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THE LENGTH BIASED NEW QUASI LINDLEY DISTRIBUTION: STATISTICAL PROPERTIES AND
APPLICATION
N. W. Andure (Yawale)1 and R. B. Ade2* •
1 Department of Statistics, Government Vidarbha Institute of Science and Humanities,
Amravati, Maharashtra, India, neetayawale@gmail.com 2*Department of Statistics, Government Vidarbha Institute of Science and Humanities, Amravati, Maharashtra, India, rajeshwarb.sc@gmail.com
Abstract
In this paper, a new distribution namely the length biased new quasi-Lindley distribution is proposed with the different weight function. The different mathematical and statistical properties of the proposed distribution are derived and discussed. The survival function, hazard rate function and mean residual life function for the length biased new quasi Lindley distribution is discussed. Also, concepts like stochastic ordering and entropy for proposed distribution are studied. The parameters of the proposed distribution are estimated by using the method of maximum likelihood estimation. The performance of the newly introduced distribution is studied using a real- life data set.
Keywords: Length Biased Distribution, New Quasi Lindley Distribution, Reliability Analysis, Stochastic ordering, Maximum Likelihood Estimation.
I. Introduction
The concept of weighted distributions was first introduced by [8] to model ascertainment bias, weighted distributions were later formalized in a unifying theory by [16]. Weighted distribution is used in a variety of research fields related to reliability, environment, engineering and biomedicine. The weighted distribution reduces to length-biased distribution when the weight function considers only the length of the units. The concept of length-biased sampling was first introduced by [5] and [24]. [14] studied size-biased sampling and related form-invariant weighted distributions. Refer [15] for a general statistical discussion of weighted distributions. [12] proposed a useful result by giving a relationship between the original random variable X and its length-biased form Y when X is either Inverse Gaussian or Gamma distribution. Several researchers have studied length biased versions of different distributions see, [10], [6], [7], [17], [13] and [19]. Initially Quasi Lindley distribution was proposed by [22]. Later New Quasi Lindley (NQL) distribution was studied by [21] for modelling various data sets with probability density function (p.d.f) as follows
f(x\G,a) =-¡^(9 + ax)e-e% ;x > 0,6 > 0,62 + a > 0 (1)
The NQL distribution in (1) is a mixture of exponential and gamma distribution [exponential (9) and gamma (2, 9)].
In the present work, length biased new Quasi Lindley distribution is proposed and discussed in next section.
II. Length Biased New Quasi Lindley Distribution
Suppose X be a non-negative random variable with pdf f(x). Let w(x) be the non-negative
weight function, and then the pdf of the weighted random variable Xw is given by:
w(x)f(x)
fw(x)=TWxy) ' X>°
where w(x) be a non - negative weight function and
E(w(x)) = j w(x)f(x)dx < ro
When w(x) = xc ,the resulting distribution is termed as weighted distribution. When w(x) = x the resultant is known as size or length biased distribution. In this paper, the length biased version of new quasi-Lindley distribution is proposed. The weight function used is as follows
w(x) = y (2)
According to [1], let X is a non-negative random variable with pdf f(x). Let w(x) be the nonnegative weight function then the pdf f (x) for a length biased distribution of X is given by:
f(x) =w'(x)w™f™ , x > 0 (3)
E(wt(x)w(x)) v '
Assuming the E(w(x)wt(x)) = /w(x)wt(x)f (x) dx < ro Provided that w„ (x) = x
Using equation (1) and (2) in (3), the pdf of length biased new quasi-Lindley (LBNQL) distribution is
fi(x; d,a) =■ 6
where
E (w(x)w,(x))
nx x82 2n(82 + 3a)
w (x)w,(x)) = I + ax)e~9xdx = ^2+^
f
E(w(x)wt(x)) = I Jo
a4
fl(x-,d,a) = —7-x2(6 + ax)e-9x ;x > 0,6 > 0,a> 0 (4)
J 2(82 + 3a) K J v '
and the cumulative distribution function (CDF) of LBNQL distribution is obtained as
Jrrn
fl(t;9,a)dt
X
Q4 Ç00
F(x) = 1-2(WTlâ)lt2(d + at)e-0tdt
a4
F(x) = 1--\-(9 J™ t2 e-etdt + afœt3 e-etdt)
K J 2(92 + 3a)K Jx Jx J
after the simplification, the CDF of the LBNQL distribution is
F(x) = l-( 1 + a03x3+x(6a9 + 283) + (3a82 + 94)x2\ e-gx
( ) ( 2(82 + 3a) J ( )
Figure 1: PDF of LBNQL distribution for different values of d and a.
