A New Pi-Exponentiated Method for Constructing Distributions with an Application to Weibull Distribution Текст научной статьи по специальности «Химические технологии»

A New Pi-Exponentiated Method for Constructing Distributions with an Application to Weibull Distribution

M. A. Lone* •

Department of Statistics, University of Kashmir, Srinagar, India murtazastat@gmail.com

T. R. Jan

Department of Statistics, University of Kashmir, Srinagar, India drtrjan@gmail.com

Abstract

A novel method for generating families of continuous distributions is presented by introducing a new parameter referred as Pi-Exponentiated Transformation (PET). Various properties of the PET method have been obtained. The method has been specialized on two-parameter Weibull distribution, and a new distribution called Pi-Exponentiated Weibull (PEW) is attained. A comprehensive mathematical treatment of the new proposal is provided. Closed-form expressions for the density function, distribution function, reliability function, hazard rate function have been provided. The PEW distribution is quite flexible, and it can be used to model data with decreasing, increasing or bathtub shaped hazard rates. Simulation study has been carried out to assess the behavior of the model parameters. Finally, the effectiveness of the suggested method is demonstrated by examining two real-life data sets.

Keywords: Pi-Exponentiated Transformation; Quantile Function; Reliability Function; Mean Waiting Time; Maximum Likelihood Estimation.

1. Introduction

Classical distributions are extensively employed in many applicable domains, including engineering, environmental studies, medical sciences, economics, actuarial, finance, insurance etc. to represent lifetime data. These distributions have been successfully implemented in all the fields listed above. However, in many domains, like reliability engineering and medical science, these conventional distributions do not offer the perfect fit when the data follow non-monotonic failure rates. As a result, generalized versions of these classical distributions are required to model reliability engineering and medical science data. Therefore, researchers became inspired to develop new modifications to theses existing distributions. These modified distributions offer more flexibility to the baseline model by introducing one or more extra parameters. In recent advances in distribution theory, researchers have shown a keen interest in proposing new methods for expanding the family of lifetime distributions. This has been accomplished through a variety of methods. Some well-known methods are:

• The exponentiated transformation initiated by Mudholkar and Srivastava [16], and is given

by

F(x;a) = ($(x))a; a > 0,x E R. Where f(x) is the cumulative distribution function (cdf) of baseline model.

M. A. Lone, T. R. Jan

A New Pi-Exponentiated Method for Constructing RT&A, No 1 (72)

Distributions with an Application to Weibull Distribution Volume 18, March 2023

• The beta-generated technique was proposed by Eugene et al. [7] that makes use of the beta distribution as the generator with parameters a and b to establish the beta generated distributions.

p(x)

F(x) = J r(s)ds. 0

Where r(s) is the probability density function (pdf) of a beta random variable (rv) and p(x) is the cdf of any rv X.

• The quadratic rank transmutation map approach proposed by Shaw and Buckley [19] and is given as

F(x; I) = (1 + I)p(x) - £p(x)2, | < 1, x e R. Where p(x) is the cdf of an existing distribution.

• Minimum Guarantee distribution proposed by Kumar et al. [9] and is given by

i 1

F(x) = e P(x), x e R. Where p(x) is the cdf of an existing distribution.

• Log-transformation proposed by Maurya et al. [15] and is given by

F(x) = 1 - log(2,- fx)), x e R. log2

Where p(x) is the cdf of an existing distribution.

• A new transmuted cumulative distribution function based on the Verhulst logistic function proposed by Kyurkchiev [10] and is given by

F(x) — 2^(x)

1 + p(x)'

Where p(x) is the cdf of an existing distribution.

Marshall and Olkin [14] proposed a general method for generating a new family of life distributions defined in terms of survival function as:

N ad (x) aip (x) n ^

F (x; a) = -—. = , , . ; a > 0, x e R.

1 - aip(x) $(x) + aip(x)

Where a = 1 — a and Pp(x) = 1 — p(x) is the survival function of the random variable X.

