A new class of exponential transformed Lindley distribution and its Application to Yarn Data. Sandeep K. Maurya, Arun Kaushik∗ , Sanjay K. Singh and Umesh Singh Department of Statistics Banaras Hindu University Varanasi, India ∗
[email protected]
ABSTRACT In the present paper, we proposed a new class of distribution taking Lindley distribution as baseline distribution. The proposed distribution named as exponential transformed Lindley is very applicable in the situation of increasing hazard rate and also some of the important statistical properties have been studied. Further, we have provided maximum likelihood and Bayesian estimation procedure for parameter estimation of the proposed distribution. Finally, we have analysed a real data set to explain the suitability of our distribution in comparison to other well-known distributions. Keywords: Lindley Distribution, maximum likelihood and Bayesian estimation. 2000 Mathematics Subject Classification: 62E10, 60E05, 62H10, 62H12.
1
Introduction
Lifetime distributions are very useful in the various field of applied science especially in medical, engineering, finance, marketing, banking, etc. To explaining real life phenomenon, there are a lot of lifetime distributions. Among all of them, some frequently used distributions are exponential, gamma, Weibull, lognormal, etc. Each has their own merits and demerits due to their flexibility of shapes and hazard rates like increasing, decreasing or constant hazard rate, depends on the nature of distributions. Lindley distribution is also one of the favourite lifetime distribution. Lindley distribution is a mixture of exponential distribution and length biassed the exponential distribution and proposed by (Lindley, 1958). The cumulative distribution function (cdf) of Lindley distribution is defined as, F (x) = 1 − e−θx
1 + θ + θx . 1+θ
(1.1)
Latter (Ghitany, Atieh and Nadarajah, 2008) derived various statistical properties of Lindley distribution. Lindley distribution is very useful in case of study of stress-strength reliability. After that many generalisations have been done by taking Lindley distribution as baseline distribution. In this context (Ghitany and Al-Mutairi, 2008) proposed size biased Poisson Lindley
distribution and (Ghitany, Al-Mutairi and Nadarajah, 2008) proposed zero-truncated PoissonLindley distribution. In similar context (Elbatal, Merovci and Elgarhy, 2013) proposed threeparameter generalised Lindley distribution on the concept of mixtures of the gamma distribution, (Deniz and Ojeda, 2011) proposed discrete Lindley distribution, (Nadarajah, Bakouch and Tahmasbi, 2011) proposed two parameter Lindley distribution by using the concept of exponentiated generalisation of distribution, called generalised Lindley distribution. (Hassan, Elbatal and Hemeda, 2016) proposed Weibull Quasi Lindley distribution and discussed its statistical properties, (Rashid and Jan, 2016) proposed Lindley power series distribution. Recently, (Maurya, Kaushik, Singh, Singh and Singh, 2016) proposed a method of developing new distributions and provided its application to real life phenomena. It may account that nearly all generalisation based on baseline distribution add some additional parameters to existing one. So that, the generalised distribution add complexities in inferences. Taking this point in mind, we proposed exponential transformed Lindley (ETL) distribution which is based on transformation proposed by (Kumar, Singh and Singh, 2015) and derived its various statistical properties and we take a real data set to shows model suitability for that data set in comparison to seven other well-known distributions. Since Lindley distribution is one of increasing widespread distribution so in this paper, we propose a new lifetime distribution with Lindley as baseline distribution, which also incorporate all the properties of its baseline distribution along with more flexibility in term of fitting without any additional parameter. Let X be random variable having baseline cdf is Lindley distribution, then by using the transformtion propound by (Kumar et al., 2015), the cdf of our proposed ETL distribution is given as: h exp 1 − e−θx 1 + F (x) = e−1
θx 1+θ
i
−1 .
