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THE NEW LENGTH BIASED QUASI LINDLEY DISTRIBUTION AND ITS APPLICATIONS
N. W. Andure (Yawale)1 and R. B. Ade2
1 Department of Statistics,Government Vidarbha Institute of Science and Humanities,
Amravati, Maharashtra, India, [email protected] 2Department of Statistics,Government Vidarbha Institute of Science and Humanities, Amravati, Maharashtra, India, [email protected]
Abstract
In this paper, Length biased Quasi Lindley (LBQL) distribution is proposed. The different properties of the proposed distribution are derived and discussed. The parameters of the proposed distribution are estimated by using method of maximum likelihood estimation and also the Fisher's Information matrix is obtained. The performance of the proposed distribution is studied using reallife data sets.
Keywords: Length Biased Distribution, Quasi Lindley Distribution, Reliability Analysis, Maximum Likelihood Estimation, Likelihood Ratio test.
I. Introduction
The Quasi Lindley (QL) distribution was introduced by Shanker and Mishra (2013). The QL distribution has two parameters a and 0. The Quasi-Lindley distribution reduces to one parameter Lindley distribution if a =0. If a =0, it reduces to the gamma distribution with parameter (2, 0). The probability density function of QL distribution is a mixture of exponential (0) and gamma (2, 0). The Probability density function of Quasi Lindley distribution (QLD) with parameters a and 0 is given by
f(x;a,0) = (a + 6x)e~9x ;x > 0,9 > 0, a > -1 (1)
and the cumulative distribution function of the two parameter Quasi Lindley distribution is given by
F(x; a,B) = 1 - e ~0x ; x > 0,9 > 0, a > -1 (2)
II. Length Biased Quasi Lindley Distribution
Length biased distribution is a particular case of weighted distributions that were first introduced by Fisher (1934) to model the ascertainment bias. These weighted distributions were later developed by C R Rao (1965) in a unifying manner. Weighted distributions arise when the observations generated from a stochastic process are not given equal chances of being recorded and moderately, they are recorded in accordance to some weight function. When the weight function depends only on the length of units of interest, the resulting distribution is called as length biased. Length biased concept was firstly given by Cox (1969) and Zelen (1974). The study of weighted distributions helps us to deal with model description and data interpretation problems. In the study of distribution theory, weighted distributions are useful because it provides a new understanding of existing standard probability distributions and also it provides methods for extending existing standard probability distributions for modelling lifetime data due to introduction of additional parameter in the model which creates flexibility in their nature. Much work has been done to characterize the relations between original distributions and their length biased versions.
various researchers have reviewed and studied different weighted distribution and found its applications in different fields such as reliability, biomedicine, ecology, and branching processes [refer Lappi et al. (1987), Mir et al. (2013), Mudassir et al. (2015), Shenbagaraja et al. (2019)]. Definition: The non-negative random variable X is said to have weighted distribution, if the probability density function of weighted random variable Xw is given by
w{x)f(x) f'(x) = -E(Md) ' X>0
Where w(x) be a non - negative weight function and
e(w(x)7 = J w(x)f(x)dx <
For different weighted models, different choices of the weight function can be done. When w(x) = xc ,the resulting distribution is termed as weighted distribution. In this Paper, the Length biased version of Quasi Lindley distribution is studied, here choice of c = 1 is done as a weight, in order to get the Length biased Quasi Lindley distribution and the probability density function of Length biased Quasi Lindley distribution (LBQLD) is given by
fL (x;a,0)= ^—f (3)
Where E(x) = j/ x f(x;a,9)dx
E(x) = (4)
v J 0(a +1) v '
After substituting from equation (1) and (4) in equation (3), the probability density function of Length biased Quasi Lindley distribution is obtained as
fL(x-,a,e)= ^ (a + Ox)e~0x ;x > 0,6 > 0,a > -2 (5)
and the cumulative distribution function (cdf) of LBQL distribution is obtained as
h (x) = J0X fL (t;a,9)dt Fl (x)=foX($£0) (a + 0t)e ~et dt Fl (x) = ($#0) /(X t(a + et)e~!t dt
after simplification, the cumulative distribution function of Length biased Quasi Lindley distribution is
Fl(X) = $r(M#r(4.!x) ;x > 0,9 > 0,a > —2 (6)
The graph of the probability density function and cumulative distribution function of length biased quasi-Lindley distribution (LBQLD) for different values of parameters, are shown in Figure 1 and 2.