Anwar et al. [8] presented a new method based on trigonometric function called Sine-Exponentiated-Transformation (SET). The cdf of SET family of distributions for x e R is defined as

Fset(x, a) = p(x) sin (f pa (x)) ; a > 0. Where p(x) is the cdf of a continuous rv X.

• Lone et al. [11] proposed a new method for generating a family of continuous distributions called ratio trasformation (RT) method. The cdf of RT method for x E R is defined as

p(x)

frt(x;a) = , , p(x) ;a > 1 + a — ap(x)

Where p(x) is the cdf of a continuous rv X.

• Recently, Lone et al. [12] introduced an innovative method for generating a family of continuous distributions called the MTI method. They employed MTI method on Weibull distribution and derived a new three-parameter MTI Weibull (MTIW) distribution. The cdf of MTI method for x E R is defined as

r r \ ap(x)

Fmti(x; a) =- yv ' ; a > 0.

a — loga p(x)

Where p(x) = 1 — p(x) is the survival function of the random variable X.

In this manuscript a novel method for introducing greater flexibility to a family of distribution functions by bringing in new parameter to the given family has been introduced. This novel method has been refereed as PET. The proposed PET transformation is very simple and efficient method for introducing a new parameter to generalize the existing distributions. Some general properties of this class of distribution functions have been discussed. Then PET method has been specialized to a two-parameter Weibull distribution and generated a three-parameter PEW distribution, several statistical and reliability measures of PEW distribution have been obtained.

In section 2, the pdf and the cdf of the novel method have been obtained and various general properties of this method have been discussed. In section 3, the method has been specialized on two-parameter Weibull distribution and its structural properties as well as reliability measures have been obtained. In section 4, estimates of unknown parameters and simulation study have been performed. In section 5, two real data sets were analyzed to illustrate the efficacy of the suggested model. In section 6, the conclusion is stated.

2. General properties of PET method

Let X be a continuous rv, then the cdf of PET for x E R, is defined as

n(F(x))a — 1

Fpet(x) =-:— ; a > 0. (1)

n — 1

Obviously, Fpet(x) is a valid cdf only if F(x) is a valid cdf. The corresponding pdf of PET for x E R, is defined as

fpET(x) = ^n(F(x))a(F(x))a—1f (x) ; a > 0. (2)

n — 1

Clearly, fPET(x) is a weighted version of f (x), the weight function is given by

v(x) = n(F(x))a (F(x))a—1. Therefore, fPET(x) can be written as

fPET (x) = ff(^.

M. A. Lone, T. R. Jan

A New Pi-Exponentiated Method for Constructing RT&A, No 1 (72)

Distributions with an Application to Weibull Distribution Volume 18, March 2023

Where, k = E[v(X)] is the normalizing constant. By using the following power series

a" = £ j-u', (3)

j=0 j

the linear representation for the cdf and the pdf in (1) and (2) are respectively given by

1

Fpet(x)

n - 1 and

oo

£ a-(F(x))aj - 1 j=0

CO

fPET(x) = b £ a-(F(x))a(j+1)-1 f (x). j=0

Where, a- = j- and b = ^.

The reliability function RpET(x) is given by

Rpet(x) = ^ (1 - n(F(x))a-1) ; a > 0. (4)

The hazard rate function hPET(x) is given by

, , , alognf(x)(F(x))a-1

hPET(x) = n1-(F()x)r _1 ; a >0. (5)

If h(x) and R(x) are the hazard rate function and reliability function of f then the hazard rate

hPET(x) is given by

(F(x))a-1

hPET(x) = alognh(x)R(x) n1-(F(x))* _ 1 ; a > 0. (6)

From (6), it is clear that

'0 V a > 1

logn n-1

V a< 1

and

lim hPET(x) = <{ 1 ^lim^ h(x) V a = 1

lim hPET(x) = lim h(x). If F-1 (x) exists, then for a > 0, a random sample from FPET(x) can be obtained as

-if (log(1 + U(n - 1)) N

X = F- , ,

logn

where U is a uniform rv, 0 < u < 1.