θx θ2 −θx −θx 1+ . (1 + x)e exp 1 − e f (x) = (e − 1)(θ + 1) 1+θ
(1.2)
(1.3)
The associated hazard rate is, h(x) =
θ2 (1 + x)e−θx . 1 + θ + θx (θ + 1) exp eθx −1 1+θ
(1.4)
The rest of the paper is organised as follows: In section 2, we have discussed the shapes of distribution and hazard rate. In section 3, some properties of the proposed distribution have been derived. Section 4, discusses the maximum likelihood and Bayesian inferential procedures for the proposed distribution. In section 5, a real data of yarn data is considered for model superiority. In section 6, compression of estimator is given and at last, in section 7, the conclusions of the whole paper is abridge out.
2
Shapes
To know about the shape of the distribution by using equations (1.2) and (1.3), the cdf and pdf plots for different values of θ are given in figure 1. Nature of hazard rate is plotted in figure 2. This figure shows that the proposed distribution has increasing hazard rate.
Mathematically we can find the shape of hazard rate by using the result of (Glaser, 1980). According to him, if f (t) is density function and f 0 (t) is the first order derivative of f (t) with respect to t and a new term η(t) which is equal to −f 0 (t)/f (t). Than he proved that if η 0 (t) > 0 for all t > 0, then the distribution has increasing hazard rate. Since in case of our proposed distribution, we see that; η(t) = θ −
1 θ2 − (1 + t)e−θt (1 + t) (1 + θ)
and θ3 θ2 θ −θt 1 1 −θt −θt = − η (t) = [(1+t)θ−1] e + (1+t)e + e . (2.1) 1+θ (1 + t)2 (1 + θ) (1 + t)2 (1 + θ) 0
The second part of equation (2.1) is decreasing function of t which has minimum value zero when x → ∞, which shows that η 0 (t) > 0 for θ ≥ 1 and also η 0 (t) > 0 for θ ≤ 1 for all t ≥ 0. Hence, the proposed distribution has increasing hazard rate.
3 3.1
Some Statistical Properties Moments of the distribution
We are using a lemma to obtain the expression of the moments, which is given below: Lemma 3.1. ∞
Z
θx L1 (θ, r, δ) = x (1 + x)e exp 1 − e 1+ 1+θ 0 ∞ k l m+1 X X X X (−1)l k l m+1 = k! l m n −δx
r
−θx
dx, (3.1)
k=0 l=0 m=0 n=0 θm (n
×
(1 +
θ)l
+ r)! . (δ + θl)n+r+1
P i Proof. Using the expansion of ex = ∞ i=0 x /i!,we get k Z ∞ ∞ X 1 θx r −δx −θx L1 (θ, r, δ) = x (1 + x)e 1−e 1+ dx k! 1+θ 0 k=0 P i b y i , when b be a real number now, by the result of expansion of series, (1−y)b = ∞ i=0 (−1) i Pb b b i i and (1 − y) = i=0 (−1) i y , when b be an integer number then simplifying, we have, =
∞ X k X l m+1 X X (−1)l k l m + 1 θm (n + r)! . k! l m n (1 + θ)l (δ + θl)n+r+1 k=0 l=0 m=0 n=0
(See (Graham, Knuth and Patashnik, 1989) for detail expression of binomial series). According to the above lemma 3.1, we have the rth moments as, E(X r ) =
θ2 L1 (θ, r, θ). (θ + 1)(e − 1)
Hence, the first moment i.e. the mean of the distribution is, E(X) = Similarly, other higher order moments can be obtained.
(3.2) θ2 L1 (θ, 1, θ). (θ + 1)(e − 1)
3.2
Conditional Moments of the distribution
Again we derive another lemma to get the expression of the conditional moments as: Lemma 3.2. ∞
−θx 1 + θ + θx x (1 + x)e exp 1 − e dx L2 (θ, r, δ, t) = 1+θ t ∞ X k X l m+1 X X n+r X (−1)l k l m + 1 = k! l m n Z
×
Proof.
−δx
r
k=0 l=0 m=0 n=0 p=0 θm (n + r)!
(1 + θ)l p!(δ + θl)n+r+1
Z
∞ r
x (1 + x)e
L2 (θ, r, δ, t) = t
−δx
e−(δ+θl)t [(δ + θl) t]p .
