WEIGHTED INTERVENED EXPONENTIAL DISTRIBUTION AS A LIFETIME DISTRIBUTION Текст научной статьи по специальности «Математика»

WEIGHTED INTERVENED EXPONENTIAL DISTRIBUTION AS A LIFETIME DISTRIBUTION

Vilayat Ali Bhat1, Sudesh Pundir2 •

1,2 Department of Statistics, Pondicherry University, Kalapet, Puducherry-605014, INDIA

vilayat.stat@gmail.com1 sudeshpundirl9@gmail.com2

Abstract

This study proposes and investigates the weighted intervened exponential distribution, which is demonstrated as a generalized extension of the intervened exponential distribution. The form of the weighted intervened exponential distribution is obtained by considering a specific non-negative weighted function. The probability density function and cumulative density function of the proposed model are given, and its generalized form of reliability function and the hazard rate function is also derived. By choosing a different set of parametric values, the graphical demonstrations of the probability density function of weighted intervened exponential distribution are given where it acquires different curve shapes. The weighted intervened exponential distribution density function is then further studied in the limited form as a special case called the length-biased intervened exponential distribution. Along with the distribution of order statistics, stochastic ordering, stress-strength reliability, and entropy measure, several distributional and reliability aspects of the length-biased intervened exponential distribution are derived. For estimating the unidentified parameters of the length-biased variant, the most suggested approach known as the maximum likelihood estimation technique is implemented. To explore the behavior of the parameter estimates for various sample sizes, a sample data generation technique is required to carry out the process. Since the quantile function of the length biased intervened exponential distribution is not in closed form. So, the alternative data generation algorithm is employed which is known as the acceptance-rejection algorithm technique, and a Monte-Carlo simulation study is done. The absolute average bias and mean square error of the estimated parameters of the length-biased version model are calculated and it is noticed that both the calculated measures decrease simultaneously on increasing the sample size. In order to determine if the model is appropriate, a real-life time-to-event data set is examined as an example, and length biased distribution is juxtaposed with several other common available lifetime distributions for comparison purposes.

Keywords: Acceptance-rejection algorithm; Entropy; Weighted intervened distribution; Monte Carlo simulation

l. Introduction

The literature witnesses the dominance of exponential distribution among all existing lifetime models over data analysis in essential fields like reliability theory, survival analysis, and several other branches of statistical as well as the applied sciences. As it is well observed the exponential model has been extended in different ways with the help of new model development methods and transformation techniques, such as inverted gamma distribution by Lin et al., [ll], generalized exponential distribution by Gupta and Kundu [9], weighted exponential distribution by Gupta and Kundu [8], etc. are few examples. Apart from these extensions, a new development in statistics called the intervention was brought into existence by Shanmugam [2l] in the form of a new discrete intervention model called the intervened Poisson distribution. Later on, in another

successful attempt Shanmugam et al. [22] developed the continuous intervened exponential model, which was later introduced as a lifetime distribution in the field of reliability theory and survival analysis by Bhat and Pundir [2]. The cumulative and probability density functions (c.d.f & p.d.f) of IvED are given by

-(x-a)/pe_ e-(x-a)/0

- P = 1

and,

FivED (x; ©)

fivED(x; ©)

1

Pe

(p-1)

1 - (1 + ^ e-(x-a)/e

e-(x-a)/p6-e-(x-a)/6

(P-I)e (x-a) e-(x-a)/e e2 e

P

1

P = 1 P = 1

(1)

(2)

where, a < x < ro, and © = {(a,e,p) : a > 0,e > 0,p > 0} which is commonly known as the parameter space with intervention parameter p and rate parameter e. The wider applicability of the model motivated us to extend the intervened exponential distribution (IvED) further in the direction of the weighted distributions and the distribution obtained is known as weighted intervened exponential distribution (WIvED). Thus, the newly proposed WIvED is a type of generalization of IvED. To trace the history of weighted distributions, Fisher [7] investigated the influence of ascertainment procedures on frequency estimates and developed the notion of weighted distributions. When diversifying Fisher's core theories, Rao ([18], [19]) realized the need for a unifying notion by attempting to identify numerous sampling conditions that might be handled according to what he named weighted distributions. To know the history and applicability of weighted distributions Patil published several articles that he wrote with co-authors refer Taillie et al. [23], Patil and Taillie [12], Patil and Rao [15], Denis and Patil [5], Laird et al. [10], Patil and Ord [14], etc. moreover, see more references Patil [16]. In a definition, Patil et al. [13] proposed the methodology used to determine the weighted version probability density function (p.d.f) is depicted below:

f (x; ©)

w(x) ft (x; ©)