3. PEW distribution and its properties

A rv X has a three-parameter PEW distribution denoted by PEW(a, j8, A) with parameters a, ft and A, if the cdf and the pdf of X for x > 0, are respectively, given by

n(1-e-Axii )a _ 1

Fpew(x) =-,-; a, ft, A > 0 (7)

n — 1

and

fpEW (x) = aAßognxß-1 e-^^1-^ (1 - e-^ r1; a, ß, A > 0.

The linear representations for the cdf in (7)is given by (9).

F

1

PEW

n- 1

£ ame-kAxß - 1

(8)

(9)

\k=0

Where

kf]a\ (logny

am = £ (-1)\kl /, . j=0 Vk/ j!

The linear representations for the pdf in (8) is given by (10).

TO

fpEW = E bmg(x). k=0

(10)

Where

£

(-1)ka(logn)j+1 fa(j + 1) - 1

and

= (n - 1)(k + 1)j!\ k

g(x) = (k + 1)Aßxß-1 e-(k+1)Axß.

Clearly, g(x) is the Weibull distribution with scale parameter (k + 1)A and shape parameter j8.

The reliability and the hazard rate of PEW distribution for x > 0 are given by (11) and (12),

respectively

Rpew (x)

n

n- 1

(1 - n(1-e-Axß )a; a, ß, A > 0

and

, , aAßlognxß-1 e-Axß (1 - e-Axß )a-1 „ , „ hpEw(x) = —t—ä-ß---; a, ß, A > 0.

71

1-(1-e-Axß)a _ 1

(11)

(12)

Figure 1 shows some PEW density graphs for various selected parameter values. Figure 2 depicts graphs of the hazard rate of the PEW distribution for different parameter values.

3.1. Simulation and Quantile

The PEW distribution can be simulated using inverse cdf method

X =- A ^

1

flog(1 + U(n - 1))

logn

Where U is a uniform rv, 0 < u < 1. The qth quantile of PEW distribution is given by

xq = \- Al°g

1

log(1 + q(n - 1)) logn

The median can be obtained as

x0.5

- Alog

1

'log( 2 (n + 1))'

logn

CO

b

m

ß

a

ß

a

x

Figure 1: Density plots of PEW for different combinations of a, ft and A = 1.

x x

Figure 2: Hazard rate plots of PEW for different combinations of a, ft and A = 1.

3.2. Moments and generating function

The rth moment of PEW distribution is obtained by using the following series representation.

ax = £ (13) k=0 k! tt /fa _ 1 \

(1 - x)b-1 = £ (-1)m 1xm ; \x\ < 1, b > 0. (14)

m=0 V m J

The rth moment of X can be obtained as

tt

E(Xr) = J xrf (x)dx 0

tt

= aAftlognr xr+ft-1e-Axftn(1-e-Axft )a (1 - e-Axft )a-1 dx. (15)

n - 1 J 0

Using (13) and (14) in (15), we have

-1 tt

E ) = jOft £ (l°gny+(-1)m (a(a + 1) - 1W xr+ft-1e-A(m+1)xftdx. (16)

( ) n - 1 a,m=0 a\ \ m )J y 1

By applying the transformation xft = y in (16), we get the final expression as E(Xr)= a £ (-1)m (logn)a+1 (a(a + 1) - 1N r + 1)

E(X) = n-1 a,£=0 ^al(m + 1) ft A m )T( ft + ^

The moment generating function of PEW distribution is obtained as

tt

mx (t) = j etxf (x)dx.