−θx 1 + θ + θx exp 1 − e dx 1+θ
A similar procedure have been taken as in lemma 3.1, the expression can be written as: ∞ X k X l m+1 X X (−1)l k l m + 1 = k! l m n k=0 l=0 m=0 n=0 (3.3) Z ∞ θm × xr+n e−(δ+θl)x dx. (1 + θ)l t R∞ Here we use complementary incomplete gamma function Γ(a, x) = x ta−1 e−t dt which can be P l rewritten as (a − 1)!e−x a−1 l=0 x /l!. By using this function in equation (3.3) and after simplifica-
tion, we get ∞ X k X l m+1 X X n+r X (−1)l k l m + 1 L2 (θ, r, δ, t) = n m l k!
×
k=0 l=0 m=0 n=0 p=0 θm (n + r)!
(1 + θ)l p!(δ + θl)n+r+1
e−(δ+θl)t [(δ + θl) t]p .
By the lemma 3.2, the rth conditional moments can be obtained as, E(X r |X > x) =
3.3
θ2 L2 (θ, r, θ, x). (e − 1)(θ + 1)(1 − F (x))
(3.4)
Moment generating function, characteristics function and cumulant generating function of the distribution
The moment generating function (MGF) of random variable X of proposed distribution is given as follows: MX (t) =
θ2 L1 (θ, 0, θ − t) (e − 1)(θ + 1)
f or
t < θ.
(3.5)
Similarly the characteristic function (CHF) of X can be found as, φX (t) =
θ2 L1 (θ, 0, θ − it) (e − 1)(θ + 1)
(3.6)
where i =
3.4
√
−1 denotes imaginary and the cumulant generating function (CGF) of X is, θ2 + log L1 (θ, 0, θ − t). (3.7) KX (t) = log (e − 1)(θ + 1)
Mean deviation about mean and median of the distribution
The Mean deviation about mean and mean deviation about median are defined by, δ1 (X) = R∞ R∞ 0 (x − µ)f (x)dx and δ2 (X) = 0 (x − M )f (x)dx respectively, where µ denotes the mean and M denotes median. Then the mean deviation about mean is derived as, Z ∞ Z µ (x − µ)f (x)dx (µ − x)f (x)dx + δ1 (X) = µ
0
Using integral by part we have, Z
∞
δ1 (X) = 2µF (µ) − 2µ + 2
xf (x)dx, µ
where F (·) be the proposed cdf. Now by the lemma 3.2, Z ∞ θ2 f (x)dx = L2 (θ, 1, θ, µ) (e − 1)(θ + 1) µ and thus, δ1 (X) = 2µF (µ) − 2µ +
θ2 L2 (θ, 1, θ, µ). (e − 1)(θ + 1)
(3.8)
In the similar way the mean deviation about median is defined as, M
Z
Z
∞
(M − x)f (x)dx +
δ2 (X) = 0
(x − M )f (x)dx M
the steps are same as in the mean deviation about mean, we have Z ∞ δ2 (X) = −µ + 2 xf (x)dx. M
From the lemma 3.2, Z
∞
xf (x)dx = M
θ2 L2 (θ, 1, θ, M ). (e − 1)(θ + 1)
Hence, the mean deviation about median is δ2 (X) = −µ +
3.5
θ2 L2 (θ, 1, θ, M ). (e − 1)(θ + 1)
(3.9)
Quantile function of the distribution
The quantile function Q(p) of pth quantile is obtained by solving the equation F (Q(p)) = p. Hence from the equation (1.2), we have e−θQ(p)
1 + θ + θQ(p) = 1 − log(1 + p(e − 1)) 1+θ
(3.10)
for 0 < p < 1, we put Z(p) = −1−θ−θQ(p) in equation (3.10) and put T (p) = 1−log(1+p(e−1)), we get Z(p)eZ(p) = −(1 + θ)e−(1+θ) T (p) then solution for Z(p) is, h i Z(p) = W −(1 + θ)e−(1+θ) T (p)
(3.11)
where W (·) is Lambert W function, for more detail see (Corless, Gonnet, Hare, Jeffrey and Knuth, 1996). Hence, from equation (3.11), quantile function is, Q(p) = −1 −
3.6
i 1 1 h − W −(1 + θ)e−(1+θ) T (p) . θ θ
(3.12)
Order Statistics of the distribution
