CC BY
32
Power Length biased weighted lomax distribution
SHAMSHAD UR RASOOL •
Department of Statistics, University of Kashmir, Srinagar, India [email protected]
S.P.AHMAD* •
Department of Statistics, University of Kashmir, Srinagar, India [email protected]
Abstract
In this research paper, we have proposed the Power Length Biased Weighted Lomax Distribu-tion(PLBWLD) as a new probability model . Moments, moment generating function, characteristic function, cumulant generating function, and reliability analysis such as survival function, hazard rate, reverse hazard rate, cumulative hazard function, and mills ratio are among the statistical features of PLBWLD that have been obtained here. Order statistics and PLBWLD's generalized entropy are also calculated. Maximum likelihood estimation is used to estimate the parameters of the model. Finally for demonstration purposes an application to the real data sets is provided to understand the new probability model's performance and flexibility.
Keywords: Length biased weighted Lomax distribution, power length biased weighted Lomax distribution, hazard rate function, moments, maximum likelihood estimation, order statistics, generalized entropy.
1. Introduction
Pareto distribution of second type is another name for the Lomax distribution. Lomax distribution was first used to model the failure rate of businesses by Lomax [8]. In the literature, the Lomax distribution has been employed in a variety of ways. According to Balkema and de Haan[3], it has been extensively utilized for life testing and reliability modeling including insurance, actuarial, demographics, economics, medical sciences, finance and engineering. The number of novel models with a high degree of flexibility is growing year after year. As a result, the researchers have shifted their focus to create new families of distributions and propose a variety of new families of distributions in order to better examine and investigate real-world data in various applications. Statistical distributions have gained a lot of attention recently as researchers try to figure out how to create flexible models for modelling a variety of data sets. It is because the classical distributions aren't very good at modelling data sets with a lot of variation. As a result, generalised probability models continue to grow and expand. In recent years, designing a new probability model from previously established models using various methodologies has gained a lot of attention. The power transformation technique, in which an extra parameter is added to the parent distribution, is one such strategy employed by several researchers. The addition of an extra parameter to the parent model usually increases the goodness of fit and gives more flexibility. Krishnarani [6] , Zaka and Akhter [9] are few of the researchers who have worked on power generalization of probability models. The concept of weighted distributions was first developed by Fisher [5].
If X is a non-negative random variable with the probability density function f (x), then the probability density function of the weighted random variable Xw is given by
, , w(x)f(x) fw(x) = E(w(x)) ; x - 0
When w(x) = x, the resultant distribution is clearly length biased, with a probability density function given as;
fL (x) = f); x -0
If the random variable X has the length biased weighted Lomax distribution with shape parameter n and scale parameter A respectively, then it's probability density function(pdf) and cumulative distribution function (cdf) proposed by Ahmad et al. [1] , are respectively given as
f (x; n, A) = x (1 + A) ^^ ; x > 0, n > 1, A > 0
F(x;n, A) = 1 - (1 + A) n (1 + A) ; x > 0,n > 1, A > 0 2. POWER LENGTH BIASED WEIGHTED LOMAX DISTRIBUTION(PLBWLD)
The primary goal of this research paper is to improve the flexibility of the length biased weighted Lomax distribution by developing an expanded version of the model using power transformation
technique. Suppose the random variable X assumes the length biased weighted Lomax distribu-
1
tion with parameters n and A, then the transformed variable V = X p will follow power length biased weighted Lomax distribution with parameters n, P and A.
The probability density function of the power length biased weighted Lomax distribution is
obtained as;
f (v;n,P, A) = n(n~1)Pv2p-1 (1 + VP) ^ ) ; x > 0,n > 1,A,p > 0 (1)
The cumulative distribution function of the power length biased weighted Lomax distribution is obtained as
-n
1 in ti \
(2)
Fiv-n.^A) = 1 - (1 + A)" (1 +
AA
For the visual illustration of the possible shapes of pdf and cdf of PLBWLD, Figure 1 and Figure 2 have been plotted. Plots of the survival function and hazard rate function of the PLBWLD distribution for different parameter values are also displayed in Figure 3 Remark: For p = 1 in 1, we obtain the length biased weighted Lomax distrbution.