For -2 < a < 0 and в = 1
For a > 0 and 6 = 1
For -2 < a < 0 and в > 0
For a > 0 and в > 0
Figure 1: Pdf plot of LBQLD for the different values of a and 9.
8=1.4, a=6 4 6=1.6 ,a=6.5 6=1.8, a=6.6 6=2.6, a=6.7
Figure 2: cdf plot of LBQLD for the different values of a and 9.
III. Reliability Analysis
In this section, the survival function, hazard rate, reverse hazard rate function and mill's ratio are discussed for the Length biased Quasi Lindley distribution.
The survival function is also known as reliability function and the Survival function of Length biased Quasi Lindley distribution is defined as
S(x) = 1 -
S(x) = 1 - Fl(X)
(ay(2,8x) + y(3,9x)
V (« + 2)
The hazard function is also known as hazard rate or instantaneous failure rate or force of mortality and the hazard function of Length biased Quasi Lindley distribution is given by
/$ (X; a,8)
h(x) =
h(x) =
S(x)
x82(a + dx)e~6x
(a+ 2)- (ay(2, dx) + y(3,8x))
where (a+ 2)- (ay (2, Gx) + y(3, Ox)) > 0
The reverse hazard function of Length biased Quasi Lindley distribution is given by
f$(x; a, 8)
hr (x) =
hr(x) =
F$ (x)
x82(a + 8x)e~6x (ay(2,8x) +y(3,8x))
The Mills ratio of Length biased Quasi Lindley distribution is given by,
1
Mills ratio = -——
hr 00
Mills ratio —
(ay(2,9x) + y(3,9x)< x92(a + 9x)e-8x
)=1.2,a=0.5 }=1-4 ,a=0.6 )=1.6,a=0.7
Fig.3: Graph of survival function of LBQLD.
Fig.4: Graph of Hazard function of LBQLD.
Figure (4) shows the behavior of hazard function. For different choices of a and 9 it shows decreasing failure rate.
IV. Statistical Properties
In this section, the statistical properties of Length biased Quasi Lindley distribution are discussed. I. Moments
Let X denotes the random variable of LBQL distribution with parameters 0 and a, then the rth order moment of LBQL distribution is defined as
E(Xr) -
— I xr fL (x;a,d)dx Jo
-f ,
o
9
(a+ 2)
02 r +
(a + 6x)e 'xdx
(a+ 2)
fo
o
xr+1(a + 9x)e 'xdx
92
(a+ 2)
K
xr+2-1e-0xdx + 9 I xr+3-1 e~°xdx
f xr+3-1e-6xdx^
after simplification,
E(Xr) =ß'r =
02 far(r+2)+r(r+3)\
(a+2) V 8r+2 )
(7)
putting r = 1 in equation (7), the mean of LBQL distribution is given by
2(a + 3)
V. = E(X) =
9(a + 2)
After putting r = 2,3 and 4 in equation (7), the second, third and fourth raw moments of Length biased Quasi Lindley distribution are obtained as,
6(a + 4)
V% = E(X%)
- .
62(a + 2)
V/ = E(X3) =
24(a + 5)
-
Therefore,
V4 = E(X4)
Variance = a2 =
93(a + 2)
120(a + 6) B4(a + 2)
2(a2 + 6a + 6) 82(a + 2)2
V2(a2 + 6a + 6) S. u.= a =-—---
8(a + 2)
a J2(a2 + 6a + 6) C.V = - =
C.D.(y)= — =
ß 2(a + 3)
(a2 + 6a + 6)
ß B(a + 2)(a + 3)
II. Moment generating Function and Characteristic Function LBQLD
Let X follows LBQL distribution, then the moment generating function (MGF) of X is,
n +
Mx (t) = I etx fL(x-,a,e)dx
(tx)2 (tx)3
f OO
= J (1 + (tx)+^2r +^3r+-)fL (x,a,6)dx
+
=j: z
(tx)r
fL(x,a,6)dx
r= 0 +
Z(t)r r
—¡- I xr fL(x,a,8)dx
r=o ' 0 (t)"
Mx (t) = T1?=o(rr E(%r ) (8)
Substituting value of E(xr) from equation (7) in equation (8),
Mx(t)
+
A, r! \(a + 2) 3
ai(r + 2)+ r(r + 3) 8_T2
)}
Similarly, the characteristic function of LBQL distribution can be obtained as, <Px(t) = Mx(it)
+ (ar(r + 2) + r(r + 3)>
2-j r! \(a + 2)(
er+2
III. Harmonic Mean
Let X follows LBQL distribution then, the harmonic mean is obtained as
r+ 1
H = J ~fL (x,a,8)dx
= f
Jo
e2
■(a + 8x)e 'xdx
e2
(a + 2)
(a + 2)
a I e-dx dx + 9 I xe-dxdx
oo
after simplification,
H =
8(a + 1) (a + 2)
V. Order Statistics for LBQL Distribution
Order statistics have central role in statistical theory. It deals with the ordered data that is necessary to take for quality control, reliability, hydrological and extreme values analysis.