Let we take n random sample from the proposed distribution say, X1 , X2 , . . . , Xn and the corresponding order statistics is, X1:n < X2:n < · · · < Xn:n . Let F(x) and f(x) be the population cdf and pdf respectively, then for r = 1, 2, . . . , n the pdf fr (x) of rth order statistics Xr:n is, n! F r−1 (x)[1 − F (x)]n−r f (x) (r − 1)!(n − r)! n−r X n! i n−r = (−1) F r+i−1 (x)f (x). (r − 1)!(n − r)! i fr (x) =
(3.13)
i=0
Now by using equations (1.2) and (1.3) in (3.13) we have, n−r X θ2 1 + θ + θx −θx n! exp 1 − e fr (x) = (r − 1)!(n − r)! (1 + θ)(e − 1) 1+θ i=0 i h r+i−1 exp 1 − e−θx 1 + θx −1 1+θ n − r . × (−1)i (1 + x)e−θx i e−1
(3.14)
And corresponding rth order statistic of cdf Fr (x) is, n X n
F i (x)[1 − F (x)]n−i i i=r n n−i X X n n − i = (−1)j F i+j (x). i j Fr (x) =
(3.15)
i=r j=0
Using equation (1.2) in equation (3.15) we have, n X n−i X n n−i 1 + θ + θx −θx i+j Fr (x) = 1− e . i j 1+θ
(3.16)
i=r j=0
3.7
Entropies for the distribution
Entropy is a measure of the average amount of information contained in random variable X. Shannon entropy proposed by (Shannon, 1951) is defined as, E[− log f (x)]. θ2 1 + θ + θx −θx − log f (x) = − log − log(1 + x) + θx − 1 − e (1 + θ)(e − 1) 1+θ
(3.17)
and hence, E[− log f (x)] = − (log K + 1) + θKL1 (θ, 1, θ) − K
∞ X (−1)i+1 i=1
i
L1 (θ, i, θ) (3.18)
θ KL1 (θ, 1, 2θ). + KL1 (θ, 0, 2θ) + θ+1 θ2 and L1 (·, ·, ·, ·) has been define in lemma 3.1. Another entropy is (e − 1)(θ + 1) R 1 ´ entropy which is proposed by (Renyi, 1961) and defined as, JR (γ) = Renyi log[ f γ (x)dx] 1−γ where γ > 0 and γ 6= 1, which is generalization of Shannon entropy. Now using equation (1.3) where K =
we have, Z ∞ Z γ f (x)dx = 0
∞
0
θ2 (e − 1)(θ + 1)
γ
γ −γθx
(1 + x) e
1 + θ + θx −θx γ e dx exp 1 − 1+θ
(3.19)
this can be simplified as, Z
∞ γ
f (x)dx = 0
αθ2 (e − 1)(θ + 1)
γ X γi X j k+γ ∞ X X
(−1j )θk iγ i!(1 + θ)j j i=0 j=0 k=0 l=0 j k+γ l! × k l [(j + γ)θ]l+1
So that, γ θ2 1 JR (γ) = log + 1−γ (e − 1)(θ + 1) 1−γ iγ j ∞ XXX k+1 j )θ k X iγ j k + γ l! (−1 . × × log i!(1 + θ)j j k l [(j + γ)θ]l+1
(3.20)
i=0 j=0 k=0 l=0
4 4.1
Estimation procedure for the distribution Maximum likelihood estimation
In classical setup, we consider maximum likelihood estimation (MLE) method for the estimation of parameter θ of the proposed distribution which is obtained by maximizing the logarithm of the likelihood function. The logarithm likelihood function of the proposed distribution is, n
log L = n log
n
X X θ2 −θ xi + log(1 + xi ) (1 + θ)(e − 1) i=1 i=1 n X 1 + θ + θxi −θxi + 1− e . 1+θ
(4.1)
i=1
Differentiating it with respect to the parameter θ we get, n
d log L n(θ + 2) X θ = − xi + [(2 + θ) + xi (1 + θ)] xi e−θxi . dθ θ(θ + 1) (1 + θ)2
(4.2)
i=1
Now equating the equation (4.2) to zero, we have a nonlinear equation and after solving this we get MLE θˆ of parameter θ. Since this equation is not in closed form and cannot be solved