E [w(X)]

where, f (x; ©) is the natural density function of existing distribution with parameter space © and w(x) is the non-negative weighted function by choosing w(X) = Xr, then E [w(X)] = E [Xr] is the rth moment about origin. The rth raw moment of IvED derived by Bhat and Pundir [2] is presented in a simplified form as given by

U'r

1 ^ (r \Jker-k+1 (pr-k+1 , ae (p -1

(3)

Therefore, according to the definition of weighted distributions, the c.d.f and p.d.f of WIvED(a, e, p) derived from a r.v X ~ IvED(a, e, p) are given by

F

WIvED

(x;©)

1

er+l{pr+l ea/pe r(r+1,x/pe)-ea/e r(r+1,x/e)}

1 er+1 {er(r+2,x/e)-ar(r+1,x/e)} ea/e

1 Ek=0 (k)aker-k+2r(r—k+2) e

p = 1

(4)

and,

-(x-a)/pe_ xre-(x-a)/e

fWIvED (x; ©)

u

=0 (k)aker-k+1(pr

xr (x-a)

uk=0 (k)aker-k+2r(r-k+2)

k+1 -1)r(r-k+1) e-(x-a)/e

p = 1 p = 1

(5)

where, a < x < ro, and © is parameter space. The reliability function of the WIvED(a, e, p) is obtained as

RwIvED (x; ©)

er+1 {pr+1ea/pe r(r+1,x/pe)-ea/e r(r+1,x/e)}

er+1{er(r+2,x/e)-ar(r+1,x/e)} a/e Ek=0 (k)aker-k+2r(r-k+2) e

p=1

(6)

1

p

x e

1

p

and, its hazard rate function is

lWIvED

xre-(x-a)/p6 _xre-(x-a)/e

( _ J 0r+l {pr+lea/P0r(r+1,x/pe)-ea/er(r+1,x/0)}

( ' ) ' _xr(x-a)_ x/0

er+1 {er(r+2,x/e)-a-r{r+i,x/e)}

p _ 1 p _ 1

(7)

further, we study the special case density function of WIvED obtained at r _ 1 from equation (5), that is, defined as Length(size)-biased intervened exponential distribution (LBIvED). The mean, variance several statistical and reliability properties, parameter estimation, simulation, stochastic ordering, order statistics, and real-life applicability of LBIvED are studied in proceeding sections and subsections.

2. Distribution and its Properties The cumulative and probability density functions (c.d.f & p.d.f) of LBIvED are given by

F

LBIvED

(x; 0)

1 {p(x+pd)e-(x-a)/P0-(x+9)e-(x-a)/e} _ 1

1 (p-1)[a+e(p+1)] p _ 1

i e2+ (x+e)(x-a+e) (x-a)/9 p _ i

1 0(a+20) e p _ 1

(8)

and,

-(x-a)/p6_

-(x-a)/e

f LBIvED (x; 0)

e(o-1)[a+e(p+1)]

x(x-a) e-(x-a)/e e2 (a+2e) e

p _ 1 p _ 1

(9)

where, 0 < a < x < to and © being its parameter space. The attempt is made to derive the expressions of mean x) and its variance (a?). Also, some of the measures of LBIvED(a, d,p = 1) which are not in closed form are the quantile function, median and the mode. Note, for simplicity here on-wards, we take t = (p — 1) [a + d (p + 1)].

Ux

2e2 (p3 - 1) + 2ae (p2 - 1) + a2 (p - 1)

(10)

and

a2 = 203 (p4 — ir + 2ad2 (p3 — ir + (3a + 2) ad (p2 — ir + a3 (p — 1) )2

ax =-t--) . (11)

In Figure 1 and Figure 2, the p.d.f of LBIvED(a, d, p = 1) is plotted graphically. This is evident from the graphical representation that the distribution is positively skewed, where multiple p.d.f curves are displayed for various chosen sets of parameter values.

0.05

20

{a, e, p) = (0.011, 4.220, 2.01 (<7, e, p) = (0.020, 4.855, 1.308) (a, e, p) = (0.030, 6.055, 2.041) (a, e, p) = (0.045, 8.450, 2.116)

40 60 80 100

x

Figure 1: P.d.f Plot

0

Figure 2: P.d.f Subplots

2.1. Moments

In this subsection, the derivation of moments particularly central and non-central are derived. So, if a random variable X ~ LBIvED(a,6,p = 1), then central moments (^r) expression is obtained

by

I {• ro f -

Vr = eTj (x - Vx)rx{e-(x-a)/pe - e-(x-a)/e} dx

to solve the integral, first, we expand the term (x - vx)r by using binomial expansion. After, integration and simplification the resulting equation is