Suppose X(1),X(2),............,X(n) be the jth order statistic and it is denoted by X^ .
The probability density function of the jth order statistics X^) for 1 < j <n is
fxa) (x) = u-rnn-» [p(x)]i~1[1 - F(x)]n~: fi(%) 9
Substitute the value from equation (5) and (6) in equation (9), the probability density function of jth order statistics of LBQL distribution is given as
f {X) 9 WYM^MV- x ¡1- ay(2M+y(3M}n-: + ex
Jx(j)K J (j-1)'.(n-j)ll (a+2) J L (a+2) J (a+2) J v '
For j = 1 in equation (10), therefore the probability density function of first order statistics of LBQL distribution is obtained as
fxu, W = n[1- ^-¡y^-1 x (a + 9x)e~ex (11)
Put j = n in equation (10), the probability density function of nth order statistics of LBQL distribution is given by,
(X) = n[-L-H-)-J X M (a + 8x)e
0x
(12)
VI. Entropy
The concept of Entropies points out the diversity, uncertainty, or randomness of a system and the entropies have large application in several fields such as probability & statistics, physics, communication theory and economics. Entropy of a random variable X is a measure of variation of the uncertainty.
I. Renyi Entropy
The entropy termed as Renyi entropy is important in ecology and statistics as index of diversity. Renyi entropy is an extension of Shannon's entropy. Renyi (1961) give an expression of the entropy function is defined by
(j fi (x;a,Û)dx7
e(S) = i—jlog ¡I f$ (x;a,8)dx
where S > 0 and 5^1
e(S) = 1—^log
r fe
-x(a + 6x)e
dxj
(a+ 2)
(S) = ^ logrjj'-)}0 Ç xs (a + 8x)s e~eSx dx
(13)
Using binomial expansion in equation (13),
e(5) = 1—"¿log
jfer+2) i % Q(a)'-k"fer- e-"x
e(S) =---log
-kak I (x)S+k+l-l0-8Sx
e~'°x dx
}
e(S) = ---log
1 — 5l0g V(« + 2)
e2
+ \ % o>?-k °k r-Bm]
II. Tsallis Entropy
Tsallis entropy was introduced by Tsallis (1988) as a basis for generalizing the standard statistical mechanics. For a continuous random variable X, Tsallis entropy is defined as follows.
= (l - C *5 (a + Ox)5e~WxdxA (14)
Using binomial expansion in equation (14),
*=i—iji—% 0 (a)A-k°k 0x)A+k e-Aex ^ = I—Ijl ■— % 0 (a)A-kek fy)A+k+l-le-A'x ^
= I^jl — ((â+2))À % Q(a)A-k ek "W^1)
VII. Likelihood Ratio Test
Let xl,x2,x3,.....,xn be a random sample from the LBQL distribution. To test the hypothesis
H0: f(x) = f(x; a, 9) Against Hl:f(x) = fL(x; a, 9)
In order to test whether the random sample of length n has been drawn from length biased Quasi Lindley distribution or not the following test statistics is used
= 1~\ (x'; a,8) L0 I 1 f(x; a, 9)
î=I
-mi n *
v ' i=l
Reject the null hypothesis, if
n »
v ' i=l
i=1 v '
k*= W=1 Xi > k* where k* = k (j^f > 0
For large sample size n, 2logA is distributed as chi-square distribution with 1 degree of freedom (df) and also p-value is obtained from the chi-square distribution. Thus, reject the null hypothesis, when the probability value is given by
P(A*> ¡3*)
Where p* = nHi=1 xi is less than specified level of significance and ^1=. xi is observed value of the statistics A*.
