Sciresol Sciresol https://indjst.org/author-guidelines Indian Journal of Science and Technology 10.17485/IJST/v13i23.178 Beta Lehmann-2 power function distribution with application to bladder cancer susceptibility and failure times of air-conditioned system Zaka Azam azamzka@gmail.com 1 Akhter Ahmad Saeed 2 Jabeen Riffat 3 Government College of Science Lahore, +93-300-4364368 Pakistan College of Statistical and Actuarial Sciences, University of the Punjab Lahore Pakistan COMSATS University Islamabad Lahore Campus Lahore Pakistan 13 23 2020 Abstract

Objectives: Probability distributions have great use in reliability engineering where the researchers try to find the distribution of the different processes. To meet the needs of the reliability engineers, we have proposed a simple probability distribution named as Beta Lehman-2 which may be proved more useful as compared to already existing models of the probability distributions. The aim of the study is to show the performance of the proposed distribution over already existing distributions. Methods: In this study, a new Beta Lehmann-2 Power function distribution (BL2PFD) is proposed. We suggest a new generator that will modify the Power function distribution called Beta Lehmann-2 generator (BL2-G). Findings: The various properties of the new distribution have been discussed in detail such as moments, vitality function, conditional moments and order statistics etc. We have also characterized the BL2PFD based on conditional variance. This distribution can be used for approximately symmetric data (normal data), positive and negative skewed data. Application: The application of this distribution is illustrated by using data sets from medical and engineering sources. The shape of the new distribution has been studied for applied sciences. After analyzing data, we conclude that the proposed model BL2PFD perform better in all the data sets while compared to different competitor models.

Keywords Beta Lehmann-2 Power function distribution Characterization of truncated distribution Lehmann alternatives Percentile estimator Power function distribution None
Introduction

The researchers in Engineering sciences mostly study the reliability of different components by taking the help from probability distributions that are simple in mathematical expression instead of using mathematically complex probability distributions. In Dallas (1976) introduced the power function as the inverse of Pareto distribution. Meniconi and Barry (1996) showed that power function distribution is better to fit for failure data over exponential, log ﻿normal ﻿and Weibull because it provides a better fit.

More studies about the application of this distribution and its applications can be found inAhsanullah, Shakil, and Kibria (2013); Chang (2007); Dorp and Kotz (2002). For modeling heterogeneous population, Saleem, Aslam, and Economou (2010) talked about the two component mixture of one-parameter Power function distribution. Estimation of the parameters of the two-parameter Power function distribution was studied by Zaka and Akhter (2013) through the methods of the least squares, relative least squares and ridge regression. According to its applicability in real life situations for modeling survival data, Tahir, Alizadeh, Mansoor, Cordeiro, and Zubair (2014) proposed the modification of the Power function distribution as Weibull-Power function distribution. By using the Bayesian inference, Hanif, Al-Ghamdi, Khan, and Shahbaz (2015) estimated the parameter of the one-parameter Power function distribution. In Shahzad and Asghar (2016) introduced the Transmuted Power function distribution by following Shaw and Buckley Shaw and Buckley (2009). In Okorie, Akpanta, Ohakwe, and Chikezie (2017) proposed the modification of the Power function distribution by using Marshall and Olkin Marshall and Olkin (1997) technique. In Haq, Usman, Bursa, and Ozel (2018) proposed the McDonald Power function distribution and Ibrahim (2017) proposed the Kumaraswamy Power function distribution. In Jabeen and Zaka (2019) discussed the parameters estimation for continuous uniform distribution using modified percentile estimators. Further Zaka, Akhter, and Jabeen (2020) introduced the exponentiated generalized class of power function distribution.

Materials and Methods

Lehmann alternatives were introduced by Lehman (1953) in the two-sample hypothesis testing context and are useful in survival analysis.

x=1-1-Gxα       (Lehmann2 relationship)

In Eugene, Lee, and Famoye (2002) proposed the Beta generator (Beta-G).

Fx=Bx(a,b)B(a,b)

Then the mixture of these two techniques is known as Beta Lehmann-2 generator (BL2-G). The probability density function (pdf) and cumulative distribution function (cdf) of the BL2-G are given as

Fx=B1-1-Gxα(a,b)B(a,b)

And

fx=1-1-Gxαa-11-Gxαb-1α1-Gxα-1gx B(a,b)

Where Gx:cdf and gx:pdf of any probability distribution

In this work, we suggest a new distribution that will generalize the Power function distribution (PFD) by using the above mentioned technique. We have derived some of the main structural properties of this distribution. We have also characterized the distribution by conditional moments (Right and Left Truncated mean), doubly truncated mean (DTM) and conditional variance. Maximum likelihood method (MLM) and Percentile estimation (P.E) method are used to estimate the shape and scale parameters of BL2PFD. The application of this distribution is illustrated by using data sets from medical and engineering sources.

2.1 Model Identification for Beta Lehmann-2 Power function distribution (BL2PFD)

The pdf and cdf of Power function distribution are given as follows

gx=γxγ-1βγ;    0<x<β,     γ>0

and

Gx=xβγ

Where γ and β  are the shape and scale parameters.