3. RELIABILITY ANALYSIS OF THE POWER LENGTH BIASED WEIGHTED
LOMAX DISTRIBUTION(PLBWLD)
This section focuses on obtaining the reliability (survival function), hazard rate (failure rate), reverse hazard function, cumulative hazard function and mills ratio expressions respectively for PLBWLD.
3.1. Survival function
The survival function or reliability function is the complement of the cumulative distribution function and it is defined as the probability that a system will survive beyond a specified time
v
v
Figure 1: Pdf Plots of the PLBWLD density for various values of n , ft and X.
v
v
Figure 2: Distribution function Plots of the PLBWLD for various values of n , ft and X.
and is obtained for the PLBWLD as
R(v; n,ft, X) = 1 - F(v; n, ft, X) = (l + X) ' (l + (3)
3.2. Hazard Rate
Hazard rate also known as hazard function , force of mortality or failure rate. The Hazard rate assess the ability of a lifetime component to fail or to expire depending on the life completed and thus has wide variety of applications in lifetime distributions. Using (1) and (3), the expression for the hazard rate of PLBWLD is obtained as
h(v;n,ft,x) = M = '^ft-1 (i + if (i + if «
v v
Figure 3: Survival function and Hazard Rate Plots of the PLBWLD for various values oft], / and A.
3.3. Reverse Hazard function
The concept of reversed hazard rate of a random life is defined as the ratio between the life probability density to its distribution function . It is expressed as
hr (v;n, a) =
F(v;j/, A)
Using equation (1) and (2), the reverse hazard function for the Power length biased weighted Lomax distribution is obtained as
n(n-1)3v2/-i (1 + A \-(n+1) hr(v;n,3,A) = ^-a/A) / x (5)
' 3 v " ' vPfJ
1 - il + î)il +
3.4. Cumulative Hazard function
The cumulative hazard function can be thought of as providing the total accumulated risk of experiencing the event of interest that has been gained by progressing to time t. The cumulative hazard function for the PLBWLD is defined as
aplbwldj8, A) = - logR(v;q,fi, A)
i (1 + AI'!
APLBWLD (- ^ ft A) = log < )--¿r > (6)
I i1 + ^J
3.5. Mills Ratio
The mills ratio for the power length biased weighted Lomax distribution is defined as
M.R = = _-J , „ x - 1 (7)
R(-;n,ft A) ^ + A^j-n ^ + -Al^
4. RESIDUAL AND REVERSED RESIDUAL LIFE FUNCTIONS OF THE POWER LENGTH BIASED WEIGHTED LOMAX DISTRIBUTION(PLBWLD)
4.1. Residual life function
In life testing situations, the additional lifetime given that a component has survived until time t is called residual life function. More specifically, if v is the life of a component, then the random variable r(t) = (v — t|v > t); t > 0 is used to explain the residual life of a lifetime component. For the PLBWLD, the survival function of the residual life time r(t), t > 0 is defined as
Rr{t)(v, n, ß,X) = (l +
Rr{t)(v; n, ß X) (v + t)ß )-n
1 +
R(v +t)
R(v)
(v + t)ßn
1 + v r il + tßn
(8)
A/\ X J\ X J \ X
For the residual life time random variable r(t), t > 0, the cdf and pdf are respectively obtained as