Figure 2: CDF of LBNQL distribution for different values of d and a.
III. Reliability Analysis
In this section, the reliability function or survival function, hazard rate function and mean residual life for the LBNQL distribution is discussed.
I. Survival Function of LBNQL Distribution
The survival function or the reliability function of Length biased new quasi-Lindley distribution (LBNQLD) is defined as
S(x) = 1-F(x) (6)
Substituting from equation (5) in (6),
s(x) = (± + a93x3+x(6a9 + 293) + (3a92 + 94)x2\ e-0x ( ) ( 2(92 + 3a) J ( )
Figure 3: survival function of LBNQL distribution for different values of d and a.
II. Hazard Rate Function of LBNQL Distribution
The basic tool for studying the aging and reliability characteristics of the system is the hazard rate (HR). The hazard function is also known as the hazard rate. Thus, the hazard rate function of the LBNQL distribution is given by
fl(x-,e,a)
h(x) =;
S(x)
(8)
Substitute the value of (4) and (7) in (8), h(x) =
х2в4(в + ax)
2(в2 + 3a) + ав3х3 + (вав + 2в3)х + (Зав2 + в4)х2
Figure.4: Hazard rate function of LBNQL distribution for different values of d and a.
Figure (4) shows the behavior of hazard function. For different choices of a and 8 it shows increasing failure rate.
III. Mean Residual Life Function of LBNQL Distribution
The mean residual life function is defined as
1 Гт
т(х) = Е[Х>х] = 1_F(x)J [1-F(t)]dt
_ 1 X
= , ав3х3 + х(6ав + 2в3) + (Зав2 + в4)х2\ _вт (1+ 2(в2 + 3а) )е
ав3Ь3+ Ь(6ав+ 2в3) + (3ав2+в4)Ь2\ а„
Х1 (1+---^ \ N-—)e-etdt
Ц 2(в2 + 3а) J
After simplifying, we get
, ч 24а + 6в2 + ав3х3 + х(18ав + 4в3) + (6ав2 + в4)х2 т(х) =
в(ва + 2в2 + ав3х3 + (Зав2 + в4)х2 + х(вав + 2в3))
3 (02+4а)
It can be easily verified that m(0) = -^-z—{ = и'*
7 4 J 9(92 + 3а)
IV. Moments and Associated Measures
Let X denotes the random variable of LBNQL distribution with parameters a and 6 then the rth order moment of LBNQL distribution can be defined as
E(Xr) = u'r = i xrfl(x;в,a)dx
0
4
= хГ+2 lîïh^i9 + ax)e-9xdx = J7*r+2 УЗ + ax)e-exdx
2(92 + 3a)J0 4 J
04 (в Ç xr+3-1 e-exdx + a Ç xr+4—1 e-exdx)
(Г(Г+3) «Г(Г+4)Ч , .
( ДГ+2 + ar+4 I (9)
2(в2 + 3а)
' _ в4 ßr
2(82+3a) \ er+2 er+4 J Putting = 1 in (9), the mean of LBNQL distribution is given by
, , 3(92+4a)
^ = E(x) = mT3;r)
and putting r = 2,3,4 in (9), the second, third and fourth raw moments are
V2 =E(X ) = Q2(Q2 + 3a) ' V3 = E(x ) = g3(92 + 3a)' = E(X )= g4(g2 + 3a)
Therefore,
Tr . 7 3(12a2 + 8a92 + 94)
Variance = a2 =-r-—-——
92(92 + 3a)2
j3(12a2+8 ав2+в4)
Standard Deviation = a =-—т—;-г-
в(в2 + 3a)_
a J3(12a2 + 8ав2 + в4)
Coefficient of Variation(c. v) = — =-————--
^L 3 (в I )
a2 (12a2+8ав2+в4) Coefficient of Dispersion(y) = — = ■
U в(в2 + 3а)(в2 +4a)
N.W. Andure (Yawale) and R.B. Ade
THE LENGTH BIASED NEW QUASI LINDLEY RT&A, No 2 (68) DISTRIBUTION: STATISTICAL PROPERTIES AND APPLICATION_Volume 17, June 2022
I. Moment Generating Function and Characteristic Function
Let the random variable X follows the LBNQL distribution. By definition of moment generating function of X and using equation (4), we get
Mx(t) = E(etx) = I etx fl(x;8,a)dx
= /"(l + ( tx) + +(tf+...)fi(x; 8, a)dx = CT,Zo(^ffi(x;e,a)dx
= T,Zo(J;)-Cxrfl(x;8,a)dx
= Z™=o(^-E(Xn (10)
Substituting from equation (9) in (10),
0ri 04 fr(r + 3) ar(r + 4) My x 1 ' '
v ' ¿^ r\ [2(92 + 3a) ( Sr+2 Sr+4
r=0 K v
Similarly, the characteristic function of LBNQL distribution is obtained as
$x(t) = Mx(it)
00 _
Z(it)r ( 04 r(r + 3) ar(r + 4)
—Ue2 + 3a)(-B^ + -er+4-
r=0
V. Entropy
The concept of entropy is important in a variety of topics such as probability and mathematics, physics, communication theory and economics. The entropy of random variable X is a measure of the variability of uncertainty.