Following the generator (1), the BL2PFD is obtained by putting (3) and (4) in (2) and simplifying, we get﻿

fx=1-1-xβγαa-11-xβγαb-1α1-xβγα-1γxγ-1βγ B(a,b)  ;   0<x<β

and associated cdf is obtained by putting (4) in (1) as

Fx=B1-1-xβγα(a,b)B(a,b)

We may observe α, a and b are the tuning parameters. γ as the shape and β as scale parameter.

By definition, the survival function is

Sx=1-Fx=1-B1-1-xβγα(a,b)B(a,b)

And the Hazard Rate Function (HRF) of probability distribution is given as

2.2 Asymptotic behavior

The behavior of the pdf, cdf, hazard and survival functions of BL2PFD are being investigated as x → 0 and x → ∞.

i. limx0f(x)=0; possible values of α,a,b,γ and

ii. limxf(x)=; possible values of α,a,b,γ and

iii. limx0F(x)=0; possible values of α,a,b,γ and

iv. limxF(x)=1; if x=β and  possible values of α,a,b,γ and β

v. limxF(x)=0; if xβ if γ=0 and α0

vi. limx0F(x)=1; if xβ if γ>0 and α=0

vii. limx0S(x)=1; if xβ if γ=0 and α0

viii. limxS(x)=0; if xβ if γ>0 and α=0

ix. limx0H(x)=0; possible values of α,β,γ,φ and θ

x. limxH(x)=; possible values of α,β,γ,φ and θ

2.3 Characteristics of hazard function using glaser method

In Glaser (1980) had defined the conditions of increasing, decreasing, and upside-down bathtub-shaped failure rate. We use these conditions in our proposed distribution.

ηx=-f'xfx ηx=-βγ(γ-1)x-αb-1γxγ-1βγ1-xβγ+α(a-1)1-xβγα-11-1-xβγαγxγ-1βγ

If x > 0, then the values of \acuteηx under the following conditions are given in Table 1 .

<bold id="strong-5baf265b52ab46c5b8842e09673a361e"/>Values of <inline-formula id="if-0e1e7e3eba96"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>\acute</mml:mi><mml:mi>η</mml:mi><mml:mfenced><mml:mi>x</mml:mi></mml:mfenced></mml:math></inline-formula> under the following conditions
a b γ α β η´x
1 1 1 ≥1 ≥1 0
≥2 1 1 ≥1 ≥1 >0
1 ≥2 1 ≥1 ≥1 >0
1 1 ≥2 ≥1 ≥1 >0
0 1 1 ≥1 ≥1 <0
1 0 1 ≥1 ≥1 <0
1 1 0 ≥1 ≥1 <0

The above conditions shows that the hazard function of BL2PFD is increasing but if (a,b or γ)0, then it will be decreasing function. (See Figure 1)

2.4 Shapes

The BL2PFD can be approximately Normal Curve, whereas the HRF can be bathtub, monotonically increasing and decreasing shapes. (See Figure 1)

Plots of PDF, CDF and HRF of BL2PFD

The rth moments about zero of any distribution is described below

μr'=0βxrf(x)dx

By solving we get

μr'= aj ai al βrBa,b (a+l+j)

where aj=j=0-1jΓ(b)Γb-j j!, ai=i=0-1iΓ(rγ+1)Γrγ+1-i i! and al=l=0-1lΓ(iα+1)Γiα+1-l l! Plots of moments underdifferent parametric values of BL2PFD

2.6 Moment generating function

Apart from generating functions, the moment generating function can be utilized to describe the characteristic of the random variable.

Mot=0βetxf(x)dx

If X follows BL2PFD, the moment generating function may be derived as,

Mot=r=0trr! aj ai al βrBa,b (a+l+j)

where aj=j=0-1jΓ(b)Γb-j j!, ai=i=0-1iΓ(rγ+1)Γrγ+1-i i! and al=l=0-1lΓ(iα+1)Γiα+1-l l! 2.7 Random number generator

The random number of BL2PFD may be obtained from

F(x)=B1-1-xβγα(a,b)B(a,b)

After simplifying we get,

x=β1-1-rbeta(n,a,b)1α1γ

Where “ rbeta(n,a,b)" is the random numbers generated from Beta distribution.