analytically. So we have to use some numerical technique for the solution. Here, we propose to the use of Newton-Raphson method. In the case of choice of the initial guess, contour plot method is used (see (Kaushik, Singh and Singh, 2016) for more detail). In the case of large samples, we can obtain the confidence intervals based on the diagonal elements of Fisher ˆ which provides the estimated asymptotic variance for the parameter information matrix I −1 (θ) q ˆ θ. Thus, two-sided 100(1 − η)% confidence interval of θ can be defined as θˆ ± Zη /2 var(θ), where Zη /2 stands for the upper η/2% points of standard normal distribution. The Fisher Information matrix can be estimated by, 2 −d log L ˆ I(θ) = dθ2 θˆ
(4.3)
where, n d2 log L n(θ + 4θ + 2) X (θxi )2 2 θ−1 −θxi =− + xi + . xi e θxi + + dθ2 (θ(θ + 1))2 (θ + 1)2 θ+1 (θ + 1)3 i=1
4.2
Bayesian estimation
An essential element in Bayesian estimation problem is the specification of the loss function. The choice depends on the problem in hand. For more discussion on the selection of a suitable loss function, readers may refer to (Singh, Singh and Kumar, 2011) and (Singh, Singh and Yadav, 2014). Another important element is the choice of the appropriate prior distribution that covers all the prior knowledge regarding the parameter of interest. For the criteria of choosing an appropriate prior distribution, see (Singh, Singh and Kumar, 2013). With the above philosophical point of view, we are motivated to take the prior for θ as Gamma(α, β) distribution with the pdf π(θ) =
αβ −αθ β−1 e θ Γβ
θ > 0,
(4.4)
where α > 0 and β > 0 are the hyper parameters. These can be obtained, if any two independent informations on θ are available, say prior mean and prior variance are known (see also (Singh et al., 2013) and (Singh, Singh and Kumar, 2014)). The mean and variance of the prior distribution defined in equation (4.4) are M =
β α
and V =
β α2
giving α =
M 2 /V
β α
and
β α2
respectively. Thus, we may take
and β = M/V . For any finite value of M and V to be
sufficiently large prior defined in equation (4.4) behaves like as non-informative prior. For more applications regarding the use of gamma prior, readers may refer to (Singh, Singh, Yadav and Viswkarma, 2015). The posterior pdf of θ given X corresponding to the considered prior pdf π(θ) of θ is given by L(θ)π(θ) ψ(θ|X) = R ∞ 0 L(θ)π(θ)dθ
(4.5)
hence h i P P β+2n−1 i + xi ) θ(1+θ)n exp n − θ(α + ni=1 xi ) − ni=1 e−θxi 1+θ+θx 1+θ h ψ(θ|X) = R Q Pn Pn −θx 1+θ+θxi i ∞ n θβ+2n−1 i dθ i=1 (1 + xi ) (1+θ)n exp n − θ(α + i=1 xi ) − i=1 e 1+θ 0 Qn
i=1 (1
(4.6)
The loss functions considered here are general entropy loss function (GELF) and squared error loss function (SELF), which are defined by, ˆ θ) = LG (θ, and
θˆ θ
!δ
θˆ θ
− δ log
! −1
(4.7)
2 ˆ θ) = θˆ − θ LS (θ,
(4.8)
respectively. The Bayes estimator of θ under GELF given in equation (4.7) and under SELF given in equation (4.8) are given by h i −1 δ θˆG = E(θ−δ |data)
(4.9)
θˆS = E(θ|data)
(4.10)
respectively. It is easy to see that when δ = −1, the Bayes estimator (4.9) under GELF reduces to the Bayes estimator (4.10) under SELF. Now, the Bayes estimator of the parameter θ of proposed distribution having pdf defined in equation (1.3) under GELF is obtained as follows h R ∞ Q Pn Pn −θxi 1+θ+θxi i −1/δ n θβ+2n−δ−1 exp n − θ(α + x ) − dθ (1 + x ) n i i i=1 i=1 i=1 e 1+θ 0 (1+θ) h i θˆG = R Q P P ∞ n n n θβ+2n−1 −θxi 1+θ+θxi dθ exp n − θ(α + x ) − e (1 + x ) i (1+θ)n i=1 i i=1 i=1 1+θ 0 (4.11) Further if θˆS denotes the Bayes estimator of θ under SELF, then it can be obtained by putting δ = −1 in equation (4.11) and therefore the same is given by, h R ∞ Qn Pn −θxi 1+θ+θxi i Pn θβ+2n (1 + x ) x ) − exp n − θ(α + dθ n i i i=1 e i=1 i=1 1+θ 0 (1+θ) h i . θˆS = R Q P P ∞ n n n θβ+2n−1 −θxi 1+θ+θxi dθ i=1 e i=1 xi ) − i=1 (1 + xi ) (1+θ)n exp n − θ(α + 1+θ 0