I. R'enyi Entropy
R'enyi entropy [18] is important in nature and mathematics as an indicator of diversity. R'enyi entropy is also important in quantum data, where it can be used as a catch measure. R'enyi entropy is provided by
Re(S)=j^log (J fls(x;0,a)dx)
where 5 > 0 and 5^1
Re(5) = 1-5° (J" {w^)X2(d + aX)e-9X) dX) Re(s) = i—5log{{2(ê2+3a)) l"x2S(1+^x)Se-eSxdx)
putting (l^xf = ZT=0(«)&y
Re(5) = i^log Q $ £(x)2^1-1 e--dx}
. ✓ r- V ft 00
1 \ 05 IV iS\ r(2S +j + !)\
Retf) =—sloa\ [2{e2 + 3a)) X (j) Q) m26+j+
j=0
II. Tsallis Entropy
A generalization of Boltzmann-Gibbs (B.G) statistical properties initiated by Tsallis has received great attention. This generalization of BG statistic was proposed firstly by introducing the mathematical expression of Tsallis entropy by [23] for a continuous random variable and is defined as follows
^Ä^^rl1-! fiX(x-,e,a)dx
'T^i'-lo
p / Q4
A\1-J0 (2(e2 + 3a)
sÄ = T-A[1-C\2ïê2+3âjx2(e + ax)e) dxt sä=T—Ä(1- fco^î fx2Ä (1+ax)Ä e-Äexdx]
putting (l+-ex) =Tr=o(Ä^x)1
1-X\ (2(Q2 + 3a)
V ft 00
1 e5 \Y (X\,a^ir(2X+j+1)
Ä 0
' | I-V/I/II 0ÄX.
) X Q gy l (x)2Ä+j e-Ä*dx }
sä=-—
m®
1-A) \2(e2 + 3a)l ¿-i\jJ\e; (ex)2Ä+i+
=0
VI. Order Statistic of LBNQL Distribution
Order statistic have a central role in statistical theory. Suppose X(1),X(2),............,X(n) be the
continuous ascending order statistic. The probability density function of the jth order statistic X^) for 1 < j < n is
= (-¿-J! [F(X)]J'~1 [1 - F(x)]n-JA(x) (11)
Substitute the value of pdf and cdf of LBNQL distribution in (11), we get
f M- n h ii I «03x3+x(6«0+2e3)+(3ae2+e4)x2\-gxy-1 >X(.i)(X) = (j-mn-j)d1 (1+ 2(92 + 3a) )£ ]
* [(i + .'3'3*'(6.^'2*'')'2)e-'' ]n- x_g_xH, + ax)e-'' (12)
Put j = 1 in equation (12), the probability density function of first order statistics of LBNQL distribution.