2.8 Inverse moments

By definition Inverse moments may be obtained as

μ-r'=0βx-rf(x)dx

We get inverse moments for BL2PFD as

μ-r'= aj ai al β-rBa,b (a+l+j)

where aj=j=0-1jΓ(b)Γb-j j!, ai=i=0-1iΓ(-rγ+1)Γ-rγ+1-i i! and al=l=0-1lΓ(iα+1)Γiα+1-l l! 2.9 Vitality function

The vitality function is obtained for BL2PFD as

Vx=1Sxxβx fxdx

That may be obtained as

Vx=βB(a,b) aj ai al1-1-1-xβγαa+l+ja+l+j1-B1-1-xβγα(a,b)B(a,b) where aj=j=0-1jΓ(b)Γb-j j!, ai=i=0-1iΓ(1γ+1)Γ1γ+1-i i! and al=l=0-1lΓ(iα+1)Γiα+1-l l!
2.10 Information function

The Information Function is given as

IF=0βf(x)sdx

For BL2PFD the information function is given as

IF=(αγ)s-1β(γ-1)(s-1)ajaialβ(γs-γ)B(a,b)s(a-1)+j+l

where aj=j=0-1jΓαsb-1-s-1α+1Γαsb-1-s-1α+1-j j!, ai=i=0-1iΓγ-1-s-1γ+1Γγ-1-s-1γ+1-i i!  and al=l=0-1lΓ(iα+1)Γiα+1-l l!

2.11 Order statistics

The pdf of the order statistic may be written as

f1:nx=1B1,nf(x)1-F(x)n-1

For BL2PFD, we may write the lower and upper order statistics as

f1:nx=1B1,n1-1-xβγαa-11-xβγαb-1α1-xβγα-1γxγ-1βγBa,b*1-B1-1-xβγα(a,b)B(a,b)n-1

and

fn:nx=1B1,nf(x)F(x)n-1

fn:nx=1B1,n1-1-xβγαa-11-xβγαb-1α1-xβγα-1γxγ-1βγBa,b*B1-1-xβγα(a,b)B(a,b)n-1 2.12 Incomplete moments

The incomplete moments are given as

μX|a,b,α,β,γ;r(p)=0Pxrfxdx

By simplifying for BL2PFD we get

μX|α,β,γ,φ,θ;r(p)=aj ai al βr1-1-pβγα(a+l+j)Ba,b(a+l+j)

where aj=j=0-1jΓ(b)Γb-j j!, ai=i=0-1iΓ(rγ+1)Γrγ+1-i i! and al=l=0-1lΓ(iα+1)Γiα+1-l l! 2.13 Conditional moments

The conditional moments may be obtained as

EXr|X>t=1F-(t)tβxrf(x)dx

The conditional moments for BL2PFD may be obtained by using above expression as

EXr|X>t=1F-(t)aj ai al βr1-tβγα(a+l+j)Ba,b(a+l+j)

where aj=j=0-1jΓ(b)Γb-j j!, ai=i=0-1iΓ(rγ+1)Γrγ+1-i i! and al=l=0-1lΓ(iα+1)Γiα+1-l l! 2.14 Lorenz and Bonferroni curve

The Lorenz and Bonferroni curve may be obtained as

L(p)=1μ0qxl=0tlhl+1(x)dx L(p)=1μaj ai al β1-1-qβγα(a+l+j)Ba,b(a+l+j)

where aj=j=0-1jΓ(b)Γb-j j!, ai=i=0-1iΓ(1γ+1)Γ1γ+1-i i! and al=l=0-1lΓ(iα+1)Γiα+1-l l! B(p)=1Pμaj ai al β1-1-qβγα(a+l+j)Ba,b(a+l+j)

2.15 Characterization of BL2PFD

Let “X” be Beta-Lehmann2- Power function variable with Probability density function

fx=1-1-xβγαa-11-xβγαb-1α1-xβγα-1γxγ-1βγ B(a,b);      0<x<β

And let F-x be the survival function respectively. Then the random variable “X” has BL2PFD if and only if

V(Xxt)=ajaha1β2F(t)B(a,b)1-1-tβγa+j+1αa+j+1-ajaia1βF(t)B(a,b)1-1-tβγαa+j+1a+j+12 where $\quad \mathrm{V}(\mathrm{X} \mid \mathrm{x} \leq \mathrm{t}):$ Conditional variance \begin{array}{l}Also\;a_j=\sum_{j=0}^\infty\frac{\left(-1\right)^j\Gamma\left(b\right)}{\Gamma\left(b-j\right)\;j!},\;a_i=\sum_{i=0}^\infty\frac{\left(-1\right)^i\Gamma\left(\frac1\gamma+1\right)}{\Gamma\left(\frac1\gamma+1-i\right)\;i!},\;a_l=\sum_{l=0}^\infty\frac{\left(-1\right)^l\Gamma\left(\frac i\alpha+1\right)}{\Gamma\left(\frac i\alpha+1-l\right)\;l!}\;and\\\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;a_h=\sum_{i=0}^\infty\frac{\left(-1\right)^i\Gamma(\frac2\gamma+1)}{\Gamma\left(\frac2\gamma+1-i\right)\;i!}\end{array}

Proof:

