(𝑥; 𝛼, 𝜃) = 𝐹 (𝛼 𝑥 ) (See Table 1 in Appendix) Characterization based on Conditional moment (Doubly Truncated Mean)

Let “X”

Weighted Power function Variable with Probability density function 𝑔(𝑥) = 2𝛾 𝑥 𝛽 ; 0 < 𝑥 < 𝛽 And let

𝐺̅(𝑥) be the survival function respectively. Then the random variable “X ” has Weighted Power function distribution if and only if 𝐸(𝑋|𝑥 < 𝑋 < 𝑦) = 2𝛾𝛽 {𝐺(𝑦) − 𝐺(𝑥)} [𝑦 − 𝑥

2𝛾 + 1 ] where 𝐸(𝑋|𝑥 ≤ 𝑋 ≤ 𝑦): Doubly Truncated Mean .

Proof: Necessary part:

𝐸(𝑋|𝑥 ≤ 𝑋 ≤ 𝑦) = 1𝐺(𝑦) − 𝐺(𝑥) ∫ 𝑥 2𝛾 𝑥 𝛽 𝑑𝑥 𝐸(𝑋|𝑥 < 𝑋 < 𝑦) = 2𝛾𝛽 {𝐺(𝑦) − 𝐺(𝑥)} [𝑦 − 𝑥

2𝛾 + 1 ] (5.1)

Now Sufficient Part:

𝐸(𝑋|𝑥 ≤ 𝑋 ≤ 𝑦) = 1{𝐺(𝑦) − 𝐺(𝑥)} ∫ 𝑥 𝑦𝑥 𝑔(𝑥)𝑑𝑥 𝐸(𝑋|𝑥 ≤ 𝑋 ≤ 𝑦) = 𝑦𝐺(𝑦) − 𝑥𝐺(𝑥) − ∫ 𝐺(𝑋) 𝑦𝑥 𝑑𝑥𝐺(𝑦) − 𝐺(𝑥) (5.2) Equate (5.1) and (5.2), we get 𝑦𝐺(𝑦) − 𝑥𝐺(𝑥) − ∫ 𝐺(𝑥) 𝑦𝑥 𝑑𝑥𝐺(𝑦) − 𝐺(𝑥) = 2𝛾𝛽 {𝐺(𝑦) − 𝐺(𝑥)} [𝑦 − 𝑥

2𝛾 + 1 ]

After differentiating the above equation, we get 𝑔(𝑦) = 2𝛾𝑦 𝛽 This is the pdf of WPFD. Comparison between Maximum Likelihood and Percentile Estimation Methods of the Parameters of WPFD

Maximum Likelihood Method (MLM)

Let x , x ,..., x n be a random sample of size n from the WPFD. The log-likelihood function for the WPFD is given by 𝐿(𝛾, 𝛽) = 𝑛𝑙𝑛(2𝛾) + (2𝛾 − 1) ∑ 𝑙𝑛(𝑥 𝑖 ) 𝑛𝑖=1 − 2𝑛𝛾 𝑙𝑛(𝛽) The score vector is 𝑈 𝛽 (𝛾, 𝛽) = 𝑛𝛾𝛽 (6.1) 𝑈 𝛾 (𝛾, 𝛽) = 𝑛𝛾 + 2 ∑ 𝑙𝑛 𝑥 𝑖𝑛𝑖=1 − 2𝑛 𝑙𝑛(𝛽) (6.2) The parameters of Weighted Power Function distribution can be obtained by solving the above equations resulting from setting the two partial derivatives of L( 𝛾 , 𝛽 ) to zero; β does not exist, but the likelihood function can be maximize by taking 𝛽̂ = 𝑥 𝑛 ; Where “x n ” is the maximum value in the data. (6.3) 𝛾̂ = ( 𝑛2(𝑛 𝑙𝑛(𝛽) − ∑ 𝑙𝑛 𝑥 𝑖𝑛𝑖=1 )) Modified Maximum Likelihood Method (MMLM)