VIII. Bonferroni and Lorenz curves
The Bonferroni and Lorenz curves are given as
1 fD
B (p) = — J xfL (x;a,8)dx 1 fD
L(p) = pB(p) = — I xfL(x,a,8)dx ß J0
Where E(x)=ß= and q = F~1(p)
8(a + 2) fD % 82 '
B(p) = —;-r I x% --7 (a + 8x)e 'xdx
p2(a + 3) J0 (a + 2J
83
¡■D
B(p) = —--r I x2 (a + 8x)e~'xdx
(y) p2(a + 3)J0 ( )
B(p) = —-- X (a J x3~1e~6x dx + 8 J x4~1e~6x dx7
p2(a + 3) V I Jo J
aY(3,8q)+Y(4,8q) B(p) =-
L(p) = pB(p) =
2(a + 3)p
ay(3,8q) + y(4,8q)
2(a + 3)
IX. Maximum Likelihood Estimation
In this section, the maximum likelihood estimation of the parameters of Length biased Quasi Lindley (LBQL) distribution is discussed. Let x1,x2, ...,xn be a random sample of size n from the LBQL distribution, then the corresponding likelihood function is given by
9 (xt 62(a + 6xi )e~6xi (a+ 2)
L(x;a,e) = ~(-
i=i ^
L(x; a, 6) =
e%
(a + 2)
; IL
(a + dxje~eY-i=
Takin log and solving likelihood function is obtained as follows
dlogL _ —n
da (a-2)
+ rn=
(a+exi)
= 0
(15)
di^-L = 2n + X'n xi v = 0
de = 8 i=la+8xi L7 Xi~ 0
(16)
The MLE of the parameters cannot be obtain in close form. The exact solution of above equation for unknown parameters is not possible manually. So, we can solve above equations with the help of R Software using (optim function, nlminb (), nlm ()).
To obtain confidence interval we use the asymptotic normality tests. If as A = (a, 8) denote the MLE of A = (a, 8), state the results as follows:
^n(A-A)^N(0,I~1(A))
Where 1(A)is Fisher's Information Matrix is
I(X) = _~ n
E[d% logl\Efd2 log I
d28
ddda
d2 logl 7 E(d% log I
dad8
d2a
Where
d2 logl n
d2a (a + 22)2 ¿—t (a + ext)2
i=l
d2 logl _2n
d2e =~
n 2 (a + 0xi )
d2 logl d2log I
ddda dadd Z—i (a + 6xb)
=_Z
Since A being unknown, I .(A) is estimated by I .(A)and this can be used to obtain asymptotic confidence intervals for a and 6.
X. Application
In this section, three real life data set are studied for the purpose of illustration to show the usefulness
x
2
and flexibility of the LBQL distribution.
To compare the length biased Quasi Lindley (LBQL) distribution with QL, Power Lindley (PL), Exponential (Exp.) distributions, the criteria like Bayesian information criterion (BIC), Akaike Information Criterion (AIC), Corrected Akaike Information Criterion (AICC), HQIC are used and parameters are estimated using ML method of estimation. The real-life data sets are given as follows:
Data set I: The first real life data set represents the breaking stress of carbon fibres (in Gba) observed and reported by Nichols and Padgett (2006) and is executed below in table 1.
Table 1. Data consists of breaking stress of carbon fibres (in Gba) observed by Nichols and Padgett (2006).
Data set I
3.70 2.74 2.73 2.50 3.60 3.11 3.27 2.87 1.47
3.11 3.56 4.42 2.41 3.19 3.22 1.69 3.28 3.09
1.87 3.15 4.90 1.57 2.67 2.93 3.22 3.39 2.81
4.20 3.33 2.55 3.31 3.31 1.25 4.38 1.84 0.39
3.68 2.48 0.85 1.61 2.79 4.70 2.03 1.89 2.88
2.82 2.05 3.65 3.75 2.43 2.95 2.97 3.39 2.96
2.35 2.55 2.59 2.03 1.61 2.12 3.15 1.08 2.56
2.85 1.80 2.53
Data set 2: The second real life data set represent the fatigue life of some aluminum's coupons cut in specific manner (see, Birnbaum and Saunders, 1969). The dataset (after subtracting 65) is given below in table 2.
Table2. The fatigue life of some aluminum's coupons cut in specific manner (Birnbaum and Saunders, 1969).
Data set II
5 25 31 32 34 35 38 39 39
40 42 43 43 43 44 44 47 47
48 49 49 49 51 54 55 55 55
56 56 56 58 59 59 59 59 59
63 63 64 64 65 65 65 66 66
66 66 67 67 67 68 69 69 69
69 71 71 72 73 73 73 74 74
76 76 77 77 77 77 77 77 79
79 80 81 83 83 84 86 86 87
90 91 92 92 92 92 93 93 94
97 98 98 99 101 101 103 105 109
139 147
Data set 3:
This data set presented in Murthy et al. (2004) and used by some researchers. This data set present failure times for a particular windshield model including 85 observations that are classified as failure times of windshields.