Necessary part:

$$\mathrm{E}\left(\mathrm{X}^{\mathrm{r}} \mid \mathrm{x} \leq \mathrm{t}\right)=\frac{1}{\mathrm{F}(\mathrm{t})} \int_{0}^{\mathrm{t}} \mathrm{x}^{\mathrm{r}} \frac{\left(1-\left(1-\left(\frac{\mathrm{x}}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{\mathrm{a}-1}\left(\left(1-\left(\frac{\mathrm{x}}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{\mathrm{b}-1} \alpha\left(1-\left(\frac{\mathrm{x}}{\beta}\right)^{\gamma}\right)^{\alpha-1} \frac{\gamma \mathrm{x}^{\gamma-1}}{\beta^{\gamma}}}{\mathrm{B}(\mathrm{a}, \mathrm{b})} \mathrm{d} \mathrm{x}$$ Put\;1-{\left(1-{\left(\frac x\beta\right)}^\gamma\right)}^\alpha=z $$\mathrm{E}\left(\mathrm{X}^{\mathrm{r}} \mid \mathrm{x} \leq \mathrm{t}\right)=\frac{1}{\mathrm{F}(\mathrm{t}) \mathrm{B}(\mathrm{a}, \mathrm{b})}\left[\int_{0}^{1-\left(1-\left(\frac{\mathrm{t}}{\beta}\right)^{\gamma}\right)^{\alpha}} \beta^{\mathrm{r}}\left\{1-(1-\mathrm{z})^{1 / \alpha}\right\}^{\mathrm{r} / \gamma}(\mathrm{z})^{\mathrm{a}-1}(1-\mathrm{z})^{\mathrm{b}-1} \mathrm{d} \mathrm{z}\right]$$ $$\mathrm{E}\left(\mathrm{X}^{\mathrm{r}} \mid \mathrm{x} \leq \mathrm{t}\right)=\frac{\mathrm{a}_{\mathrm{j}} \mathrm{a}_{\mathrm{i}} \mathrm{a}_{1} \beta^{\mathrm{r}}}{\mathrm{F}(\mathrm{t}) \mathrm{B}(\mathrm{a}, \mathrm{b})}\left[\frac{1-\left\{1-\left(\frac{\mathrm{t}}{\beta}\right)^{\gamma}\right\}^{\alpha}}{\mathrm{a}+\mathrm{j}+1}\right]^{\mathrm{a}+\mathrm{j}+1}$$ $$\mathrm{E}(\mathrm{X} \mid \mathrm{x} \leq \mathrm{t})=\frac{\mathrm{a}_{\mathrm{j}} \mathrm{a}_{\mathrm{i}} \mathrm{a}_{1} \beta}{\mathrm{F}(\mathrm{t}) \mathrm{B}(\mathrm{a}, \mathrm{b})}\left[\frac{1-\left\{1-\left(\frac{\mathrm{t}}{\beta}\right)^{\gamma}\right\}^{\alpha}}{\mathrm{a}+\mathrm{j}+\mathrm{l}}\right]^{\mathrm{a}+\mathrm{j}+1}$$ Where\;a_j={\sum_{j=0}^\infty{\frac{{\left(-1\right)}^j\Gamma(b)}{\Gamma\left(b-j\right)\;j!}}},\;a_i={\sum_{i=0}^\infty{\frac{{\left(-1\right)}^i\Gamma(\frac1\gamma+1)}{\Gamma\left(\frac1\gamma+1-i\right)\;i!}}}\;and\;a_l={\sum_{l=0}^\infty{\frac{{\left(-1\right)}^l\Gamma(\frac i\alpha+1)}{\Gamma\left(\frac i\alpha+1-l\right)\;l!}}}

Put r=2

$$\mathrm{E}\left(\mathrm{X}^{2} \mid \mathrm{x} \leq \mathrm{t}\right)=\frac{\mathrm{a}_{\mathrm{j}} \mathrm{a}_{\mathrm{h}} \mathrm{a}_{1} \beta^{2}}{\mathrm{F}(\mathrm{t}) \mathrm{B}(\mathrm{a}, \mathrm{b})}\left[\frac{1-\left\{1-\left(\frac{\mathrm{t}}{\beta}\right)^{\gamma}\right\}^{\alpha}}{\mathrm{a}+\mathrm{j}+\mathrm{l}}\right]^{\mathrm{a}+\mathrm{j}+1}$$ Where\;a_j={\sum_{j=0}^\infty{\frac{{\left(-1\right)}^j\Gamma(b)}{\Gamma\left(b-j\right)\;j!}}},\;a_h={\sum_{i=0}^\infty{\frac{{\left(-1\right)}^i\Gamma(\frac2\gamma+1)}{\Gamma\left(\frac2\gamma+1-i\right)\;i!}}}\;and\;a_l={\sum_{l=0}^\infty{\frac{{\left(-1\right)}^l\Gamma(\frac i\alpha+1)}{\Gamma\left(\frac i\alpha+1-l\right)\;l!}}} \mathrm V(\mathrm X\mid\mathrm x\leq\mathrm t)=\frac{{\mathrm a}_\mathrm j{\mathrm a}_\mathrm h{\mathrm a}_1\beta^2}{\mathrm F(\mathrm t)\mathrm B(\mathrm a,\mathrm b)}\left[\frac{1-\left\{1-\left(\frac{\mathrm t}\beta\right)^\gamma\right\}^\alpha}{\mathrm a+\mathrm j+\mathrm l}\right]^{\mathrm a+\mathrm j+1}-\left[\frac{{\mathrm a}_\mathrm j{\mathrm a}_\mathrm i{\mathrm a}_1\beta}{\mathrm F(\mathrm t)\mathrm B(\mathrm a,\mathrm b)}\left[\frac{1-\left\{1-\left(\frac{\mathrm t}\beta\right)^\gamma\right\}^\alpha}{\mathrm a+\mathrm j+1}\right]^{\mathrm a+\mathrm j+1}\right]^2