In this modification of the MLM, the (6.2) equation is replaced by the co-efficient of variation of WPFD. 𝑐. 𝑣 = 1√4𝛾(𝛾 + 2)

By solving the above expression, we get 𝛾̂ = −1 + √1 + 𝑥̅ 𝑆 𝛽̂ = 𝑥 𝑛 ; Where “ x n ” is the maximum value in the data Estimation of Weighted Power Function Distribution Parameters from “common percentiles” (P.E)

Dubey (1967) proposed a percentile estimator of the shape parameter, based on any two sample percentiles. Marks (2005) also discussed it, in which he estimated the parameters of Weibull distribution with the help of percentiles. Let 𝑥 , 𝑥 , 𝑥 , … , 𝑥 𝑛 be a random sample of size n drawn from Probability density function of Weighted Power function distribution. The cumulative distribution function of a Weighted Power function distribution with shape and scale parameters 𝛽 and 𝛾 , respectively 𝑥 = 𝛽(𝑅) ; 𝑅 = 𝐺(𝑥) (6.4) Let P and P are the 75 th and 25 th Percentiles, therefore (6.4) becomes 𝑃 = 𝛽(. 75) (6.5) 𝑃 = 𝛽(. 25) (6.6) Solving the above equations, we get 𝛾̂ = 𝑙𝑛 ( .75.25 )2 ∗ 𝑙𝑛 ( 𝑃 𝑃 ) and 𝛽̂ = 𝑃 (. 75) generally 𝛾̂ = 𝑙𝑛 ( 𝐻𝐿 )2 ∗ 𝑙𝑛 ( 𝑃 𝐻 𝑃 𝐿 ) and 𝛽̂ = 𝑃 𝐻 (𝐻) Where H= Maximum Percentage, L= Minimum Percentage and P = Percentile

Modified Percentile Estimator (M.P.E)

In this modification of the percentile estimators, (6.6) is replaced by the Median of Weighted Power function distribution. 𝑥̃ = 𝛽2 ⇒ 𝛽̂ = 𝑥̃2

From (6.5) 𝛽̂ = 𝑃 (.75) therefore 𝑥̃2 = 𝑃 (.75) ⇒ 𝛾̂ = 𝑙𝑛 (2∗.75)2∗𝑙𝑛 ( 𝑃75𝑥̃ ) 𝛾̂ = 𝑙𝑛 (2 ∗ 𝐻)2 ∗ 𝑙𝑛 ( 𝑃 𝐻 𝑥̃ ) 𝑎𝑛𝑑 𝛽̂ = 𝑃 𝐻 (𝐻) Where H= Maximum Percentage and P = Percentile. A simulation study is used in order to compare the performance of the proposed estimation methods. We carry out this comparison taking the samples of sizes as n = 40 and 100 with pairs of ( β , γ ) = {(1, 2), (3, 2) and (4, 3)}. We generated random samples of different sizes by observing that if R i is random number taking (0, 1), then x i = βR i1 2γ⁄ is the random number generation from Weighted Power function distribution with ( γ , β ) parameters. All results are based on 5000 replications. Such generated data have been used to obtain estimates of the unknown parameters. The results obtained from parameters estimation of the 2-parameters of Weighted Power function distribution using different sample sizes and different values of parameters with mean square error M.S.E. 𝑀. 𝑆. 𝐸 (𝛽̂) = 𝐸 [(𝛽̂ – 𝛽) ] , 𝑀. 𝑆. 𝐸 (𝛾̂) = 𝐸[(𝛾̂– 𝛾) ] If we study the two results of the Table 2 and Table 3 from Appendix, in which sample sizes are (40 and 100) and the combinations of the values of ( 𝛽 , 𝛾 ) = {(1, 2), (3, 2) and (4, 3)}. Then we get the results that MLM is the best for the estimation of 𝛽 and 𝛾. After MLM, the MMLM and Percentile method are best for the estimation of scale and shape parameters of the Weighted Power function distribution. Application

In this section, we illustrate the usefulness of the WPFD and its modifications. We fit these distributions on real life data and compare the result with the existing distributions.