Fr(f)(v; n, ß, X) = 1 - Rrjv, n, ß, X)
(t)
F^ „ ß, X) = 1 - (l (l + ^ )(l + ^ )' (l + X )-1
XX On differentiating the above equation w.r.t v, we obtain
it a i\ l -m
fr(t)(v; n,ß, x) = ^t-
(9)
(10)
where
i = (1 + -n (l +
1 + X
1 1 + xJ I1 + X
m
(l+X) ' (l+3) (l) ' >+t)ß-11 -(
(v + t)ß-
(v + t)ßn\ fa (v + t)ß
1 -l 1 + I1 + X
1+£) (l+Ç -
Also, the associated failure rate of T(ty t > 0 for the power length biased weighted Lomax distribution is given by
hr(t)(v; n, ß X)
= fr{t)(v; n, ß, X) = Rr(t) (v; n, ß, X)
K)(v; n, ß, X)
r(t) where
l - m\ fa fa + (v + t)ßn)
tß\ -n X ) V1+ X
1 + x) (ll)
i=(l" (l(l+X) " Xßtß-1
1 -<1 + xJ I1 + X
1
1
n
9
l
X
l
2
n
1
m
1 + A
Y O + S)O + '> + ^ i-(
(v + t)ß
1 - 1 +
(v + t)M ( (v + t)ß
A J
A
i+A) O+f
4.2. Reversed Residual life function
The random variable r("t) = (t — v|v < t); t > 0is used to explain the residual life of a lifetime component. For the power length biased weighted Lomax distribution , the survival function of the reversed residual life time r("f), t > 0 is defined as
Rr(t)(v; n ßA)
F(t - v) F(t)
Rr(t) (v; n, ß, A)
1 - 1 +
(t-v)ß
1+
(t-v)ßn
i - ii + $)(i + A
(12)
For the reversed residual life time random variable r(t), t > 0, the cdf of power length biased weighted Lomax distribution is obtained as
Fr[t)(v; n, A) = 1 — Rr(t) (v; n, A)
Fr(t)(v; n ßA)
i +
(t-v)ß
i +
(t-vfy
i - (i + A
i + A
i + A
n 'i + tAn
i - (i + A
i + A
(i3)
5. STATISTICAL PROPERTIES OF PLBWLD
This section is devoted to discuss the related measures of the new formulated model like raw moments, central moments, measures of skewness, kurtosis, coefficient of variation, index of dispersion, mode and harmonic mean.
5.1. Raw Moments
The rthmoment of the PLBWLD about origin is given by
¡ir = E(Vr) = J vrf (v; n, ß, A)dv
Using (1) and further simplification, r moment of the PLBWLD about origin is obtained as
Fr =
A? ( f + D!
(n - ß - i)(n - ß - i)...(n - ß - i)
ß
(i4)
Using equation(14) and substituting r = 1,2,3,4 , the first four moments about origin of the
PLBWLD are obtained as
Fi
Aß (ß + (n - ß -1)
(i5)
2
n
n
A
The equation (15) represents the mean of the PLBWLD.
Aß
H2
+ 1 !
(n - ß - i)(n -1 -1)
(16)
Aß
H3
+ 1 !
(n -1 - 1)(n - ß - 1)(n - ß -1)
(17)
Aß
Hi
+ Л!
(n - ß - 1)(n - ß - 1)(n -3 - 1)(n - ß -1)
ß
(18)
5.2. Moments about Mean (Central Moments)
The moments about the mean, also known as central moments is defined as
H2 = Hi - (H1)2
using equations (15) and (16), we have
Hi
Aß (I +1)!
(n - ß - 1)(n - i -1)
\
A1 (1 + - 1 - 1)
The equation(19) represents the variance of our new formulated model.
Aß (ß +1)!
Нз
-3
Aß [ß + 1)!
Aß (1 + 1)!
(n - ß - 1)(n - i - 1)(n - ß - 1) (n - 1 - 1)(n - ß - 1) (n - ß - 1)
+ 2
(19)
H4 =
Aß (ß + 1)!
(n - ß - 1)(n - ß - 1)(n - 3 - 1)(n - i -1)
(n - 1 - 1)(n - i - 1)(n - 3 - 1)
+6
-4
Aß [ß +1)!
A3 (ß + 1)!
Aß (1 + ^ (n - ß - 1)
(20)
\ Лß (^i).\
Aß (ß + ^
(n - ß -1)
\
\
(n - ß - 1)(n - i - 1)
ß
'Aß (1 + 1)!