Also Sufficient part

$$\mathrm{V}(\mathrm{X} \mid \mathrm{x} \leq \mathrm{t})=\frac{1}{\mathrm{F}(\mathrm{t})} \int_{0}^{\mathrm{t}} \mathrm{x}^{2} \mathrm{d} \mathrm{x}-\left\{\frac{1}{\mathrm{F}(\mathrm{t})} \int_{0}^{\mathrm{t}} \mathrm{x} \mathrm{dx}\right\}^{2}$$ $$\mathrm{V}(\mathrm{X} \mid \mathrm{x} \leq \mathrm{t})=\mathrm{t}^{2}-2 \int_{0}^{\mathrm{t}} \frac{\mathrm{xF}(\mathrm{x})}{\mathrm{F}(\mathrm{t})} \mathrm{d} \mathrm{x}-\left\{\mathrm{t}-\int_{0}^{\mathrm{t}} \frac{\mathrm{F}(\mathrm{x})}{\mathrm{F}(\mathrm{t})} \mathrm{d} \mathrm{x}\right\}^{2}$$

Equate (7) and (8), we get

t^2-2{\int_0^t{\frac{xF\left(x\right)}{F\left(t\right)}}}dx-\;{\left(t-{\int_0^t{\frac{F\left(x\right)}{F\left(t\right)}}}dx\right\}}^2=\frac{a_j\;a_h\;a_l\;\beta^2}{F\left(t\right)B(a,b)}{\left({\frac{1-\left(1-{\left(\frac t\beta\right)}^\gamma\right\}}{a+j+l}}^\alpha\right]}^{a+j+l}-{\left(\frac{a_j\;a_i\;a_l\;\beta}{F\left(t\right)B(a,b)}{\left({\frac{1-\left(1-{\left(\frac t\beta\right)}^\gamma\right\}}{a+j+l}}^\alpha\right]}^{a+j+l}\right]}^2 t-{\int_0^t{\frac{F\left(x\right)}{F\left(t\right)}}}dx=\frac{a_j\;a_i\;a_l\;\beta}{F\left(t\right)B(a,b)}{\left({\frac{1-\left(1-{\left(\frac t\beta\right)}^\gamma\right\}}{a+j+l}}^\alpha\right]}^{a+j+l}

Therefore

$$\mathrm{t}^{2}-2 \int_{0}^{\mathrm{t}} \frac{\mathrm{xF}(\mathrm{x})}{\mathrm{F}(\mathrm{t})} \mathrm{dx} \frac{\mathrm{a}_{\mathrm{j}} \mathrm{a}_{\mathrm{h}} \mathrm{a}_{1} \beta^{2}}{\mathrm{F}(\mathrm{t}) \mathrm{B}(\mathrm{a}, \mathrm{b})}\left[\frac{1-\left\{1-\left(\frac{\mathrm{t}}{\beta}\right)^{\gamma}\right\}^{\alpha}}{\mathrm{a}+\mathrm{j}+1}\right]^{\mathrm{a}+\mathrm{j}+1}$$

Differentiate w.r.t “t”

t^2f\left(t\right)= \frac{a_j\;a_h\;a_l\;\beta^2}{B\left(a,b\right)}{\left({1-\left(1-{\left(\frac t\beta\right)}^\gamma\right\}}^\alpha\right]}^{a+j+l-1}\alpha{\left(1-{\left(\frac t\beta\right)}^\gamma\right\}}^{\alpha-1}\frac{\gamma t^{\gamma-1}}{\beta^\gamma}

As

{a_j\;a_h\;a_l\left({1-\left(1-{\left(\frac t\beta\right)}^\gamma\right\}}^\alpha\right]}^{a+j+l-1}={\left(1-{\left(1-{\left(\frac t\beta\right)}^\gamma\right)}^\alpha\right)}^{a-1}{\left({\left(1-{\left(\frac t\beta\right)}^\gamma\right)}^\alpha\right)}^{b-1}{\left(\frac t\beta\right)}^2