By using the same procedure as above, we get the final expression for moment generating function as

M () " £ (-1)"''Cog*)"*1/'(' + 1) - 1\r(' +1) Mx(t)=m r~ft+V)

3.3. The Mean residual life of PEW distribution The mean residual life function, say p.(t) of PEW distribution can be obtained as

F(t) = Rt) ( E(t) - J xf (x)dx\ - t. (17)

Where

and

eW = £ "J* +1- 0+1) (18)

n 1 a,m=0 Afta!(m + 1)ft+1 V m J ft

hf(x)dx =£ (-1)m(l°gn)a+1

0 n 1 a,m=0 A ft a!(m + 1) ft+1

a(a + 1) - ^ , lUft 1 m

+ 1)tft,ft + 1

(19)

x

M. A. Lone, T. R. Jan

A New Pi-Exponentiated Method for Constructing RT&A, No 1 (72)

Distributions with an Application to Weibull Distribution Volume 18, March 2023

Substituting (11), (18) and (19) in (17), we have

a E (—1)m (logn)a+1 (a (a + 1) — 1

n — n^—e^ )a a,m=0 A P a!(m + 1) P+1V m 1 .) (, , « 1

T{ P + 0 — 7(A(m + + 0

-1.

p

Where y(p, q) = J xq—1e—xdx, is called lower incomplete gamma function. 0

The mean waiting time \1 (t) of PEW distribution, can be obtained as

t

1

U (t) = t — YJTj J xf (x)dx. (20)

Substituting (7) and (19) in (20), we get

a ~ (—1)m (logn)a+1 u(t)=t--P- E 11—

n(1—e )a — 1 a,m=0 APa!(m + 1)P

a(a +1— > (AC* + 1)P P +1)

3.4. Renyi Entropy

Renyi entropy of PEW distribution, sayREX(u) can be obtained as

REX(u) = y—ulog ^ J f (x)udxj ; u > 0, u = 1.

(to u

J ^Wog*^ xu(P—1)e—uAxp x(1 — e—AxP)u(a—1) nu(1—e-Axii)adx) . (21)

Using (13) in (21), we have

(ulogn)a

REX (u

(u)=a-A—n) - 'og(p+'og (e

TO

,.u(P—1) uAxp /-t „—Axp \a{a+u)— ur

x J pxu(p—1)e—uAx (1 — e—Ax)a(a+u)—udxj . (22)

Using (14) and applying the transformation y = xp in (22), then the final expression for REx(u) is given by

REx(u)- u r, JaAlogn) J E (—1)m(ulogn)a

:(«)= T—X n—n) - >og(P) + ^ a!

ra(a + u) — u^ r lu + ^

m

(A(m + u))"+"T

x

x

3.5. Order Statistics

Let X1, X2,..., Xn be a random sample of size n, and let Xr:n denote the rth order statistic, then,

the pdf of Xr:n, say fr:n (x) is given by

n' 1

frn(x) = {r _ 1)n(n _ r)' F(x)r-1f (x)(1 _ F(x))n_r. (23)

Substituting (7) and (8) in (23) and using(14), we get

aA?logn nf (_1)a(n_r) fn_e_^V^1 fr:n (x)= B(r, n _ r + 1) f (n _ 1)a+r ^ V

X x?_1 )a (1 _ e_Ax? )a_1.

Where B(a, m) is a beta function.

3.6. Stress Strength Reliability

If X1 ~ PEW(a1, A1, ?) and X2 ~ PEW(a2, A2,?), where X1 and X2 are independent strength and stress rv's respectively, then, the stress strength reliability P(X1 > X2), say SSR, can be obtained as

CO

SSR = J fi(x)Fi(x)dx. (24)

— CO

Using (7) and (8) in (24), we have

CC

SSR = [ (OiM^nx?_ie_xixpnd_e_Aix?)ai

J V (n _ 1)2

;(1 _ e_A! x? )«i_1n(1_e_A2 x?f2 ^ dx _

n- 1

(25)

Using (13), (14) and applying the transformation y = x? in (25), then the final expression for SSR is given by

ccR ( «1A1 f f (_1)m+n(logn)a+b+1 fa (a + 1) _ Afba2\_ D

n _ H (n _ 1) afo mf=o a'b'(A1 (1 + m)+ nA2)\ m J{nJ )