(n - ß -1)
V
'Aß (1 + 1)!
/
-3
(n - 1 - 1)
/
The following four coefficients are obtained for the PLBWLD based upon the first four moments about the mean and using the above expressions defined as:
ft1=4
F2
71 = Vlk
a = Hi P2 = ~2 Hi
72 = ß2 - 3
2
ß
3
ß
4
ß
2
3
3 / \
A ß(ß + 1> (n-ß-1)(n-ß-1)(n-ß-1)
-3
A ß ( ß + 1\! (n-ß-1)(n-ß-1)
A^ Q + 1)! (n-ß-1)
A2 (ß + 1\! (n-ß-1)(n-ß-1)
A^ (ß + 1)! (n-ß-1)
+ 2
1 / \ A ß(ß + 1>
(21)
We need another measure that is dependent on the sign of the third central moment since the nature of skewness cannot be estimated using this relation.
71
vß
A 3 ( 3+1\! (n-ß-1)(n-ß-1)(n-ß-1)
-3
Y1
Also,
Aß (ß+1\!' (n-ß-1)
(n
Aß (ß+1)! -ß-1)(n-ß-1)
(22)
ß2
¥4 (¥2 )2
(23)
where
¥4
4
Aß
+1)!
/
(n - ß - 1)(n - ß - 1)(n - ß - 1)(n - ß - 1)
/
+6
-4
Aß
Aß
+1)!
\
(n - ß - 1)(n - 2 - 1)(n - 3 -1)
+ 1 !
(n - ß - 1)(n - 2 -1)
\ /a 1
ß +1\!
(n - ß - 1)
Aß (ß+^ (n - ß - 1)
\
/
3
Aß
+ 1 !
(n - 1 - 1)
And,
¥2
A2 (2 +1\!
Again,
1A1 (ß + 1)!
(n - ß - 1)(n - 2 -1)
72 = ß2 - 3
(n - ß - 1)
5.2.1 Coefficient of variation
2
3
(n-ß-1)
3
3
ß
2
CV
VF2 ¥'1
On using the equations (15) and(19), the coefficient of variation can be obtained for PLWLD.
C.V
(n
A^ (J+1)' -ß-1)(n-2-1)
Aß (ß (n-ß-1)
Aß (ß+1)! (n-ß-1)
(24)
2
5.2.2 Index of Dispersion
The index of dispersion is defined as
D = ^ H1
Using the formula we obtain the index of dispersion for the PWLBWLD as
A^ (ß/ Aß (ft+1)!
(n-ft-1)(n-2-1) \ (n-1 -1) D =-1-^-(25)
X1 (1+1)! (n-ft-1)
5.2.3 Mode
To discuss PLBWLD's monotonicity, we use the logarithm of its probability density function as;
log f (v; n,ß, A) = loJ n(n-1)ß v2ß-1 (1 + A I
vß \-(n+1y
In order to find the value of mode, we differentiate the above equation w.r.t v and equate to zero, it yields
(2ft - 1)X 1 ^ (26)
(П - 2)0 + 2,
Equation (26) represents the modal value for the PLBWLD. 5.2.4 Harmonic Mean
The harmonic mean for the PLBWLD is defined as
CC
11 E(V-1 ) = E(-) = J vf (v; n, ß, A)dv
0
=/v *X2^ (1+X P *
on solving the integral and further simplification, we obtain the harmonic mean for PLBWLD as
1
H =-1-7-) (27)
1 -ft+1, .Vr^,-1+U X X
n(n - 1)A-1Ö+1 (-1)k+1 (-(-ф+1)
k=0 \ > V k ) 1 -n-k+1
6. Moment Generating function, Characteristic function and cumulant
Generating function OF PLBWLD
6.1. Moment Generating Function
The moment generating function of PLBWLD distribution is defined as
Mv (t) = J etvf (v)dv
2
using the following series expansion
ex = 1 + x + — + ...