Therefore

t^2f\left(t\right)=\frac{\beta^2}{B\left(a,b\right)}{\left(1-{\left(1-{\left(\frac t\beta\right)}^\gamma\right)}^\alpha\right)}^{a-1}{\left({\left(1-{\left(\frac t\beta\right)}^\gamma\right)}^\alpha\right)}^{b-1}{\left(\frac t\beta\right)}^2\;\alpha{\left(1-{\left(\frac t\beta\right)}^\gamma\right\}}^{\alpha-1}\frac{\gamma t^{\gamma-1}}{\beta^\gamma} f\left(t\right)=\frac1{B\left(a,b\right)}{\left(1-{\left(1-{\left(\frac t\beta\right)}^\gamma\right)}^\alpha\right)}^{a-1}{\left({\left(1-{\left(\frac t\beta\right)}^\gamma\right)}^\alpha\right)}^{b-1}\;\alpha{\left(1-{\left(\frac t\beta\right)}^\gamma\right\}}^{\alpha-1}\frac{\gamma t^{\gamma-1}}{\beta^\gamma}

The pdf of BL2PFD

Results 3.1 Maximum Likelihood Method (MLM)

Let x1, x2 ,..., xn be a random sample of size “n” from the BL2PFD. The log-likelihood function for the BL2PFD is given by

L\left(a,b,\alpha,\;\beta,\gamma\right)=nln\left(\frac{\alpha\gamma}{\beta^\gamma}\right)+n\left(a-1\right){{ln}{\left(1-{\left(1-{\left(\frac{x_i}\beta\right)}^\gamma\right)}^\alpha\right\}}}+n\left(\alpha b-1\right)ln\left(1-{\left(\frac{x_i}\beta\right)}^\gamma\right)+n\left(\gamma-1\right)lnx_i

The score vector is

U_a\left(a,b,\alpha,\;\beta,\gamma\right)=\frac\partial{\partial a}L\left(a,b,\alpha,\;\beta,\gamma\right)

U_b\left(a,b,\alpha,\;\beta,\gamma\right)=\frac\partial{\partial b}L\left(a,b,\alpha,\;\beta,\gamma\right)

U_\alpha\left(a,b,\alpha,\;\beta,\gamma\right)=\frac\partial{\partial\alpha}L\left(a,b,\alpha,\;\beta,\gamma\right) U_\beta\left(a,b,\alpha,\;\beta,\gamma\right)=\frac\partial{\partial\beta}L\left(a,b,\alpha,\;\beta,\gamma\right) U_\gamma\left(a,b,\alpha,\;\beta,\gamma\right)=\frac\partial{\partial\gamma}L\left(a,b,\alpha,\;\beta,\gamma\right)

The parameters of BL2PFD can be obtained by solving the above equations resulting from setting the five partial derivatives of L\left(a,b,\alpha,\beta,\gamma\right) equals to zero.

3.2 Estimation of BL2PFD Parameters from 'common percentiles' (P, E)

InDubey (1967) proposed a percentile estimator of the shape parameter, based on any two sample percentiles. After Dubey (1967), Marks (2005) also discussed it, in which he estimated the parameters of Weibull distribution with the help of percentiles.

Let x_{1\;},x_2,x_3,\dots,\;x_n be a random sample of size n drawn from Probability density function of BL2PFD. The cumulative distribution function of BL2PFD with shape and scale parameters \;\gamma\;and\;\beta\;, respectively

F\left(x\right)=\left(\frac{B_{\left(1-{\left(1-{\left(\frac x\beta\right)}^\gamma\right\}}^\alpha\right\}}(a,b)}{B(a,b)}\right\}

By solving we get

x=\beta\left(1-\left(1-rbeta(n,a,b)\right)^{\frac1\alpha}\right)^{\frac1\gamma}\;

Where “ rbeta(n,a,b)" is the random numbers generated from Beta distribution.

Let P75 and P25 are the 75th and 25th Percentiles, therefore\;\left(9\right)\;becomes\;

P_{75}=\beta\left(1-\left(1-0.75\right\}^\frac1\alpha\right\}^\frac1\gamma\; P_{25}=\;\beta\left(1-\left(1-0.25\right\}^\frac1\alpha\right\}^\frac1\gamma\;

Solving the above equations, we get

{\left(\frac{P_{75}}{P_{25}}\right)}^\gamma=\;\left(\frac{1-{\left(1-0.75\right\}}^{\frac1\alpha}}{\left(1-{\left(1-0.25\right\}}^{\frac1\alpha}\right\}}\right\} \gamma{{ln}{\left(\frac{P_{75}}{P_{25}}\right)=\;{{ln}{\;\left(\frac{1-{\left(1-0.75\right\}}^{\frac1\alpha}}{\left(1-{\left(1-0.25\right\}}^{\frac1\alpha}\right\}}\right\}}}}} \widehat\gamma=\;\frac{{{ln}{\;\left(\frac{1-{\left(1-0.75\right\}}^{\frac1\alpha}}{1-{\left(1-0.25\right\}}^{\frac1\alpha}}\right\}}}}{{{ln}{\left(\frac{P_{75}}{P_{25}}\right)}}} and\;\;\;\;\;\;\;\;\;\;\;\;\;\;\widehat\beta=\;\frac{P_{75}}{{\left(1-{\left(1-0.75\right\}}^{\frac1\alpha}\right]}^{\frac1{\widehat\gamma}}} generally\;\;\;\;\widehat\gamma=\;\frac{{{ln}{\;\left(\frac{1-{\left(1-H\right\}}^{\frac1\alpha}}{1-{\left(1-L\right\}}^{\frac1\alpha}}\right\}}}}{{{ln}{\left(\frac{P_H}{P_L}\right)}}} and\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\widehat\beta=\;\frac{P_H}{{\left(1-{\left(1-H\right\}}^{\frac1\alpha}\right]}^{\frac1{\widehat\gamma}}}