4. Estimation

4.1. Maximum Likelihood Estimation

Let x1, x2,..., xn be a random sample from PEW distribution, then the logarithm of the likelihood function is

n- 1

l =nlog(aA?) + nlogi * ) +(? _ 1) f xi _ A f x[

\ n 1 / _1 _1

i=1 i=1

+ logn £ (1 _ e_Ax?^j +(a _ 1) tlog (1 _ e_Ax?^j . (26)

1

The MLEs of a, A and ft are obtained by partially differentiating (26) with respect to the corresponding parameters and equating to zero, we have

dl- = n + p log (l - e-Axft^ (logn(1 - e-Axi)a + = 0 (27)

dl n n n ft

dft = n + P ^ - A P Xil0gXl

+ p AX>l0gX (a + alogn(1 - e-Axi )a - l) = 0 (28)

i=i e-Axi - 1 V

dA = A - txft + t^h- (a + alogn(1 - e-Axft)a - l) = . (29)

dA A =1 =1 e- i -1 V J

Since, the above equations (27), (28) and (29) are not in closed form and are difficult to solve analytically. As a result, it is difficult to calculate the estimates of the parameters a, ft and A.

However, R software can be used to solve the equations numerically.

4.2. Simulation study

The simulation study has been conducted using R Software to demonstrate the behaviour of the MLEs in terms of the sample size. Two sets of sample (n=50, n=100) each repeated 1000 times with different combinations of parameters A = (1,2), a = (0.5,1.5,3) and ft = (0.5,1.5,3,5) were achieved from PEW. In each setting, the average values of MLEs and the corresponding empirical mean squared errors (MSEs) were obtained. The simulation results are presented in tables 1 and 2. Tables 1 and 2 show that the estimates are stable and reasonably close to the true parameter values. As the sample size increases the MSE decreases in all the cases.

5. Applications

In this section, we examine two data sets in order to describe the significance and flexibility of PEW distribution. The first data set has been taken from (Cordeiro and Brito [6]), consist of 48 rock samples from a petroleum reservoir. The dataset corresponds to twelve core samples from petroleum reservoirs that were sampled by four cross-sections. Each core sample was measured for permeability and each cross-section has the following variables: the total area of pores, the total perimeter of pores and shape. We analyze the shape perimeter by squared (area) variable. The observations are: 0.0903296, 0.2036540, 0.2043140, 0.2808870, 0.1976530, 0.3286410, 0.1486220, 0.1623940, 0.2627270, 0.1794550, 0.3266350, 0.2300810, 0.1833120, 0.1509440, 0.2000710, 0.1918020, 0.1541920, 0.4641250, 0.1170630, 0.1481410, 0.1448100, 0.1330830, 0.2760160, 0.4204770, 0.1224170, 0.2285950, 0.1138520, 0.2252140, 0.1769690, 0.2007440, 0.1670450, 0.2316230, 0.2910290, 0.3412730, 0.4387120, 0.2626510, 0.1896510, 0.1725670, 0.2400770, 0.3116460, 0.1635860, 0.1824530, 0.1641270, 0.1534810, 0.1618650, 0.2760160, 0.2538320, 0.2004470.

The second set of data is taken from (Aydin [2]) representing a random sample of average daily wind speed data for March, collected in 2015 from the Turkish Meteorological Services for Sinop, Turkey.The data are recorded as follows

2.8, 1.8, 3.2, 5.0, 2.4, 4.8, 2.9, 2.9, 2.3, 3.2, 2.3, 2.0, 1.9, 3.3, 4.4, 6.7, 4.3, 1.9, 2.2, 3.3, 2.1, 4.0, 2.0, 3.1, 3.8, 3.1, 3.2, 3.4, 2.8, 2.1, 3.1.

We compare the fit of the proposed PEW distribution with its sub-model Weibull (W) (see [20]) and a number of other competing models, namely Alpha Power Weibull (APW) (see [13]), Alpha Power Inverse Weibull (APIW) (see [3]), Modified Weibull (MW) (see [18]), Transmuted

Table 1: mean values of ML estimates and their corresponding mean square errors(n=50).