œ tr t
Mv (t)= E vrf (v; n, ß, X)dv
r=0 r! <
Using equation (14) we obtain the moment generating function for PLBWLD as
Mv (t) = E-
r=0 '
Xß + 1)!
(n -1 - l)(n - 2 - l)...(n - ^ -1)
(28)
6.2. Characteristic Function
The characteristic function for the PLBWLD can be obtained using the relation <pv(t) = Mv(it)
(it)
Xß +1)!
$(t) = E , 1 2
E0 r! (n - 1 - l)(n - ß - l)...(n - r - 1)
(29)
6.3. Cumulant Function
The cumulant function for the PLBWLD is obtained by using the relation kv(t) = logMv(it)
œ tr kv (t) = log e
Xß [ß +1)!
r=0
■ (n - ß - l)(n - ß - l)...(n - r -1)
(30)
7. Order Statistics of PLBWLD
The order statistics connected to the power length biased weighted Lomax distribution is devoted in this section. Let V(f„) be the tth order statistics with the random sample V(i), V(2), V(3), ...V(m) derived from the PLBWLD having the probability density function (pdf) f (v; n, ß, X) and cumulative distribution function (cdf) F(v; n, ß, X). Therefore, the probability density function (pdf) and cumulative distribution function (cdf) of V(t;n) say f(t-n) (v) and F(t;n) (v) are respectively defined as
n ■ +1 +
f(tn) (v) = {t - l)!(n - t)! [F(v; n, ß, X)]t-1 [1 - F(v; n, ß, X)]n-t f (v; n,ß, X)
(31)
r=t\l
F(t;n)(v) = E ' [F(v;n,ß,X)]' [1 - F(v;n,ß,X)]
n-j
(32)
Using equation(l) and equation(2) in equation(31) and equation(32), the pdf and cdf of tth ordered statistics for the PLBWLD are derived and are expressed as
f(t;n) (v)
(t - l)!(n - t)!
vß
1 - 11 + X 1 + x
vßn
t-1
„ vß
1 + vX
1 +
vßn
nt
n(n - l)ßv2ß-l (l
X2
+ -
-(n+1)
F<"^ = Ë (n)
1 -11 + XI I1 + ?
„ vß
1 + vX
1 +
vßn
n-j
2
n
n
n
n!
X
vß
n
X
In order to obtain the expression for pdf of smallest(minimum) order statistics V(j) and the largest (maximum) order statistics V(m) of PLBWLD , we assume t = 1 and n respectively and are expressed in the form as
f(1;n) (v)
1 +
1 +
Vf n
n — 1
n(n - 1)f ™—1 L + vf
A2 1 + A
-(n+1)
(33)
f(n;n) (v)
1 — 1 +
1 +
vf n
n1
n(n — 1)/? v2r—1 L + vf
A2 V 1 + A
-(n+1)
(34)
7.1. Median order statistics
The pdf of median order statistics, V(n+1) is defined as
(2n +1)!
f(n+1;n)(v) =
[F(v; n, /, A)]n [1 — F(v; n, /, A)]n f (v; n, /, A)
f(n+1;n)(v) =
(2n +1)! (n)!(n)!
—n
1 — I 1 + ^ I I 1 +
v n
1 + A
—n
v
n(n — i)/v2/—1 L + ±
A2 1 + A
1+
v n
(n+1)
8. Characterization of PLBWLD
Theorem 1. Let V(j), V(2), •••V(n) be n independently and identically distributed random samples selected from PLBWLD having a sample mean of vn and sample variance of sn then,
nlima E( I) =( f )2
Proof: E(vn) = f and var(vn) We know that
E(vn) = var(vn) + [E(vn )]2
E(vn)2 = -n
A?
+ 1 !
(n — / — 1)(n — f — 1)
A?
+ 1 !
' — f — 1)
+
Af (f +!)!
(n — 1 — 1)
E(s2n)
2
A/
f + D!
(n — f — 1)(n — f — 1)
'af (f + 1)!^
' — f — !)