Where H= Maximum Percentage, L= Minimum 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 150 with pairs of ( \beta, \;\gamma) = {(1, 2), (2, 1) and (1.5, 1.5)}. We generated random samples of different sizes by observing that if R_i is random number taking (0, 1), then x_i=\;\beta{\left(1-{\left(1-rbeta(n,a,b)\right\}}^{\frac1\alpha}\right\}}^{\frac1\gamma} is the random number generation from BL2PFD with ( a,\;b,\alpha,\;\beta\;and\;\gamma) 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 (shape and scale parameters) of BL2PFD using different sample sizes and different values of parameters with mean square error MSE.

$$\text { M. S.E }(\widehat{\beta})=\mathrm{E}\left[(\widehat{\beta}-\beta)^{2}\right], \text { M. } \text { S. } \mathrm{E}(\hat{\gamma})=\mathrm{E}\left[(\hat{\gamma}-\gamma)^{2}\right]$$

<bold id="strong-f6515214669841558896b43d310cd542"/>Estimates for the parameters of BL2PFD with different estimation methods under the sample size 40 when <inline-formula id="inline-formula-d10181c430924836b3650dbd1aa4bfb0"> <tex-math>a=1,\;b=2\;and\;\alpha=3</tex-math></inline-formula>
METHODS True Values Estimated Values M.S.E
\beta \gamma \widehat\beta \widehat\gamma \widehat\beta \widehat\gamma
MLM 1 2 0.9434932 1.9464116 0.1030683 0.059389
2 1 2.3387613 0.8781634 0.5333875 0.0111197
1.5 1.5 1.401726 1.440550 0.2055521 0.0289
P.E 1 2 0.753597 2.039228 1.56648 0.1911323
2 1 1.572147 1.012443 0.8186342 0.04862102
1.5 1.5 1.33692 1.518119 0.9712043 0.09801185
<bold id="strong-8a0b3e7458724b89acda0f107f5bf913"/><bold id="strong-eeafacfc103c42c1b0dcbb99465ff6be"/>Estimates for the parameters of BL2PFD with different estimation methods under the sample size 150 when <inline-formula id="inline-formula-231c1163ef5e44309436750e4bd0dc7f"> <tex-math>a=1,\;b=2\;and\;\alpha=3</tex-math></inline-formula>
METHODS True Values Estimated Values M.S.E
\beta \gamma \widehat\beta \widehat\gamma \widehat\beta \widehat\gamma
MLM 1 2 1.022541 1.896632 0.05892636 0.014971
2 1 2.033640 1.055156 0.19859548 0.0155028
1.5 1.5 1.485175 1.429680 0.11660383 0.01384723
P.E 1 2 0.7660154 1.919106 1.52632 0.04705253
2 1 1.680933 0.961805 0.7078477 0.01222795
1.5 1.5 1.351793 1.43825 0.9112946 0.02562018

If we study the results of the Table 2, Table 3, in which sample sizes are (40, and 150) and the combinations of the values of (β, γ) = {(1, 2), (2, 1) and (1.5, 1.5)}. Then we get the results that MLM is the best for the estimation of β and γ. After MLM, the P.E is best for the estimation of scale and shape parameters of the BL2PFD.

Application and Discussion

In this section, we have analyzed two real life data sets to demonstrate the performance of BL2PFD. The comparison of the Probability distributions has been made in all the data sets on the basis of Akaike information criterion (AIC), the correct Akaike information criterion (CAIC), Bayesian information criterion (BIC) and Hannan-Quinn information criterion (HQIC).

Finally, using the above mentioned criteria’s, our proposed BL2PFD is better than the different competitor models for the same data sets.