Vr = 1 E (k) (a - Vx)r er-kr (r - k + 1) {e (p'-k+2 - 1) (r - k + 1) - a (pr-k+ - 1)} . (12)

In a similar context, we derive the non-central moments (v'r) expression for a random variable X ~ LBIvED(a,e,p = 1). The procedure to obtain the final resulting expression is given by

1 cM " la

after, simplifications the resulting equation is

V = 6TJ xr+1{e-(x-a)/P6 - e-(x-a)/6J dx

vi = 1 E (r + ^ r (r - k + 2) 6r-k+1 (pr-k+2 - l) (13)

T k=o\ k

2.2. Generating Functions for Moments

In general, the moment generating function (m.g.f ) denoted by MX(t) of a r.v X is obtained by,

Mx(t) = E (VX) = J^ etxf (x)dx (14)

so, to derive the m.g.f of a r.v X ~ LBIvED(a, 6,p = 1), we have

TO

Mx (t)= etxfLBivED (x; ©)dx

Ja

Vilayat Ali Bhat, Sudesh Pundir RT&A, No 2 (73) WEIGHTED INTERVENED DISTRIBUTION_Volume 18, June 2023

on substituting the p.d.f of LBIvED, when p = 1 and proceed with the transforming technique, we get

= 6T iT (x + ^ { 6p — e—( 1 —t)^dx

on solving this further by using the gamma function, the required resulting equation of m.g.f is obtained as

MX (t) = {a (1 — A t)(12 — 6p 0 + 6(1 + p — Wpt)} eat

(1 — et)2 (1 — dpt)2 [a + 6 (p + 1)] Next, the characteristic function (c.f) of a r.v X ~ LBIvED(a, 6,p = 1) is derived by

TO

(t) = E(eitX) = eltxfLBivED (x; ©)dx Ja

= {a (1 — 6it) (1 — 6 pit) + 6 (1 + p — 26pit)} eait = (1 — 6it)2 (1 — 6pit)2 [a + 6 (p + 1)]

3. Discussion on Reliability Properties

Suppose a random variable X is such that it is non-negative, X ~ LBIvED( a, 6, p), the p.d.f of X is fLBIvED (x; ©), and its c.d.f FLBIvED (x; ©) which are given in equation (9) and (8) respectively. Then the reliability function of LBIvED(a, 6, p) is given by

{p(x+p6)e—(x—a)/p° — (x+e)e—(x—a)/e = 1

62+(x+6)(x—a+l) e—(x—a)/6 = = i (17)

6(a+26) e p 1

the essential failure rate functions of a random variable X ~ LBIvED(a, 6,p = 1) are obtained. The hazard rate and reverse hazard rate functions are given by

_xe—(x—a)/p6—xe—(x—a)/6_ p = i

hLBVED (x; Z) H ^^Hi™'*—(^^} p = (18)

e{e2+(x+e)(x-a+e)}

P = 1

in Figure 3 and Figure 4, the hazard rate function of LBIvED(a,6,p = 1) is plotted graphically. It is shown in the graphical representation the distribution is having increasing multiple hazard rate function curves that are displayed for various chosen sets of parameter values.

x

Figure 3: Hazard Rate Function Plot

0.4

-(а, в, p) = (0.01,2.52,1.02) -(а, в, p) = (0.01,3.72,1.02) " -(а, в, p) = (0.01,5.58,1.02) (а, в, p) = (0.01,6.83,1.02)

0.2

0.15

0.1

0.05

-(а, в, p) = (0.01,1.12,5.02) -(а, в, p) = (0.01,1.12,7.02)

-(а, в, p) = (0.01,1.12,8.02) (а, в, p) = (0.01,1.12,11.2)

20 40 60 80 100 0

x

20 40 60

x

80

100

Figure 4: Hazard Rate Function Subplots

3.1. Mean Residual Life Function

The Mean Residual Life (MRL) function is used to investigate the aging process phenomenon of systems. To determine the MRL for a random variable X representing the component's life, is defined by

mLBIvED(x; О) = E [X - x|X > x] =

R

LBIvED

(x; О)

RLBIvED (x; О)dx.

(19)

The function m(x; ©) is defined over any domain of a random variable, it is of special significance for a non-negative random variable explaining the lifetime of the system, and at time x for an operational system, conditional expected residual life is determined by Finkelstein [6]. Furthermore, while the shape of the hazard rate function is essential, the MRL function is considered to be much more meaningful than the hazard rate function. Since the first describes the full residual life function, whilst the other simply evaluates the possibility of immediate failure at the time x. The MRL function of the LBIvED(a, e,p = 1) is determined by using the relation (19) and is obtained as follows:

mLBIvED (x; О)

1

Г p (x + pв) e-^-^^ - (x + в) e-(x-а)/в

RlbIvED(x; О) Jx

dx.