Parameter MLE MSE

A OL £ A & £ A & £

1 0.5 0.5 1.10179 0.50379 0.50421 0.33514 0.01926 0.01998

1.5 1.09164 0.50369 1.48989 0.29139 0.02022 0.07911

3 1.09261 0.50384 2.96632 0.29186 0.01920 0.27951

5 1.10208 0.50390 4.93128 0.33049 0.01722 0.77969

1.5 0.5 1.10144 1.48760 0.50812 0.34598 0.06202 0.02153

1.5 1.10143 1.48689 1.49968 0.34594 0.05021 0.05021

3 1.10492 1.47913 3.07588 0.35744 0.07182 0.30989

5 1.09812 1.48823 4.97413 0.34323 0.06181 0.92916

3 0.5 1.06841 2.92528 0.51901 0.34669 0.26838 0.02320

1.5 1.06708 2.92446 1.53312 0.35017 0.26879 0.11011

3 1.05536 2.92289 3.06026 0.27696 0.26986 0.39581

5 1.06038 2.92293 5.08217 0.27756 0.26961 1.06087

2 0.5 0.5 2.05707 0.50408 0.50614 0.58601 0.01735 0.01989

1.5 2.0553 0.50405 1.49456 0.58328 0.02217 0.07999

3 2.05262 0.50411 2.97686 0.58133 0.01935 0.28283

5 2.06155 0.50419 4.94529 0.58885 0.01855 0.76788

1.5 0.5 2.07548 1.48263 0.51078 0.47035 0.06455 0.02193

1.5 2.07572 1.48192 1.52755 0.47079 0.06453 0.09875

3 2.07602 1.48205 3.10344 0.46824 0.06445 0.35757

5 2.06563 1.48288 5.190317 0.39804 0.06414 0.93515

3 0.5 2.08168 2.92146 0.51719 0.50571 0.27205 0.02432

1.5 2.08232 2.92217 1.5266 0.50634 0.27417 0.12172

3 2.06791 2.92024 3.04978 0.44313 0.27343 0.43695

5 2.06542 2.92144 5.07546 0.44743 0.27186 1.19552

Table 2: mean values of ML estimates and their corresponding mean square errors(n=100).