(4.12)
The integral involved in Bayes estimators do not solve analytically. In such a situation, MCMC methods namely Metropolis-Hastings algorithm (see (Hastings, 1970)) can be effectively used. After getting MCMC samples from posterior distribution, we can find the Bayes estimate for the parameters in the following way h
i −1 δ E(θ−δ |data) =
1 N − N0
N X
−1/δ θi−δ
i=N0 +1
where N0 is burn-in period of Markov chain. To obtain the MCMC samples from the posterior density of θ, using the Metropolis-Hastings (M-H) algorithm, we have considered a normal distribution as the proposal density i.e. N (µ, Σ) where Σ is the variance-covariance matrix. It may be the point here that, if we generate observation from the normal distribution, we may get negative values also which are not possible as the parameters under consideration are positive valued. Therefore, we take the absolute value of generated observation. Following this, the Metropolis-Hastings algorithm associated with the target density π(·) and the proposal density q(·) = N (µ, Σ) produces a Markov chain θi through the following steps:
Step 1. Set the initial guess of θ say θ0 . Set i = 1 Step 2. θ0 ∼ q(θ0 |θi−1 ) Step 3. r ←
π(θ0 )q(θi−1 |θ0 ) π(θi−1 )q(θ0 |θi−1 )
Step 4. u ∼ U nif (0, 1) If u < r θi ← θ0 Else
θi ← θi−1
Set i = i+1 Step 5. Repeat Step 2-4 a large number N times. i− 1 − 1 h 1 PN δ −δ θ , Obtain the Bayes estimates of θ with under GELF as E(θ−δ |data) δ = N −N i=N0 +1 i 0 where N0 is the burn in period. Substituting δ = −1, we get Bayes estimates of θ under SELF. In using the above algorithm, the problem arises how to choose the initial guess. Here, we propose the use of MLE of θ, obtained by using the method described in subsection 4.1, as an initial value for MCMC process. The choice of covariance matrix Σ is also an important issue, see (Natzoufras, 2009) for details. One choice for Σ would be the asymptotic varianceˆ While generating M-H samples by taking Σ = I −1 (θ), ˆ we noted that covariance matrix I −1 (θ). the acceptance rate for such a choice of Σ is about 27%. By acceptance rate, we mean the proportion of times a new set of values is generated at the iteration stages. It is well known that, if the acceptance rate is low, a good strategy is to run a small pilot run using diagonal Σ as a rough estimate of the correlation structure for the target posterior distribution. And then, repeat the algorithm using the corresponding estimated variance-covariance matrix; for more detail see (Gelmen, Carlin, Stern and Rubin, 1995, pp. 334-335) and (Kaushik et al., 2016). Therefore, we have also used the following described strategy for the calculations in the section 6.