On solving the above integral and further simplifications gives the following required MRL function of the model.

mLBIvED (x; О)

вp2 (x + 2pв) e-(x-&)/pi - в (x + 2в) e-(x-а)/в

t R

LBIvED

[x; О)

(20)

3.2. Stress-Strength Reliability

It is easily noticed, that a significant amount of research work had already been carried out in the field of stress-strength modeling on the estimation of reliability R = Pr. (X1 > X2), where the random variables X1 and X2 represent the strength and stress factors of the system possessing the same distributions of uni-variate family. For the vast majority of the well-known standard distributions, the algebraic representation for R has been developed. Here, X1 ~ LBIvED(a,e2,p1 = 1) and X2 ~ LBIvED(a,e^,pi = 1) are unrelated and have the same distribution, so-called LBIvED,

0

0

1

T

we calculate the reliability R. The procedure to measure R by having the c.d.f of X2 and p.d.f of Xi is described as follows

F (r.Pi) = 1 P2 (x + P262) e-(x-a)/p262 - (x + 62) e-(x-a)/62 p = 1 FLBlvED (x, ®) = 1--(p2 - 1)[a + 62 (P2 + 1)]- P = 1 (21)

and,

xe-(r-a)/p161 _ xe-(r-a)/61

fLBI"ED (x; 0) = 61 (P1 - 1)[a - 61 (P1 + 1)] p = 1 (22)

then, R is derived by

/to f /• x rM

j/ fx2 (x)dxjfx1 (x)dx = ^ FX2 (x)fx1 (x)dx (23)

= 1 6162 p1p2 fa(a + 6p ) + 6162 P1P2 (2a + 62P2) + 2 (6162P1P2)2 1

B (61P1 + 62P2 )\ ^(a + 62P2)+ (61P1 + 62P2) + (61P1 + 62P2 )2J

6162P1 \ , , a , , 6162P2 (2a + 62) , 2 (6162P2)

2

+ B (6m + 62) \^(a + 62)+ (61P1 + 62) + (61P1 + 62)2/

+ 6162P2 /,+ 6p ) + 6162P2 (2a + 62P2) + 2 (6162P2 )2 ' + bW+^Y (a + 62 P2)+ (61 + 62 p2) +(61 + 62 p^ .

6162 \ , 6162 (2a + 62) 2 (6162)

2

a (a + 62) + -V , „ , 2 +

B (61 + 62 )\K 2 (61 + 62) (61 + 62)2 J where, B = 61 p - 1) (p2 - 1) [a + 61 p + 1)] [a + 62 (p2 + 1)].

4. Entropy Measures

In science and engineering, entropy measures have already been studied as a practical application of the system. The two uncertainty measures of variation so-called Renyi entropy by Renyi [17] and Tsallis entropy by Tsallis [24] are studied in this section, the expressions of both these measures are given in the following respective subsections:

4.1. Renyi Entropy

The Renyi entropy of order a, for a non-negative r.v X ~ LBIvED( a, 6, p = 1) is derived by

1

hR(a) = ^ra log

{fLBivED (x; ©)}a dx

a > 0, a = 1

on substituting the p.d.f of LBIvED when p = 1 and making use of binomial expansion by treating a as a finite nature number. The simplified equation of Renyi entropy of order a is derived as given by

h r (a) = —1— log {£ £ C )ft)(-1)r ^ W-" r(a-k + 1) 1 (24)

R 1 -« gl£¿0 {St}' [«+ r (p - 1)]-M j ( )

4.2. Tsallis Entropy

The Tsallis entropy also known as q-entropy of order q, for a non-negative r.v X ~ LBIvED( a, 6, p = 1) is derived by

Tq (x)

1-q

1 -/ {fLBivED (x; ®)}qdx

la

; q > 0, q = 1

M

1

on substituting the p.d.f of LBIvED when p = 1 and making use of binomial expansion by treating a as a finite nature number. The simplified equation of Tsallis entropy of order q is derived as given by

1

Tq (x) 1

1 - q

(q)(q)(-1)rak (ep)q-k+1 r (q - k+1)

1 - EE

(25)

r=0kfo W [q + r (p - 1)]q-k+1 5. Stochastic Ordering and Order Statistics

In this section, the stochastic ordering of the model and its order statistics are discussed.