ÄIU« = n[{1 + ^^^r'2*''''2) ^'V * 2^;7)X2(e + ax)e-" Put j = n in equation (12), the probability density function of nth order statistics of LBNQLD.
fxw(x) = n
ae3x3 + x(6ae+ 2e3) + (3ae2 + e4)x2\ a 1-l1 +-( 2(e2+U-~)e-Bx
n-1
N.W. Andure (Yawale) and R.B. Ade
THE LENGTH BIASED NEW QUASI LINDLEY RT&A, No 2 (68)
DISTRIBUTION: STATISTICAL PROPERTIES AND APPLICATION_Volume 17, June 2022
94
X2(92 + 3a)x2(d + ax)e-eX
VII. Stochastic ordering
Stochastic ordering of positive continuous random variables is an important tool for judging their comparative behavior. A random variable X is said to be smaller than a random variable Y in the
1) Stochastic order (X <stY) if Fx(x) > FY(x) for all x
2) Hazard rate order (X <hr Y) if hx(x) > hY(x) for all x
3) Mean residual life order (X<mrlY) if if mx(x) < mY(x) for all x
4) Likelihood ratio order (X <,r Y) if fx(x) decreases in x
The following important interrelations due to [20] are well-known for establishing stochastic ordering of distributions
X <ir Y ^ X <hr Y ^ X <mrl Y
V
X< Y
The LBNQL distribution is ordered with respect to the strongest 'likelihood ratio ordering' as shown in the following theorem:
Theorem 7.1: Let X~LBNQL(91, a1) and Y~LBNQL(92, a2).lf 91 = 92 and a1 < a2 or ( a1 = a2 and 91 > 92) then X <lr Y and hence X <hr Y, X <mrl Y and X <st Y.
Proof: From the pdf of LBNQL distribution (4), we have
fx(x) = 014{022 + -3a2) (e1 + g1x\ (ei-g2)x x>0
024(012 + 3a1)\02 + a2xJ '
fr(x) 024(Si2 + 3ai)\02 + a2X
Now
logùx^ = log
fv(x) 5
This gives
014(022 + 3a2)
024(9i2 + 3ai)
j + \og(9i + aix) - \og(92 + a2x) - (Bi - 02)x
d ( fx(x)\ a1 a2
dX (l0g f(X) ) = (B1 + a1x) - (B2 + a2x) - - B2
aB-aB -iB1-B2)
(Bi + aiX)(B2 + a2X)
rfx(x) 'fy(x)
Case I: If 91 = 92 and a1 < a2, then — (logfx(x)) < 0 . This means that X <lr Y and hence X <hr Y,
dx V fY (x) J
X<mriY and X<stY.
id 9, > 9, then — (lof
'fY(x)
X<mriY and X<stY.
Thus, LBNQL distribution follows the strongest likelihood ratio ordering.
Case II: If a1 = a2 and 91>92, then — (logfx(x)) < 0 . This means thatX <lr Y and hence X <hr Y,
dx V f'y(x) J
VIII. Bonferroni and Lorenz curve
The most important inequality curves are called Bonferroni and Lorenz curve, which have some application in applied science such as economics, reliability, demography and medicine. Bonferroni and Lorenz curves are proposed by [3]. The Bonferroni and Lorentz curves for the LBNQL distribution is obtained as
1
B(p) =—I xf,(x,B,a)dx PVJo
and
where
L(p)
1 rn
= pb(p)=-\ xfl(x-,e,
r Jo
a)dx
E(x)=V=j$+$ andq = F~i(p)
after simplification,
and
B(p) =
I'
0
e4
e(e2 + 3a)
p3(e2 + 4a) I x 2(e2 + 3a)
e5
(e + ax)e-0xdx
B(p) =
p6(e2 + 4a)
r'
I x3 (e + ax)e-exdx
0
B(p) =
L(p) = pB(p) =
e2y(4,eq) + ay(5,eq) p6(e2 + 4a)
1
6(e2 +4a)
(e2y(4,eq) + ay(5,eq))
IX. Maximum Likelihood Estimation
The method of maximum likelihood is the most frequently used method of parameter estimation given in [4]. The maximum likelihood method of estimation has been adopted to estimate the unknown parameter a and 9 of the LBNQL distribution. Consider the random sample of size n from the LBNQL distribution, the likelihood function is given by
/ Q4 \ n n
L(x;a,9) = l2(-e2 + 3a)) ^^ (9 + ax^e-0^* ^ ' i=1 The log likelihood function is given by
log I = 4nlog9-n log(2 (92 +3a))+ 2 Y^=1 log xi + Y^=1 log(9 + axi) - 9 Y^=1 xi (13)
Now maximize the above log-likelihood function given in equation (13) to get maximum likelihood estimate of unknown parameters of length biased new quasi-Lindley distribution. For this purpose, take the first derivative of the above log-likelihood equation with respect to parameters a and 9 and equate to zero respectively.