Table3.failure times for a particular windshield model including 85 observations that are classified as failure times of
_windshields._
Data set III
0.040 1.866 2.385 3.443 0.301 1.876 2.481
3.467 0.309 1.899 2.610 3.478 0.557 1.911
2.625 3.578 0.943 1.912 2.632 3.595 1.070
1.914 2.646 3.699 1.124 1.981 2.661 3.779
1.248 2.010 2.688 3.924 1.281 2.038 2.822
3.000 4.035 1.281 2.085 2.890 4.121 1.303
2.089 2.902 4.167 1.432 2.097 2.934 4.240
1.480 2.135 2.962 4.255 1.505 2.154 2.964
4.278 1.506 2.190 3.000 4.305 1.568 2.194
3.103 4.376 1.615 2.223 3.114 4.449 1.619
2.224 3.117 4.485 1.652 2.229 3.166 4.570
1.652 2.300 3.344 4.602 1.757 2.324 3.376
4.663
R software is used for determining the estimation of unknown parameters and is also used for estimating the model comparison criterion values (AIC, BIC, AICC, HQIC) and -2logL. To compare the Length biased Quasi Lindley distribution with Quasi Lindley and Power Lindley, Exponential distributions, the criterion like AIC (Akaike information criterion), AICC (corrected Akaike information criterion), BIC (Bayesian information criterion) and HQIC (Hannen-Quinn information criterion) are used for comparison. The better distribution corresponds to lesser values of AIC, AICC, BIC, HQIC and -2logL.
Table 4. Estimate and goodness of fit measures under considered distribution based on data set I.
Distribution M.L. E - 21ogL AIC AICC BIC HQIC
a 8
LBQL -0.3883002 1.1745 199.7838 203.7838 203.9743 208.1631 205.5142
Distributio
n QL -0.3401128 0.9116 204.4596 208.4596 208.6501 212.8389 210.1901
Distribution
Exponential 0.3900 2.36944 249.8762 249.9397 258.2555 255.6067
Distribution 245.8762
PL 0.5781 1.1286 490.4955 494.4955 494.5590 498.8748 496.2260
Distribution
Table 5. Estimate and goodness of fit measures under considered distribution based on data set II.
Distributio M.L. E - 21ogL AIC AICC BIC HQIC
n a 8
LBQL Distribution -0.16555 0.04489 899.4582 903.4582 903.499 908.6884 905.5756
QL Distribution -0.14116 0.03144 982.2110 986.2110 986.251 8 991.4412 988.3284
Exponential 5.000 63.83168 1041.562 1045.562 1045.60 1050.792 1047.679
PL 0.1447 1.09255 1688.346 1692.346 1692.38 1697.577 1694.463
Distribution 7
_Table 6. Estimate and goodness of fit measures under considered distribution based on data set III._
Distributio _M.L. E_ - 21ogL AIC AICC BIC HQIC
n_a_0_
LBQL 0.1458 1.14416 276.632 280.632 280.680 285.5173 282.5970
Distributio 8
n
Exponential 0.040 2.5626 276.7906 280.7906 280.839 285.6759 282.7556 Distribution 4
QL 0.00362 0.77904 289.5131 293.5131 293.561 298.3984 295.4781
Distribution 9
PL 0.58946 1.18093 594.0202 598.0202 598.069 602.9055 599.9852
Distribution
From table (4), (5) and (6) it can be seen that the value of the statistics -2logL, AIC, BIC, AICC and HQIC of the Length biased Quasi Lindley distribution are comparatively smaller than the other distributions on a real-life data set. Therefore, the result shows that the Length biased Quasi Lindley distribution provides a significantly better fit than other models. So, it can be chosen to model the life testing data.
XI. Conclusion
In this paper, the Length biased Quasi Lindley distribution is proposed as a new extension of Quasi Lindley distribution. The newly introduced distribution is generated by using the Length biased techniques and taking the Quasi-Lindley distribution as the base distribution. The various statistical properties of the proposed distribution have been derived and discussed. Supremacy of the new distribution in real life is established with demonstration of real-life data sets and it is found from the results of data sets that the Length biased Quasi Lindley distribution performs better than the Quasi Lindley, Power Lindley and Exponential distributions.
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