5
Real Data Application
In the present section, we use a real data set of Yarn cycle failure data to show the applicability of proposed distribution. Further, we have compared the proposed distribution with exponential, gamma, Weibull and Lindley distributions Inverse Lindley (IL), DUS exponential (DU SE ) and generalised Lindley distributions. For model selection criterion we used AIC (Akaike Information Criterion), BIC (Bayesian information criterion) and K-S test regarding to fitting, which is defined as: ˆ AIC = 2 ∗ k − 2 ∗ log L,
ˆ BIC = k ∗ log(n) − 2 ∗ log L,
and K-S statistics is defined as n
KS = Sup |Fn (x) − F (x)|, x
where
1X Fn (x) = Ixi ≤x n i=1
is empirical distribution function, F(x) is cdf, n is sample size, k is the number of parameters ˆ is the maximum likelihood for the considered distribution. For the indication of better and L
fit of distributions, we have a smaller value of AIC, BIC and the K-S test statistic. We have also calculated the MLEs of parameters for various distributions. Here we considered the cycle of failure of yarn data for application purpose. This data set is taken from (Picciotto and Hersh, 1972) which arose in a test on the numbers of cycles at which the yarn failed. The data shows the number of cycles until failure of the yarn. This data has been analysed by (Shanker, Hagos and Sujatha, 2015) and so on. The value of AIC, BIC and K-S statistics along with associated p-value and maximum likelihood estimates of parameters by the considered real data set of compared distributions are present in table 1. We found that out of all the above eight distributions only Lindley, generalised Lindley, gamma and ETL fit to the data at 5% level of significance. However, on the basis of the value of K-S statistics or associated p-value, we may say that ETL fits better than other compared distributions. It is also noted that ETL distribution has smaller AIC and BIC in comparison to other seven considered distributions. Hence, we conclude that ETL distribution is the most suitable model for the present data as compared to other seven distributions. The empirical cumulative distribution function (ecdf) and fitted cdf plot, for the considered dataset, has been shown in figure 2. This figure also shows that the proposed ETL distribution is the better fit to the considered real data in comparison to other competitive distributions. For the purpose Bayesian estimation for the considered real data set, we used MCMC technique discussed in section 4.2. Figure 3, shows the iterations and density plot of samples generated from the posterior distribution using MCMC method. In this figure, the green line represents the ergodic mean of generated sample; the red line shows the average of generated sample after discarding first 5000 observations. The acceptance rate of our MCMC chain is 46%, which is sufficient percentage. From this figure, we see that chain has converged and are well mixed. It is, further, noted that the posterior of θ is approximately symmetric. Utilizing these MCMC samples, we computed Bayes estimates, following the method discussed in section 4.2, and got θˆS = 0.0107 under non-informative prior.