^Jn-(^ + ^n-=1i+xl-^n-=1xi = 0 (14)
+ (15)
Equations (14) and (15) are nonlinear equation. The exact solution of above equation is not possible numerically. Above nonlinear equations are solved with the help of R Software. Using the large sample property of MLE, A can be treated as being approximately normal with mean A and variance covariance matrix equal to the inverse of the expected information matrix, i.e.,
Vn(A- A) ^ N(0,I-1 (A)) 1(A) is the information matrix then its inverse matrix is I-1 (A). The I {A) variance-covariance matrix is essentially equal to the inverse of the expected information matrix I-1 (A), the observed information matrix is given by
m = —
n
E
E
d2\ogl
de2
d2\ogl dade
E
E
d2 log I deda d2 logl da2
where 1(A) is Fisher's Information Matrix.
d2logl -4n + 2n(e2 - 3a)
d e2 d2 logl
e2 (e2 + 3a)2 4-,(e + axi)2
=1
n
y
4-1 (e + axi)
9n
d a2
-y
x
(e2 + 3a)2 4-,\(e + axi)2
=1
n
d2logl d2logl 6ne v ( x,
' y((e + ax))
d a d e ( e2 + 3 a)2 ( e +
=1 -1
ddda
Since A being unknown, I-1(A) is estimated by using I-1(A) and we obtain the asymptotic confidence intervals for a and d. Hence the approximate 100(1 — xp)% confidence interval for a and 9 are respectively given by
1 ± z±Jiaa-1(A), 0 ± z±ji99-1(a)
Where is the ^th percentile of the standard distribution.
2
X. Application
In this section, one real life data set is analyzed for the purpose of illustration to show the usefulness and flexibility of the LBNQL distribution. The LBNQL model is compared with other distributions, such as, New Quasi Lindley (NQL) distribution, [21], length biased weighted New Quasi Lindley (LBWNQL) distribution [9]. The ML Estimates of the unknown parameters are determined for the LBNQL distribution and two other models along with goodness of fit test.
Data set I: Following data depicts the fatigue life of some aluminum's coupons cut in specific manner (see,[2]). The dataset (after subtracting 65) is given below:
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 ,56 ,56, 56, 58, 59, 59, 59, 59, 59, 63, 63, 64 ,64, 55, 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
The data set is modeled by LBNQL distribution and compared with the New Quasi Lindley, length biased weighted New Quasi Lindley distribution. Table 1 describes estimated unknown parameters, -log likelihood (-LL), the values of the AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion) and K-S Statistics calculated for above data using LBNQL, LBWNQL, and NQL distributions.
Table 1. Estimate and goodness of fit measures under considered distribution based on data set.
Model Estimated Parameter a e -2LL AIC BIC K-S P-Value
LBNQLD 74.01668 0.05801 945.7779 949.7779 955.0082 0.14788 0.2403
LBWNQLD 425.13398 0.04350 962.6170 966.6170 971.8473 0.18094 0.1436
NQL 168.18633 0.02900 992.7213 996.7213 1001.9516 0.23557 0.05719
From table 1 it can be seen that the value of the statistics -2LL, AIC, BIC values of LBNQL distribution are comparatively smaller than the other distributions on a data set. Therefore, the result shows that LBNQL distribution provides a significantly better fit than the other models.
Figure 5: Empirical CDF and fitted CDF plot of data set.
Figure 6: fitted PDF plot of data set.
XI. Conclusion
In the present study, a Length biased NQL distribution is proposed. Some statistical properties along with Reneyi entropy, and Tsallis entropy, Bonferroni and Lorenz curves have been discussed. Various reliability properties such as hazard rate function, mean residual life function, stochastic orderings have been obtained. It is proved that LBNQL distribution follows the strongest likelihood ratio ordering. For different choices of the parameters a and 9 increasing failure rate is observed. The parameters of the proposed distribution are obtained by using the maximum likelihood estimation technique. Finally, the new proposed distribution is tested by applying to a real-life data set and compared with new quasi Lindley distribution and length biased weighted new quasi Lindley distribution. It is observed from the table 1 that LBNQL distribution gives better fit over both distributions on a data set.
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