The Data about group of patients given Chemotherapy treatment

The first data set is reported by Bekker et al. (2000), which corresponds to the survival times (in years) of a group of patients given chemotherapy treatment alone. The data consisting of survival times (in years) for 46 patients are: 0.047,0.115, 0.121,0.132,0.164,0.197,0.203,0.260,0.282,0.296, 0.334, 0.395, 0.458, 0.466, 0.501, 0.507, 0.529, 0.534, 0.540, 0.641, 0.644, 0.696, 0.841, 0.863, 1.099, 1.219, 1.271, 1.326, 1.447, 1.485, 1.553, 1.581, 1.589, 2.178, 2.343, 2.416, 2.444, 2.825, 2.830, 3.578, 3.658, 3.743, 3.978, 4.003, 4.033. We have estimated the parameters of the model by the method of MLE. We have used five other criteria’s to compare the performance of the proposed distributions with already existing distributions. We have used Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC) and Hannan-Quinn information criterion (HQIC) for this comparison. We have compared our proposed distribution with the kumarswamy Marshal-Olkin family of distribution (Kw-MO) proposed by Alizadeh et al. (2015),

Kumaraswamy Power function distribution (KPFD) by Ibrahim (2017),

McDonald`s Power function distribution (McPFD) by Haq et al. (2018) and Power function distribution (PFD) for the same data set. The TTT-plot is displayed in Figure 4 (See Appendix), which indicates that the HRF associated with the data set has a bathtub shape, since the plot shows a first concave curvature. So, we can easily fit WPFD on the Chemotherapy treatment data. In Table 4 (See Appendix), we may see that WPFD provides better fit for the above data set as it provides minimum AIC, BIC, CAIC and HQIC.

Devices failure times data

The second data set refers to 30 devices failure times given in Table 15.1 by Meeker and Escobar (1998). The data are: 275, 13, 147, 23, 181, 30, 65, 10, 300, 173, 106, 300, 300, 212, 300, 300, 300, 2, 261, 293, 88,247, 28, 143, 300, 23, 300, 80, 245, and 266. The same data has been used by Tahir et al. (2016) for Weibull Power function distribution (WP). We have used this data set to show the performance of our proposed Weighted Power function distribution (WPFD) over Tahir et al. (2016), Kumaraswamy Power function distribution (KPFD) by Ibrahim (2017),

McDonald`s Power function distribution (McPFD) by Haq et al. (2018) and Power function distribution (PFD). The TTT-plot is displayed in Figure 5 (See Appendix), which indicates that the HRF associated with the data set has an increasing shape, since the plot shows a first concave curvature. So, we can easily fit WPFD on the Devices failure time’s data. In Table 5 (See Appendix) provide the Statistics for Devices failure times. The proposed model WPFD is showing better results by providing the smallest AIC, BIC, CAIC and HQIC for the devices failure time’s data. Concluding Remarks

We have seen from Table 4 and 5 that WPFD best describe the discussed data sets as compare to the other models in literature. So may be used in order to describe the nature of life time data of medical sciences and applied sciences by providing the lowest values of the AIC, BIC, CAIC and HQIC among all fitted probability distribution functions. In this paper, we proposed the WPFD and its modifications. We derived some of its properties. The parameters of the distribution have been estimated by the Maximum Likelihood Method (MLM), Percentile Estimators (P.E) and their modified estimators. We have also characterized the distribution by doubly truncated mean (DTM). Different criteria’s has been used as discussed above to prove that the WPFD provides a better fit than existing distributions.

It is hoped that the findings of this paper will be useful for researchers in different field of applied sciences.