5.1. Stochastic Ordering

Over the past several years, the usage of stochastic ordering has drastically increased across a wide range of statistical fields. These disciplines include reliability theory, queuing theory, survival analysis, and many more fields refer to Shaked and Shantikumar [20]. Let the r.vis X1 and X2, where X1 - LBIvED(a1, e1, p1 = 1) and X2 - LBIvED(a2, e2, p2 = 1), have p.d.f is denoted by fx1 (x), fXl (x) and c.d.fis denoted by FX1 (x), FX2 (x) respectively. According to the model, it is believed that the random variable X1 is smaller than X2 in the,

(a1) Stochastic order (mathematically X1 <st X2), if FX1 (x; ©1) > FX2 (x; ©2)Vx.

(a2) Hazard rate order (mathematically X1 <hr X2), if hX1 (x; ©1) > hXl (x; ©2)Vx.

(a3) Mean residual life order (mathematically X1 <MRL X2), if mX1 (x; ©1) > mX2 (x; ©2)Vx.

(«4) Likelihood ratio order (mathematically X1 <LR X2), if f?1 (x;©1) decreases in x. Since the following results show that, the four stochastic orders stated above are connected, X1 <MRL X2 ^ X1 <hr X2 ^ X1 <LR X2 and X1 <st X2 ^ X1 <hr X2.

When the required conditions are met, the LBIvED(a, e, p = 1) models are arranged w.r.t the strongest likelihood ratio ordering, stated in the following theorem.

Theorem 1. Suppose X1 — LBIvED(a1,e1,p1), and X2 — LBIvED(a2,e2,p2), with the condition, that if a1 = a2 = a, (p1 >p2) > 1, and (e1 >e2) then (X1 <st X2), (X1 <hr X2), (X1 <1r X2), and (X1 <mrl X2).

Proof. Proof of the specified ratio is sufficient to demonstrate the outcome.

fx, (x; ©1) = ftp - 1)[a2 + e2 p + 1)] {e-(x-a1)/p1 e1 - e-(x-a1 )/e1} fx2 (x; ©2) e1(p1 - 1)[a1 + e1 (p1 + 1)] {e^x-^^e2 - e-(x-a2)/e2 } .

Now, applying log both sides and differentiate w.r.t x.

A

dx

i fx1 (x;©1)| = emu(V1 -p2V2) - epv(u -p^) fx2 (x; ©2) J e1 e^ p2Uv

where, U = Ie-(x-a1)/p1 e1 - e-(x-a1)/e1 j, V = Ie-(x-a2)/p2e2 - e-(x-a2)/e^, u1 = e-(x-a1)/p1 e1, U2 = e-(x-a1)/e1, V1 = e-(x-a2)/p2e2, and V2 = e-(x-a)/e2. Hence, if a1 = a2 = a, (p1 >p2), and (e1>e2) then dx log { J10J } < 0, which implies that (X1 <st X2), (X1 <hr X^, (X1 </r X^, and (X1 <mrl X2). ■

5.2. Order Statistics

In order to discuss the order statistics of n random observations x = { x1, x2, x3, ..., xn } drawn from LBIvED(a,e,p = 1). The random sample is arranged in ascending order such as x1:n < x2:n <,..., < xn:n. Then, we denote the p.d.f of random variable which is of Ith order by fi:n (x,, ©); i = 1,2,..., n, and joint p.d.f of (i, j)th order variable pair by fi:j:n(x,, xj); 1 < i < j < n, whose expressions are given by

fi:n(x,; ©) = m [FLBIvED(x(i); ©)]i-1 [1 - Flexed(xw; ©)]n-fLEIvED(xw; ©) (26)

and,

fi:j:n(xi,xj) = rn[FLEIvED(x(,); ©)]i-1 [FleIvEd(x(j); ©) - FLEIvED(x(i);©)]J-i-1

[1 - FLEIvED (xj); ©)]n-jfLEIvED (x(i); ©) fLEIvED (xj); ©) (27) where F(.) and f (.) are c.d.f and p.d.f. The constants n1 and n2 are given by

__n' j __n'

K1 = (i-1)!(n-i)! and n = (i-1)'(j-i-1)'(n-j)'

on substituting i = 1 and i = n in equation (26) the p.d. fis of 1st and nth order statistics obtained are given as follows:

f1:n(x(1); ©) = n 1 - FLEIvED (x(1); ©) fLEIvED (x(1); ©)

n1

p0(1)e

-(x(1)-a)/pe _

f(1)e

(x(1)-a)/e

n1

x(1)< e (x(1)-a)/pe - e-(x(1)-a)/el

e(p - 1)n [a + e (p + 1)]n

and,

fn:n (x(n); ©) = n FLEIvED (x(n); ©) fLEIvED (x(n); ©)

n-1

(28)

t —

{p$(n)e-(x(n)-a)/pe - fwe-(x(n)-a)/e}j

n1

x(n) e

-(x(n)-a)/pe _ e-(x(n)-a)/e>

e(p - 1)n [a + e (p + 1)]n

Similarly, the joint order statistic density function of LBIvED(a, e, p = 1) is given as

fi:j:n (x(i), x(j)) e2 (p - 1)n [a+e (p + 1)]n

(p - 1) [a + e (p + 1)] - {p*(i)e-(x(i)-a)/pe - f{l)e-(x(i)-a)/e}

i-1

(29)

i)e-(x(i)-a)/pe - f(i)e-(x(i)-a)/e} -

p$(j)