Therefore
Vn 2
A? (J + 1)!
(n — f — 1)(n — f — 1)
A? 0 + 1> (0—?—1)
2 ) / 1 (, ) \ 2 r 1 (, ) 1
1 A + + A ?(} + 1)!
n (n — f — 1)(n — f — 1) (n—f—1)
n
v
A
A
n
v
A
A
n! n!
n
n
A
A
2
2
2
2
2
a
2
s
E
2
On taking the limits to both sides of the above equation , we have
lim E(--) = n^œ vn
Xß{j + 1)! _ / Xf(ß + 1)! (n-ß-l)(n-ß-1) (9-ß-1)
Xß( ß + l> (n-ß-l)
lim E(^) =(2
n^œ vn
Hence , the above theorem is proved
9. Information measure of PLBWLD
Entropy is a quantitative measures of the amount of uncertainty in a random variable. This section is dedicated to obtaining the PLBWLD generalized entropy expression.
Theorem 2. The generalized entropy for the PLBWLD is expressed as
1(a) =
i+1)!
n - ß - 1
<a - l) I (n - ß - l)(n - 2 - l)...(n - I - 1) I ( 1 + 1)!
Proof:The generalized entropy is defined as
1(a) =
vxy a -1 a(a - 1)
where
va = J vaf (v)dv
-œ
and p represents mean. For PLBWLD, we have
Xß (f +1)!
(n - ß - l)(n - ß - l)...(n - ß -1)
-1
P =
X1 ( ß + l)! (n - 1 - l)
Therefore, the expression for the generalized entropy of PLBWLD is obtained as
IW=1 J-^_^_J(n- ^ > -1
() a(a - 1) J (n - ft - 1)(n - ft - 1)...(n - I - 1) 1 (ft + 1)!
(35)
10. Estimation of Parameters
This section is devoted to maximum likelihood estimation technique for estimating the unknown parameters n, ft, X of PLBWLD.
2
2
a
1
va =
10.1. Maximum Likelihood Estimation(MLE)
Suppose v\, V2, V3, ...vm be the random sample derived from the PLBWLD having the probability density function (pdf) f (v; y,f>, A). Therefore, for m observations , the likelihood function of PLBLWD is obtained as
-(n+i)
m n — i i is . „ „ - i 7ir \
L(v; A) =
-1) ß
A2
mm ( vß
nv2ß-1 1 + X
i=1 » A
Maximizing the log likelihood function yields estimates fj, p, X estimations of the unknown parameters n, f, X. The log likelihood function is given by
m m / vf\
log L(v; n,f, X) = m log n + m log (n - 1) + m log f - 2m log X + E log (v¿)2f-1 - (n + 1) E log 1 + X
i=1 i=1 V X '
(36)
The MLE's of y, f and X are derived after partially differentiating (36) with respect to the corresponding parameters and equating to zero. We obtain the three normal equations as
^fr-M) (37)
X = (38)
mm
m + 2 E log Vi = (y + 1) E(39)
f i=1 i=1 (1 + vf )
The above three non-linear equations (37),(38) and (39) are not in closed form. Therefore, we shall solve these equations numerically using Newton-Raphson technique of solving equations iteratively and numerically .
11. SIMULATION ILLUSTRATION
The performance of maximum likelihood estimates are examined in this section. To demonstrate the behavior of maximum likelihood estimates (MLEs) in terms of random generating sample sizes n= 100, 150 and a simulation research was conducted using R software. The procedure was repeated 100 times with various parameter combinations selected. The average MLE values and accompanying empirical mean squared errors (MSEs) were calculated in each scenario. Table 1 and table 2 shows the simulation findings. The estimates are stable and near to the genuine parameter values, as shown in table 1 and 2 . In all circumstances, the MSE drops as the sample size increases.
12. APPLICATION
For illustrating the flexibility, adaptability, and suitability of the PLBWLD, we use two actual data sets to show that the power length biased weighted lomax distribution (PLBWLD) can be better model than lomax distribution (LD) and length biased weighted lomax distribution (LBWLD).