References

Acitas, S. (2019) ‘A new weighted distribution as an extension of the generalized half-normal distribution with applications’,

Journal of Statistical Computation and Simulation , Vol. 88, No.

12, pp. 2325-2341. Arnold, B.C. and Nagaraja, H.N. (1991) ‘On Some Properties of Bivariate Weighted Distributions’,

Communication in Statistics Theory and Methods , Vol. 20, No. 5-6, pp. 1853-1860. Balakrishnan, N., Koutras, M.V. and Milienos, F.S. (2017) ‘A weighted Poisson distribution and its application to cure rate models’,

Communication in Statistics Theory and Methods , Vol. 47, No. 17, pp. 4297-4310. Bekker, A., Roux, J. and Mostert, P. (2000) ‘A generalization of the compound Rayleigh distribution: using Bayesian methods on cancer survival times’,

Communication in Statistics Theory and Methods , Vol. 29, pp. 1419–1433. Dallas, A.C. (1976) ‘Characterization of Pareto and power function distribution’,

Annals of the Institute of Statistical Mathematics , Vol. 28, No. 1, pp. 491–497. Dar, A.A., Ahmed, A. and Reshi, J.A. (2017) ‘Transmuted Weighted Exponential Distribution and Its Application’,

Journal of Statistics Application and Probability , Vol. 6, No. 1, pp. 219-232. Das, K.K. and Roy, T.D. (2011) ‘Applicability of Length Biased Weighted Generalized Rayleigh Distribution’,

Advances in Applied Science Research , Vol. 2, No. 4, pp. 320-327. Dubey, S.D. (1967) ‘Some percentile Estimators for Weibull parameters’,

Journal of Technometrics , Vol. 9, No. 1, pp. 119-129. Fisher, R. (1934) ‘The effect of methods of ascertainment’,

Annals Eugenics , Vol. 6, pp. 13-25. Glaser, R. (1980) ‘Bathtub and Related Failure Rate Characterizations’,

Journal of the American Statistical Association , Vol. 75, No. 371, pp. 667-672. Gove, J.H. (2003), ‘Estimation and Application of Size-biased Distribution in Forestry’, In Modeling Forest Systems (201-211), Northeastern Research Station, USA. Haq, M.A., Usman, R.M., Bursa, N. and Ozel, G. (2018) ‘McDonald`s Power function distribution with theory and applications’,

International Journal of Statistics and Economics , Vol. 9, No. 2, pp. 70-85. Ibrahim, M. (2017) ‘The Kumaraswamy Power function distribution’,

Journal of Statistics Applications and Probability , Vol. 6, No. 1, pp. 81-90. Marks, N.B. (2005) ‘Estimation of Weibull parameters from common percentiles’,

Journal of applied Statistics , Vol. 32, No. 1, pp. 17-24. Meeker, W.Q. and Escobar, L.A. (1998) ‘Statistical Methods for Reliability Data’, (Wiley, New York). Meniconi, M. and Barry, D.M. (1996) ‘The power function distribution: A useful and simple distribution to assess electrical component reliability’,

Microelectronics Reliability , Vol. 36, pp. 1207–1212.

Mir, K.A. and Ahmad, M. (2009) ‘Size-biased distributions and their applications’,

Pakistan Journal of Statistics , Vol. 25, No. 3, pp. 283-294. Morad, A., Tahir, M.H., Cordeiro, G.M., Mansoor, M., Zubair, M. and Hamedani. G.G. (2015) ‘The Kumaraswamy Marshal-Olkin family of distributions’,

Journal of the Egyptian Mathematical Society , Vol. 23, No. 3, pp. 546-557. Para, B.A. and Jan, T.R. (2018) ‘On three parameter weighted Pareto type II distribution: properties and applications in medical sciences’,

Applied Mathematics and Information Sciences Letters , Vol. 6, No.