-(x(j)-a)/pe - f e-(x(j)-a)/e

n- j

x(j) { e-(x(j)-a)/pe - e-(x(j)-a)/e

e-(x(j)-a)/pe - f(j)e-(xa)-a)/e}

x(i) j e ~(x(i)-a)/pe - e-(x(i)-a)/e}j

j-i-1

where, ^(r) = ^x(r) + p^ and f(r) = ^x(r) + ej; r = 1, n, i, j

6. Parameter Estimation and Simulation

6.1. Estimation Procedure of Model Parameters

Let a random sample x1, x2, ..., xn, consisting of n observations is drawn from LBIvED( a, e, p = 1). The most frequently used technique called the method of maximum likelihood estimation

n

n

approach is chosen for the estimation of the parameters. So, we write the log-likelihood function of the complete data for LBIvED(a, 6, p = 1) as

log L = £ log <1 Xie

i=1

e-(xi-a)/p9 - xie-(xi-a)/9j - n log {9(p - 1) [a + 9 (p + l)]} . (30) Setting Q = e-(Xi-a)/p9 and f = e-(x-a)/9, then (30) becomes

log L = £ log {Q - f} + £ logXi - n log9 - n log(p - 1) - n log [a + 9 (p + 1)]. (31)

On differentiating equation (31) with respect to (w.r.t) a, 6, and p, the normal equations obtained are

d log L da

d log L d9 =

d log L

E

i=1

n

= E

= y Q - pf h P9 (Q - f)

(xi - a) (Q - pf) p92 (Q - f)

Q (xi - a)

n

[a + 9 (p + 1)] n (p +1)

[a + 9 (p + 1)]

n9

1 p92 (Q - f) [a + 9 (p + 1)] p - 1-

(32)

(33)

(34)

To get the maximum likelihood estimates (MLEs), we equate equations (32), (33), and (34) to zero. But the equations obtained are not in closed form. So, the recommended functions such as nlm or optim are used to maximize the log-likelihood function in the R programming language. Also, the alternative iterative technique known as the Newton-Raphson method could be employed to yield the solution for the parameters. Further to construct the Fisher information matrix there must exist the second-order partial differentials which do exists as can be proved by the continuity of first-order differentials. Also, let us suppose that the MLEs of © are given by © = { ( a, 6, p ) : a > 0, 6 > 0, p > 0 } then Fisher information matrix is defined as

I (©) = -E

a2 log l d2 log L a2 log L '

da2 dad9 dadp

a2 log l d2 log L d2 log L

d9da d92 d9dp

a2 log L d2 log L d2 log L

dpda dpd9 dp2 J

(35)

the partial derivatives of Fisher information matrix I (©) are given by

d2 log L_ n (Q - pf)2 - (Q - f) (Q - pf) + £ .,.\2 +

da2

i=1

p292 (Q - f)

[a + 9 (p + 1)]2

d2 log L A (xi - a) (Q - pf) {(1 + 3p) f - 2 (1 + p) Q} + n (p + 1)2 + n_ =1 p293 (Q - f)2 [a + 9 (p + 1)]2 92

d92

d2 log L dp2

£ Q (xi - a) {(a + 2p9) Q - 2p9f} +

n92

i=1

p492 (Q - f)2

+

[a + 9 (p + 1)]2 (p - 1)2 a2 log L_ A p9 (Q2 + pf2) - (a + p9 - 2ap + ap2 + p29) Qf n (p + 1)

k p2 93 (q - f)2 +v+np+1)\2

n9

d9da

d2 log L = " p9 (Q2 + pf2) - (a - ap + p9 + p29) Qf _

dpda = i=1 p3 9 (Q - f)2 + '[T+Hp+Ï^

a2 log L _ n (xi - a) {(a - ap + p2 9 ) Qf - p9 (p + 1) f2 - p9Q2}

dpd9

E

i=1

a

p393 (Q - f)2

[a + 9 (p + l)]2

(36)

(37)

(38)

(39)