We have adopted the data set consisting the remission time of 128 bladder cancer patients to demonstrate the performance of our proposed BL2PFD. These data were also studied byZea, Silva, Bourguignon, Santos, and Cordeiro (2012) and Lee and Wang (2003). The remission times in months are given: 0.08, 0.20, 0.40, 0.50, 0.51, 0.81, 0.90, 1.05, 1.19, 1.26, 1.35, 1.40, 1.46, 1.76, 2.02, 2.02, 2.07, 2.09, 2.23, 2.26, 2.46, 2.54, 2.62, 2.64, 2.69, 2.69, 2.75, 2.83, 2.87, 3.02, 3.25, 3.31, 3.36, 3.36, 3.48, 3.52, 3.57, 3.64, 3.70, 3.82, 3.88, 4.18, 4.23, 4.26, 4.33, 4.34, 4.40, 4.50, 4.51, 4.87, 4.98, 5.06, 5.09, 5.17, 5.32, 5.32, 5.34, 5.41, 5.41, 5.49, 5.62, 5.71, 5.85, 6.25, 6.54, 6.76, 6.93, 6.94, 6.97, 7.09, 7.26, 7.28, 7.32, 7.39, 7.59, 7.62, 7.63, 7.66, 7.87, 7.93, 8.26, 8.37, 8.53, 8.65, 8.66, 9.02, 9.22, 9.47, 9.74, 10.06, 10.34, 10.66, 10.75, 11.25, 11.64, 11.79, 11.98, 12.02, 12.03, 12.07, 12.63, 13.11, 13.29, 13.80, 14.24, 14.76, 14.77, 14.83, 15.96, 16.62, 17.12, 17.14, 17.36, 18.10, 19.13, 20.28, 21.73, 22.69, 23.63, 25.74, 25.82, 26.31, 32.15, 34.26, 36.66, 43.01,46.12 ,79.05.

We have compared our proposed BL2PFD with the Beta Exponentiated Pareto distribution (BEPD) by McDonald`s Power function distribution (McPFD) by Haq et al. (2018), Kumaraswamy Power function distribution (KPFD) by Ibrahim (2017), Beta exponentiated Pareto (BEPD) by Zea et al. (2012), Marshall-Olkin Power Lomax distribution (MOPLx) by Haq, Hamedani, Elgarhy, and Ramos (2019), and Power function distribution (PFD).

<bold id="strong-1ce3e5ea46d044199839f5ff8feb05d5"/>TTT Plot for Bladder Cancer Data

The TTT-plot of the remission time(in month) for bladder cancer patients is exhibited in Figure 3 , we may see that the Hazard rate function has little bit bathtub shape, So, we may easily fit BL2PFD on the bladder cancer data.

<bold id="strong-f7f8dfe854424a3d92d086f1a73307b0"/><bold id="strong-5de61bd5315946d0b7c264671c051493"/>“Statistics of bladder cancer data”
Models -logL AIC BIC CAIC
BL2PFD 401.2683 810.5365 822.586 810.8644
McPFD 811.5785 821.9553 811.9064 816.2008
KPFD 814.0711 822.6037 814.2662 817.5378
MOPLx 827.075 832.483 825.5162 847.3287
BEPD 826.1318 837.5085 826.4596 830.7540
PFD 942.4546 945.2988 942.4866 943.6102
Estimated pdf and cdf curves for Bladder CancerData

From Table 4 , we may see that BL2PFD provides better fit for the above data set as it provides minimum AIC, BIC, CAIC, HQIC.

4.2 Failure Times Data of Air-Conditioned System

The 2nd data set is reported by Aarset (1987) Dallas (1976), which corresponds to the 30 failure times of air-conditioned system of an airplane. The data are as follows: 23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, and 95.

We have compared BL2PFD with the alpha power transformed inverse exponential (APTIE) distribution by Dey, Alzaatreh, Zhang, and Kumar (2017), Marshall Olkin length biased exponential (MOLBE) distribution by Haq, Usman, Hashmi, and Al-Omeri (2019), APT inverted Weibull (APTIW) distribution by Ramadan and Magdy (2018), APT Pareto (APTP) distribution by Ihtisham, Khalil, Manzoor, Khan, and Ali (2019) and Alpha Power Transformed Inverse Lomax distribution (APTIL) by ZeinEldin, Haq, Hashmi, and Elsehety (2020).

<bold id="strong-af53bc3b441e4e7ba704849a6a6c1397"/>TTT Plot for Failure Times Data of Air-Conditioned System

The TTT-plot is displayed in Figure 5 , 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 BL2PFD on the failure time’s data of air-conditioned system.

<bold id="strong-f76d28f0c7434536910b538a9109cba3"/><bold id="strong-f15957af5d2b4b9f9eba19936b9178d9"/>“Statistics of air-conditioned system”
Distribution -logL AIC BIC CAIC
BL2PFD 143.0891 294.735 300.6473 296.5248
APTIL 151.910 309.819 314.023 311.652
APTIW 153.147 312.293 316.497 314.653
APTIE 153.372 310.744 313.546 312.847
APTP 156.025 314.169 316.972 316.235
MOLBE 155.336 314.673 317.475 317.984

Estimated DensityPlot for Failure Times Data of Air-Conditioned System

From Table 5 , we may see that BL2PFD provides better fit for the above data set as it provides minimum AIC, BIC, CAIC, HQIC.

Conclusion

We have proposed a new distribution called Beta Lehmann-2 Power function distribution (BL2PFD). This distribution can have applications in the fields of reliability, economics, actuaries and survival analysis. We have studied the properties of the new distribution including moments, survival function, hazard function, inverse moments, conditional moments, Lorenz curve, incomplete moments and order Statistics. We have also characterized the distribution by conditional variance. Data sets from different scenarios of applied sciences are used to show the efficiency of the proposed model over the already available models. It is hoped that the findings of this study will be useful for researchers in different field of applied sciences.