To demonstrate how the proposed distribution can be effective in a real-world situation, two real life data sets have been examined . The following models have been investigated for comparison.
• Length biased weighted Lomax distribution (LBWLD) With pdf given in
f (v;n,X) = n(nX-1) v (1 + X)-(n+1) ; v > 0,n > 1,x > 0
• Lomax distribution (LD) with pdf given as
f (v; n, X) = X (1 + X )-(n+1) ; v > 0, n > 1, X > 0
Here, several goodness-of-fit criterion such as Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Akaike Information Criterion Corrected (AICC) , Hannan Quinn Information Criterion (HQIC) and Kolmogorov -Smirnov (KS) statistics are used. The statistic with the lowest value is considered the best fit. The numerical results are produced using R programme for analysis purposes.
Table 1: Average values ofMLEs and the corresponding MSEs(n=100).
Parameter MLE MSE
n X ß n X ß n X ß
1.5 1.2 2 1.60342 1.46263 1.88991 0.10386 0.29586 0.07998
2.5 1.61165 1.52594 2.37138 0.04400 0.32747 0.11551
3 1.60230 1.44918 2.80023 0.04089 0.23487 0.18081
1.8 2 1.60342 2.13739 1.88990 0.03866 0.54490 0.07997
2.5 1.56354 2.06030 2.41250 0.01490 0.28395 0.08610
3 1.60906 2.06175 2.79079 0.039382 0.29147 0.19585
2 1.2 2 2.11303 1.37128 2.10674 0.33824 0.53606 0.17352
2.5 2.29354 1.64115 2.61302 1.94607 3.27331 0.27052
3 2.24369 1.54492 3.05438 0.88963 1.57343 0.42315
1.8 2 2.15325 2.11382 2.08488 0.40547 1.02721 0.19684
2.5 2.10288 1.95731 2.54206 0.20766 0.57590 0.21280
3 2.17188 2.13039 3.11005 0.45798 1.41945 0.32907
Table 2: Average values ofMLEs and the corresponding MSEs(n=150).
Parameter MLE MSE
n X ß n X ß n X ß
1.5 1.2 2 1.57382 1.40020 1.92791 0.02535 0.20046 0.06029
2.5 1.57460 1.35389 2.36473 0.02172 0.10933 0.08347
3 1.55781 1.36290 2.89490 0.01292 0.11236 0.08685
1.8 2 1.56615 2.00830 1.90932 0.01953 0.24642 0.04964
2.5 1.56273 1.92833 2.38188 0.01404 0.16196 0.08055
3 1.58063 2.05766 2.87854 0.02763 0.23273 0.08986
2 1.2 2 2.12305 1.36716 2.05878 0.25457 0.38323 0.12918
2.5 2.08487 1.37785 2.59013 0.15097 0.28118 0.16941
3 2.05282 1.30232 3.10483 0.14772 0.29009 0.22523
1.8 2 2.06271 1.90663 2.04340 0.14480 0.42436 0.085475
2.5 2.01103 1.84029 2.62634 0.10312 0.31950 0.20608
3 2.13476 2.06073 3.04121 0.23962 0.70488 0.26052
12.1. Data Set 1
Data set 1: The first data is on the breaking stress of carbon fibres of 50 mm length (GPa). The data has been previously used by [2] . The data is as follows:
0.39, 0.85,1.08,1.25, 1.47,1.57,1.61,1.61, 1.69, 1.80,1.84,1.87, 1.89, 2.03, 2.03, 2.05, 2.12, 2.35, 2.41, 2.43, 2.48, 2.50, 2.53, 2.55, 2.55, 2.56, 2.59, 2.67, 2.73, 2.74, 2.79, 2.81, 2.82, 2.85, 2.87, 2.88, 2.93, 2.95, 2.96, 2.97, 3.09, 3.11, 3.11, 3.15, 3.15, 3.19, 3.22, 3.22, 3.27, 3.28, 3.31, 3.31, 3.33, 3.39, 3.39, 3.56, 3.60, 3.65, 3.68, 3.70, 3.75, 4.20, 4.38, 4.42, 4.70, 4.90
12.2. Data set 2
Data set 2: The following data represent the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, observed and reported by [4]. The data are as follows:
0.1, 0.33, 0.44, 0.56, 0.59, 0.72, 0.74, 0.77, 0.92, 0.93, 0.96,1,1,1.02,1.05,1.07,1.07,1.08,1.08,1.08,1.09,1.12, 1.13,1.15,1.16,1.2,1.21,1.22,1.22,1.24,1.3,1.34,1.36,1.39, 1.44,1.46,1.53,1.59,1.6,1.63,1.63,1.68,1.71,1.72,
Table 3: -2ln(l), AIC, AICC, BICfor the first data set.