(40)

i=1 i=1

n

n

n

We drop the expectation term as it is difficult to compute for the second-order partial differential elements of the Fisher information matrix see Cohen [4]. Therefore we write I (©) as

I (©) = -

" a2 log l d2 log L d2 log L

da2 dad0 dadp

d2 log L d2 log L d2 log L

d0da d02 d0dp

d2 log L d2 log L d2 log L

dpda dpd0 dp2

(42)

(a,9,p)=(â,ê,p)

now to construct the confidence intervals (C.I/s) for parameters of LBIvED( a, Q, p = 1 ), we first find inversion matrix I(0)—1 consisting of the diagonal elements as variances whereas the off-diagonal elements represent the co-variances. The 100(1 — Z)% C.I/s for a, Q, and p are

a± Xz/2 Wo), Q ±Xz/2\JV(Q), and p ±Xz/2^V(p) respectively, where the term xz/2 is (Z/2)th upper percentile of a normal variate.

6.2. Simulation Study

It is necessary for a statistical distribution, to explore the behavior of estimated parameters by performing a Monte-Carlo simulation study at various randomly selected sample sizes, as n = { 30, 90, 150, 210, 350, 500 }. Since the quantile function of LBIvED does not exist in closed form. So, to simulate the data, an alternative technique is utilized known as the acceptance-rejection algorithm. In Table 1, the average bias (Bias = n En=1(0 — 0)) and average mean square error (MSE = n rn= 1 (© — 0)2) of the estimated parameters are cakuhted. The smuka^ous decrease of Bias and MSE of parameters a, Q, and p are reported as the sample size increases. It is clear, that the consistency property of estimated parameters holds for LBIvED.

Table 1: Parameter Bias and MSE of LBIvED(a, Q, p = 1)

K 0, p)

Bias

MSE

a

a

(0.91,0.82,0.78)

(1.01,1.87,0.83)

(1.02,1.59,1.05)

30 90 150 210 350 500

30 90 150 210 350 500

30 90 150 210 350 500

0.16580 0.06044 0.04272 0.03215 0.01838 0.01439

0.39676 0.16685 0.11098 0.08229 0.05865 0.04099

0.37336 0.15147 0.09869 0.07163 0.03737 0.03229

0.37902 0.27480 0.25322 0.23397 0.20611 0.20147

0.73710 0.58280 0.48364 0.43797 0.39767 0.36097

0.53592 0.32432 0.25286 0.22203 0.15245 0.11726

1.7e+01 2.65334 1.11791 0.97621 0.67691 0.63070

1.2e+01 2.36629 1.03614 0.69849 0.55490 0.46926

1.2e+01 1.97947 0.69786 0.45958 0.27079 0.19951

0.027489 0.003654 0.001825 0.001034 0.000338 0.000207

0.14366 0.07551 0.06412 0.05474 0.04248 0.04059

3.0e+02 7.04022 1.24971 0.95299 0.45821 0.39778

0.157419 0.54332 1.4e+02 0.027840 0.33966 5.59931 0.012317 0.23391 1.07359 0.006772 0.19182 0.48789 0.003440 0.15814 0.30791 0.001681 0.13030 0.22020

0.139400 0.28721 1.6e+02 0.022842 0.10519 3.91829 0.009740 0.06394 0.48701 0.005131 0.04930 0.21121 0.001397 0.02324 0.07333 0.001042 0.01375 0.03980

n

7. Model Applicability

7.1. Real-Life Data Based Applications

In this section, an illustration of the proposed methodology is given. Real-life examples are presented to demonstrate the superior performance of the proposed LBIvED model. The real-life data sets that are analyzed in this study are given below:

The first data set of T8 fluorescent lamps analyzed by Ahmed [1] represents the lifetime (Hours) for 50 devices given by: 0.445, 0.493, 0.285, 0.564, 0.760, 0.381, 0.690, 0.579, 0.636, 0.238, 0.149, 0.244, 0.126, 0.796, 0.405, 0.553, 0.780, 0.431, 0.184, 0.375, 0.198, 0.890, 0.192, 0.463, 0.486, 0.521, 0.366, 0.486, 0.116, 0.511, 0.612, 0.117, 0.384, 0.326, 0.057, 0.412, 0.586, 0.517, 0.570, 0.588, 0.497, 0.246, 0.234, 0.228, 0.552, 0.893, 0.403, 0.458, 0.134, 0.338.