Parameter MLE MSE

A 0L £ A & £ A& & £&

1 0.5 0.5 1.0506 0.50247 0.50289 0.23614 0.01735 0.01694

1.5 1.05062 0.50287 1.49297 0.21722 0.01631 0.05371

3 1.04942 0.50287 2.97447 0.20015 0.01601 0.17807

5 1.04802 0.50289 4.96282 0.19792 0.01496 0.47121

1.5 0.5 1.08086 1.49116 0.50075 0.28492 0.05099 0.01989

1.5 1.09562 1.49818 1.501702 0.26724 0.04504 0.08107

3 1.08072 1.48961 3.06019 0.27247 0.06647 0.30124

5 1.09101 1.49015 4.98213 0.28726 0.05136 0.76879

3 0.5 1.06714 2.95807 0.51053 0.23006 0.19014 0.02037

1.5 1.06075 2.95628 1.50918 0.19142 0.19102 0.08285

3 1.04935 2.95675 3.00808 0.19031 0.19078 0.29455

5 1.04988 2.95522 5.01794 0.14934 0.19213 0.77231

2 0.5 0.5 2.01161 0.50271 0.50106 0.46357 0.01587 0.01686

1.5 2.01208 0.50251 1.49633 0.46451 0.01733 0.05311

3 2.01008 0.50253 2.98083 0.46424 0.01524 0.17549

5 2.01285 0.50246 4.96023 0.45973 0.01634 0.46167

1.5 0.5 2.01167 1.48439 0.49989 0.39586 0.05389 0.01959

1.5 2.01133 1.48356 1.49515 0.39526 0.05386 0.07749

3 2.01735 1.48402 2.98175 0.36287 0.05382 0.27414

5 2.01093 1.48444 4.92742 0.36888 0.05417 0.72391

3 0.5 2.06175 2.95508 0.51010 0.31644 0.19261 0.02031

1.5 2.05964 2.95508 1.50872 0.31542 0.19263 0.08399

3 2.06125 2.95504 3.00326 0.31317 0.19257 0.29708

5 2.05041 2.95512 5.00925 0.28678 0.19251 0.77898

Weibull (TW) (see [1]), Odd Weibull (OW) (see [4]), Lindley Weibull (LW) (see [5]), Alpha Power Within Weibull Quantile (APWQ) (see [17]), Marshall Olkin Weibull (MOW) (see [14]) and Alpha Power exponential (APE) ([13]). The corresponding density functions for x > 0 are presented in

(ii)

(i)

x

x

Figure 3: (i) Fitted PEW density & relative histogram. (ii) Fitted PEW reliability & empirical reliability for first data set.

0 1 2 3 4 5 6 7

(ii)

Empirical Reliability Function Fitted Reliability Function

(i)

2

3

4

5

6

7

x

x

Figure 4: (i) Fitted PEW density & relative histogram. (ii) Fitted PEW reliability & empirical reliability for second data set.

the Appendix.

Tables 3, 4, 5 and 6 show that the PEW distribution has the minimum -2l(p), AIC, AICC, BIC and K-S values, as well as the greatest p-value, of all the competing models. As a result, the suggested model fits both the data sets better than the other competitive models. Also the Figures 3, 4, 5 and 6 definitely confirm the conclusions presented in Tables 3, 4, 5,& 6.

6. Conclusion

In this manuscript, a novel method known as PET has been presented. The PET approach has been applied to the Weibull distribution, and a new three-parameter PEW distribution is

(i)Q-Q Plot for PEW distribution

(ii)Q-Q Plot for PEW distribution

Theoretical Quantile

3 4

Theoretical Quantile

Figure 5: q-q plot for first and second data set.

(i)P-P Plot for PEW distribution

(i)P-P Plot for PEW distribution

Theoretical Cumuative Distribution

0.0 0.2 0.4 0.6 0.8 1.0

Theoretical Cumuative Distribution

Figure 6: p-p plot for first and second data set.

established. Various structural properties as well as reliability measures of the PEW distribution have been highlighted. The reason for adopting this method is that its cdf has a closed form and can represent data with monotone and non-monotone failure rates. It has been revealed that the three-parameter PEW distribution offers more flexibility in respect of hazard rate function and the density function. The suggested model is fitted to two distinct real-life data sets, and the figures demonstrate that it fits both data sets better than any other competing models.

o

2

5

Table 3: Estimates (standard errors) and kolmogorov smirnov test statistic for the first data set.

Model

Estimates Statistics

ß X K-S p-value

358.7757 0.5380 15.8364 0.08433 0.8844

(24.4872) (0.3304) (2.0124)

0.0320 3.4096 4.4898 0.12804 0.4108

(0.0508) (0.3606) (2.4429)

4.5086 3.0823 0.0029 0.10264 0.6927

(2.2779) (0.4720) (0.0017)

0.0010 2.7475 47.5555 0.14985 0.2313

( 2.0711) (0.3700) (7.9292)

0.6464 3.0077 0.2796 0.14075 0.2976

(0.2711) (0.3111) (0.0213)

27.13668 0.1312 3.2941 0.08862 0.8452

(15.5796) (0.0737) (5.1657)

17.0146 2.7406 1.4788 0.15011 0.2296

(22.4843) (0.2854) (1.4712)

64.6499 6.8937 65.4380 0.17289 0.1134

(9.0106) (0.2609) (0.7298)

0.0224 4.8044 2.2389 0.09189 0.8124

(0.0362) (0.6295) (9.9652)

100.4597 - 15.4005 0.10423 0.6741

(16.7779) (0.8223)

- 2.7475 47.5560 0.14990 0.2310

(0.2844) (17.9142)

PEW

APW

APIW

MW

TW

OW

LW

APWQ MOW APE W

Table 4: Information measures for the first data set.