Model -2ln(l) AIC AICC BIC HQIC K-S
PLBWLD LBWLD LD 175.004 224.008 265.990 181.004 228.008 269.989 181.391 228.199 270.180 187.573 232.388 274.369 183.6 229.731 271.720 0.078 0.250 0.358
Histogram of your_data
Figure 4: Fitted density plots for datasetl
1.76, 1.83, 1.95, 1.96, 1.97, 2.02, 2.13, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53, 2.54,2.54, 2.78, 2.93, 3.27, 3.42, 3.47, 3.61, 4.02, 4.32, 4.58, 5.55
Table 4: -2ln(l), AIC, AICC, BICfor the second data set.
Model -2ln(l) AIC AICC BIC HQIC K-S
PLBWLD LBWLD LD 187.753 195.049 226.075 193.753 199.049 230.075 194.106 199.223 230.249 200.583 203.602 234.628 196.472 200.861 231.888 0.084 0.168 0.294
13. CONCLUSION
This research paper uses power transformation to develop a novel life time probability model called power length biased weighted Lomax distribution. Ordinary moments, moment generating function, hazard rate, order statistics, and generalized entropy are among the significant aspects of PLBWLD that are obtained here. In addition, two real data sets are used to highlight the practical value.
The three-parameter PLBWLD distribution has been introduced here to have more flexibility in terms of the hazard rate function and density function. Using goodness of fit criteria, the suggested model's effectiveness is compared with other competing distributions. The new distribution can exhibit a much more flexible model for life time data . The new model was fitted to two different real-life data sets and showed that it
Histogram of your_data
0 1 2 3 4 5 6
Figure 5: Fitted density plots for datasetl
could offer a better fit than a set of extensions of Lomax distribution. We believe that the suggested model will have broader statistical applications.
References
[1] Ahmad, A. , Ahmad,S. P. and Ahmed,A.(2016). Length-biased weighted lomax distribution statistical properties and applications Pakistan Journal of Statistics, 12(2):245-255
[2] Al-Aqtash,R.,Lee,C. and Famoye,F.(2014). Gumbell-weibull distribution:properties and applications Journal of Modern Applied Statistical Methods, 13:201-225
[3] Balkema,A. and L. de Haan.(1974). Residual life time a great age. Annals of Probability, 49(5):792-804.
[4] Bjerkedal, T.(1960). Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. American Journal of Hygiene, 72: 130-148.
[5] Fisher, R. A.(1934). The effects of methods of ascertainment upon the estimation of frequencies Ann.Eugenics, 6:13-25.
[6] Krishnarani, S. D. (2016). On power transformation of half-logistic distribution. Journal of Probability and Statistics,5:1-10.
[7] Lee, E. T and Wang, J. W (2003). Statistical methods for survival data analysis . 3 edn, John Wiley, New york .
[8] Lomax, k. S.(1954). Business failure: Another example of the analysis of failure data. Journal of American Statistical Association , 49:847-852.
[9] Zaka, A. and Akhter,A. S. (2013). Methods for estimating the parameters of power function distribution Pakistan Journal of Statistics and Operation research,9:213-224.