The second data set by Bader and Priest [3] consists of 63 samples representing the strength measured in GPA for single carbon fibers with gauge lengths of 10mm given by: 1.901, 2.132, 2.203, 2.228, 2.257, 2.350, 2.361, 2.396, 2.397, 2.445, 2.454, 2.474, 2.518, 2.522, 2.525, 2.532, 2.575, 2.614, 2.616, 2.618, 2.624, 2.659, 2.675, 2.738, 2.740, 2.856, 2.917, 2.928, 2.937, 2.937, 2.977, 2.996, 3.030, 3.125, 3.139, 3.145, 3.220, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.562, 3.628, 3.852, 3.871, 3.886, 3.971, 4.024, 4.027, 4.225, 4.395, 5.020.

The output results of the analyzed data sets include the estimated parameters, the information measures such as Akaike information criteria { AIC = —2 log L(x; ©) }, Bayesian information criteria { BIC = —2 log L(x; ©) + k log L(x; ©) }, Hannan Quen information criteria { HQIC = —2 log L(x; ©) + 2k log [log(n)] }, where constants n and k represent the sample size and the number of parameters respectively. The goodness-of-fit statistics tests are the Anderson Darling test, Cramer Von Mises test, and the Kolmogorov Smirnov test statistic with a p-value. The existing distributions that are compared with LBIvED are IvED, exponential distribution (ED), weighted exponential distribution (WED), and length-biased exponential distribution (LBED). The p.d.f fs are given as,

1

ED = f (x; ©) = 1 e—x/a

a

WED = f (x; 0) LBED = f (x; 0) =

= (a+ 1) e„-ex a

[e(a +1)]

a (a + 2)

ee-2

1 - e

-aex

xe

ex

1e

-aex

(43)

(44)

(45)

Table 2: Various measures from the first data set

Models Information Measures

a e p log L AIC BIC HQIC

LBIvED 0.01975 0.13979 1.00007 7.60999 -9.21998 -3.48391 -7.03565

IvED 0.04305 0.19339 1.00033 6.10016 -6.20032 -0.46426 -4.01600

LBED 1.749e+4 4.65200 - 4.11157 -3.96781 -0.39908 -2.76691

WED 1.278e-04 4.65203 - 4.11157 -4.22313 -0.39908 -2.76691

ED 0.42990 - - -7.78987 17.5797 19.4918 18.3078

Models Goodness-of-fit Tests

a e P CVM AD KS p-value

LBIvED 0.01975 0.13979 1.00007 0.14036 0.75075 0.12723 0.3932

IvED 0.04305 0.19339 1.00033 0.17074 0.91949 0.15728 0.1685

LBED 1.749e+4 4.65200 - 0.13315 0.71331 0.16757 0.1206

WED 1.278e-04 4.65203 - 0.13315 0.71330 0.16761 0.1205

ED 0.42990 - - 0.13395 0.71750 0.23317 0.0087

Table 3: Various measures from the second data set

Models Information Measures

â 9 P -log L AIC BIC HQIC

LBIvED 1.87153 0.50522 1.00011 59.884 125.768 132.197 128.297

IvED 1.87408 0.59253 1.00026 60.663 127.326 133.756 129.855

LBED 1.58e-05 0.9806086 - 97.954 199.908 204.194 201.594

WED 1.717e-5 6.537e-01 - 110.35 224.696 228.983 226.382

ED 3.05930 - - 133.45 268.892 271.035 269.734

Models Goodness-of-fit Tests

â 9 P CVM AD KS p-value

LBIvED 1.87153 0.50522 1.00011 0.06424 0.42637 0.10492 0.4919

IvED 1.87408 0.59253 1.00026 0.06944 0.46604 0.11296 0.3975

LBED 1.58e-05 0.9806086 - 0.05891 0.36294 0.33488 1.4e-06

WED 1.717e-5 6.537e-01 - 0.05890 0.36281 0.39019 9.3e-09

ED 3.05930 - - 0.05885 0.36229 0.48600 2.3e-13

From Table 2 and Table 3, it is visible that all the existing distributions lack superiority against the new LBIvED, the information measures and the goodness-of-fit test results displayed are relatively lower for LBIvED with a higher significant p-value. Hence, the LBIvED is treated as the best-fitted model.

Conclusion

In this article, the WIvED, a generalization of the IvED that may be used to describe time-to-event data sets as a lifetime distribution is presented and examined. The length biased variant of the IvED is derived as a special instance of the WIvED and its model applicability features are thoroughly examined. We find that the LBIvED may be used as a flexible model in place of the traditional lifetime models taken in comparison that are commonly used in the literature to describe real-life time-to-event data. As evidenced by its excellent distributional, reliability, and survival qualities, we anticipate that the proposed WIvED and the length-biased variant of IvED will function as a competitive model for representing data from reliability analysis, survival analysis, and other domains of the statistics.

Acknowledgement(s)

The first author is very grateful to Pondicherry University for providing him with a research fellowship to carry out this work.

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