1, pp. 13-26. Patil, G. and Rao, C. (1978) ‘Weighted Distributions and Size-biased Sampling with Applications to Wildlife Populations and Human Families’,

Biometrics , Vol. 34, pp. 179-189. Patil, G.P. and Ord, J.K. (1976) ‘On Size-Biased Sampling and Related Form-Invariant Weighted Distributions’,

Industrial Journal of Statistics , Vol. 38, No. 1, pp. 48-61. Perveen, Z. and Ahmad, M. (2018) ‘Some properties of size -biased weighted Weibull distribution’,

International Journal of Advanced and Applied Sciences , Vol. 5, No. 5, pp. 92-98. Ramos, P.L. and Louzada, F. (2016) ‘The generalized weighted Lindley distribution: Properties, estimation and applications’,

Cogent Mathematics , Vol. 3, pp. 1-18. Tahir, M., Alizadeh, M., Mansoor, M., Cordeiro, G.M. and Zubair, M. (2016) ‘The Weibull-Power function distribution with applications’,

Hacettepe University Bulletin of Natural Sciences and Engineering Series b: Mathematics and Statistics , Vol. 45, No. 1, pp. 245-265. Zaka, A. and Akhter, A.S. (2013) ‘Methods for estimating the parameters of the Power function distribution’,

Pakistan Journal of Statistics and Operational Research , Vol. 9, No. 2, pp. 213-224. Zaka, A. and Akhter, A.S. (2014) ‘Bayesian Analysis of Power Function Distribution Using Different Loss Functions’,

International Journal of Hybrid Information Technology , Vol. 7, No. 6, pp. 229-244. Zaka, A. and Akhter, A.S. (2014) ‘Modified Moment, Maximum Likelihood and Percentile Estimators for the Parameters of the Power Function Distribution’,

Pakistan Journal of Statistics and Operational Research , Vol. 10, No. 4, pp. 361-368.

Zaka, A., Akhter, A.S. and Jabeen, R. (2020) ‘The exponentiated generalized Power function distribution: theory and real life applications’,

Advances and Applications in Statistics , Vol. 61, No. 1, pp. 33-63.

Appendix

Figure 1:

Plots of pdf of WPFD.

Figure 2:

Plots of cdf of WPFD.

Figure 3:

Plots of hrf of WPFD.

Figure 4: TTT Plot for Chemotherapy treatment Figure 5: TTT Plot for Devices failure times Table 1: The pdf and properties of the MWPFD-1 and MWPFD-2 SR Properties MWPFD-1 MWPFD-2

1 Complete pdf (𝛾𝜃 + 𝛾)(𝑥) 𝛾𝜃+𝛾−1 𝛽 𝛾𝜃+𝛾−1 ; 0 < 𝑥 < 𝛽 𝑤ℎ𝑒𝑟𝑒 𝛽 > 0 𝑎𝑛𝑑 𝜃 ≥ 0 (𝛾 𝜃⁄ + 𝛾)(𝑥) 𝛾 𝜃⁄ +𝛾−1 𝛽 𝛾 𝜃⁄ +𝛾 ; 0 < 𝑥 < 𝛽 𝑤ℎ𝑒𝑟𝑒 𝛽 > 0 𝑎𝑛𝑑 𝜃 > 0

2 Moments (𝛾𝜃 + 𝛾)𝛽 𝑟 (𝑟 + (𝛾𝜃 + 𝛾)) (𝛾 𝜃⁄ + 𝛾)𝛽 𝑟 (𝑟 + (𝛾 𝜃⁄ + 𝛾))

3 cdf (𝑥) (𝛾𝜃+𝛾) 𝛽 (𝛾𝜃+𝛾) (𝑥) (𝛾 𝜃⁄ +𝛾) 𝛽 (𝛾 𝜃⁄ +𝛾)