Model -2l(ß ) AIC AICC BIC

PEW -116.4881 -110.4881 -109.9427 -104.8745

APW -110.56961 -104.56961 -104.02416 -98.95601

APIW -113.1797 -107.1797 -106.6342 -101.5661

MW -105.4775 -99.4775 -98.9321 -93.8639

TW -107.8930 -101.8930 -101.3476 -96.2794

OW -114.7898 -108.7898 -108.2443 -103.1762

LW -105.42378 -99.42378 -98.87832 -93.81017

APWQ -111.7091 -105.7091 -105.1636 -100.0955

MOW -115.3954 -109.3954 -108.8500 -103.7818

APE -111.3370 -107.3370 -106.7915 -103.5946

W -105.48441 -101.48441 -101.21774 -97.74201

Appendix

APW f (x) = ^Xßa 1-e-Axßxß-1e-Axß

APIW f (x) = ^ \ßx-(ß+a ) e-^x-ß a e-Xx-ß MW f (x) = (a + Xßxß-1)e-

-Xxß

f{x) = X (Xr1 e-(f)ß (l -a + 2ae-(X)') f (x) = f ( X )ß e( X ^ (e( X )ß - lj-1 [l + ^ ï )ß - l)

ce-i

-2

LW f (x) = ß- Xß xß-1 + X2ß x2ß-1e-a (Xx)ß J w a + 1

Table 5: Estimates (standard errors) and kolmogorov smirnov test statistic for the second data set.

a ß X K-S p-value

PEW 48.9866 0.8570 1.8620 0.10299 0.8974

(71.8227) (0.2715) (1.1181)

APW 0.5344 1.2214 7.8545 0.10759 0.8655

(2.3730) (1.5742) (0.1492)

APIW 2.6751 4.0481 32.1024 0.13443 0.6297

(4.6311) (0.7247) (16.6057)

MW 0.0010 2.9427 0.0255 0.16492 0.3680

( 0.2108) (0.4632) (0.0249)

TW 0.7341 3.2334 4.0055 0.14982 0.4897

(0.2973) (0.4079) (0.3343)

OW 56.6837 6.9111 5.8591 0.10573 0.8789

(34.145) (4.0662) (1.8199)

LW 0.0146 2.1105 3.0945 0.15024 0.4860

(0.0189) (0.2638) (2.1745)

APWQ 7.9249 3.7269 0.0047 0.16588 0.3611

(7.7298) (0.3934) (0.0034)

MOW 0.0139 5.3051 0.1529 0.10472 0.8859

(0.0249) (0.8008) (0.0459)

APE 183.6176 - 1.0341 0.12275 0.7385

(22.3726) (0.7071)

W - 2.9413 0.0256 0.16544 0.3642

(0.3668) (0.0140)

Table 6: Information measures for the second data set.

Model -2l( ß ) AIC AICC BIC

PEW 83.70147 89.70147 90.59036 94.00343

APW 84.90786 90.90786 91.79675 95.20982

APIW 85.51669 91.51669 92.40557 95.81865

MW 92.26848 98.26848 99.15737 102.57044

TW 90.26095 96.26095 97.14984 100.56291

OW 85.10416 91.10416 91.99305 95.40612

LW 89.30294 95.30294 96.19183 99.60490

APWQ 90.44442 96.44442 97.33331 100.74638

MOW 84.89439 90.89439 91.78328 95.19636

APE 88.34464 92.34464 93.23353 95.21262

W 92.19582 96.19582 96.62439 99.06379

APWQ f (x)

MW f (x) APE f (x)

(a - 1)Xßxß-1e

-1„-Xxß

loga (l + (a - 1)(1 - e-AxP) aAp (Ax) P-1e-(Ax)l>

1 - (1 - a)e-(A-xf loga a — 1

Ae-Xx a1-e-

where a,f!>,A > 0 and T(a) = f xa 1 e xdx is the gamma function.

0

Declaration

Conflict of interest: The authors declare that they have no Conflict of interest.

Estimates

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