4 Moments Generating Function 𝑟 𝑟! ( 𝑟(𝛾𝜃+𝛾) + 1) ∞𝑟=1 𝑟 𝑟! ( 𝑟(𝛾 𝜃⁄ +𝛾) + 1) ∞𝑟=1

5 Survival Function (𝛾𝜃+𝛾) 𝛽 (𝛾𝜃+𝛾) (𝛾 𝜃⁄ +𝛾) 𝛽 (𝛾 𝜃⁄ +𝛾)

6 Hazard Function (𝛾𝜃 + 𝛾)(𝑋) (𝛾𝜃+𝛾)−1 (𝛽) (𝛾𝜃+𝛾) − (𝑥) (𝛾𝜃+𝛾) (𝛾 𝜃⁄ + 𝛾)(𝑋) (𝛾 𝛽⁄ +𝛾)−1 (𝛽) (𝛾 𝜃⁄ +𝛾) − (𝑥) (𝛾 𝜃⁄ +𝛾)

7 Random Number Generator 𝛽(𝑅) 𝛽(𝑅)

8 Inverse Moments (𝛾𝜃 + 𝛾) 𝛽 −𝑟 (−𝑟 + (𝛾𝜃 + 𝛾)) (𝛾 𝜃⁄ + 𝛾) 𝛽 −𝑟 (−𝑟 + (𝛾 𝜃⁄ + 𝛾)) Table 2: Estimates for the parameters of Weighted Power function distribution with different estimation methods under the sample size 40

Methods True Values Estimated Values M.S.E 𝛽 𝛾 𝛽̂ 𝛾̂ 𝛽̂ 𝛾̂ MLM 1 2 0.9938524 2.104826 0.00007404 0.1308768 3 2 2.981884 2.098744 0.0006498764 0.1283807 4 3 3.983295 3.154041 0.0005503625 0.2938336 MMLM 1 2 0.9936337 2.079167 0.00008128 0.1669906 3 2 2.981997 2.072701 0.0006456896 0.1585582 4 3 3.9833 3.146969 0.0005463373 0.4190948 P.E 1 2 0.9927462 2.177096 0.0008329366 0.3118459 3 2 2.97946 2.181738 0.007721837 0.3214192 4 3 3.980408 3.267518 0.006049237 0.6997704 M.P.E 1 2 0.9942221 2.249254 0.001039149 0.6753827 3 2 2.980275 2.273176 0.009635285 0.7130634

4 3 3.982908 3.418103 0.007461182 1.716589

Table 3: Estimates for the parameters of Weighted Power function distribution with different estimation methods under the sample size 100

Methods True Values Estimated Values M.S.E 𝛽 𝛾 𝛽̂ 𝛾̂ 𝛽̂ 𝛾̂ MLM

1 2 0.9974972 2.039611 0.00001267 0.0452979

3 2 2.99268 2.041316 0.0001078838 0.04401944

4 3 3.993276 3.05943 0.00009112 0.1004658

MLM-1

1 2 0.9974775 2.03012 0.00001305 0.05853799

3 2 2.992459 2.026962 0.000112158 0.05775556

4 3 3.993253 3.046534 0.00009274 0.1397898

P.E

1 2 0.9976476 2.064303 0.0003183863 0.09982225

3 2 2.992453 2.064008 0.003019171 0.09917686

4 3 3.992342 3.117756 0.002340664 0.2392223

P.E-1

1 2 0.9975793 2.101732 0.0004208572 0.1971112 3 2 2.992632 2.110028 0.003701643 0.2164466 4 3 3.993442 3.144727 0.002941814 0.4740981

Table 4: Statistics for Chemotherapy Treatment Data Distribution AIC CAIC BIC HQIC WPFD 107.5513 107.6489 109.3125 108.2008

PFD 108.7513 108.7482 110.5125 109.3788 MWPFD-1 109.5513 109.8513 113.0737 110.8503 MWPFD-2 109.6419 109.8711 113.0934 110.8704 KPFD 109.8881 109.9035 114.4717 111.1366 McPFD 111.1315 112.1841 118.1763 113.7294 Kw-MOW 119.134 120.672 128.167 122.501