6
Comparison of the estimators
In this section, we compare the considered estimators i.e. θˆM , θˆS , θˆG of the parameter θ of ETL distribution having pdf defined in equation (1.3) in terms of posterior risks (average loss over sample space) under GELF. It is clear that the expressions for the risks cannot be obtained in closed form. It may be noted that the risks of the estimators will be a function of the hyperparameters α and β of the prior distribution and the GELF parameter δ. In order to consider the variation of these values, we obtained the simulated risks for δ = ±3, M = 0.1, 0.2, 0.3 and V = 0.25, 0.5, 1, 2, 5, 10, 50, 100, 500, 1000. Table 2, provide the risk of different estimates under GELF for fixed choice of prior mean 0.01 and various choice of prior variance for real data set. Table 3, provide the risk of different estimates under GELF for fixed choice of prior mean is 0.02 and table 4, provide the risk when prior mean is 0.03. Under noninformative prior our obtained the risk under GELF are RG (θˆM ) = 0.01839, RG (θˆS ) = 0.01844, RG (θˆG ) = 0.01831, RG (θˆM ) = 0.01869, RG (θˆS ) = 0.01862 and RG (θˆG ) = 0.0184. From Tables 2-4, we observed that when overestimation is more serious than underestimation, the estimator
F(x, th)
0.4
0.3
theta=2 theta=1 theta=.5
0.0
0.0
0.1
0.2
0.2
f(x, th)
0.4
theta=.5 theta=1 theta=1.5
0.6
0.8
0.5
1.0
Cumulative distribution function plot
0.6
Probability density function plot
0
2
4
6
8
10
0
x
5
10
15
20
x
Figure 1: Probability density and cumulative distribution function plot θˆG performs better (in the sense of having the smallest risk) in comparison to θˆS and θˆM for whatever confidence in the prior of θ as its actual value. But if the guessed value of θ is either more/less than its true value, the estimator θˆG performs well for lower confidence in such guessed value of θ. Otherwise θˆM performs better (when guessed value of θ is less than its true value) or θˆS performs better (when guessed value of θ is more than its true value). Further, when underestimation is more serious than overestimation, for whatever be the confidence in the guessed value of θ for less than its true value, the estimator θˆG performs better than the estimators θˆS and θˆM . But when guessed value of θ is same as its true value, the estimator θˆG performs better for lower confidence. Otherwise θˆM performs better and when guessed value of θ is more than its true value, the estimator θˆM performs well for higher confidence; θˆS , performs batter for moderate confidence and for lower confidence value, the estimator θˆG performs better. Table 1: MLE, AIC, BIC and K-S statistics with p-value for fitted data sets. Yarn Cycle Failure Data K-S Test
ML Estimate
Distribution
AIC
BIC
P-value
Statistics
θ
α
Exponential
1282.517
1285.123
0.001
0.200
0.005
-
DUS Exponential
1269.221
1271.826
0.009
0.164
0.006
-
Lindley
1253.341
1255.946
0.193
0.108
0.009
-
ETL
1251.155
1253.761
0.562
0.079
0.011
-
Gamma
1254.489
1259.699
0.331
0.095
2.239
99.156
Generalized Lindley
1254.561
1259.771
0.316
0.096
0.010
1.141
Inverse Lindley
1308.590
1311.195
0.001
0.190
121.635
-
Weibull
1254.398
1259.608
0.000
1.000
1.604
247.916
1.5
1.0
Hazard rate plot
0.8 0.6 0.4 0.0
0.0
0.2
0.5
h(x, th)
Fitted Distribution
1.0
theta=0.2 theta=0.6 theta=1 theta=1.5
0
5
10 x
15
20
● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0
200
●● ● ● ● ●● ● ● ● ●● ● ●
●
●
●● ●
●
Exponential DUS Exponential Lindley DUS Lindley Gamma Generalised Lindley Inverse Lindley Weibull
400
600
800
1000
x
Figure 2: Hazard rate function plot for various choice of θ and empirical cdf with fitted cdf plot of considered data set.
Figure 3: Iteration and density plot of MCMC samples for real data set with non-informative prior.
Table 2: Risks of the estimators of θ under GELF for real data set with fixed M = 0.01 and δ = ±3. RG (θˆM )
δ = −3 RG (θˆS )
0.25
0.01767
0.5
RG (θˆG )
δ = +3 RG (θˆM ) RG (θˆS )
RG (θˆG )
0.01769
0.01766
0.01808
0.01799
0.01761
0.01861
0.01868
0.01844
0.01869
0.01864
0.01852
1
0.0172
0.01726
0.01706
0.0173
0.01725
0.01714
2
0.01951
0.01958
0.01936
0.01981
0.01974
0.01957
5
0.01779
0.01784
0.01771
0.01825
0.01818
0.01798
10
0.01726
0.0173
0.01722
0.01748
0.0174
0.01715
50
0.01802
0.01804
0.018
0.01892
0.01882
0.01846
100
0.01876
0.01882
0.01857
0.01889
0.01883
0.01872
500
0.01688
0.01691
0.01686
0.01742
0.01734
0.01706
1000
0.01871
0.01872
0.01871
0.01928
0.01917
0.01864
V
Table 3: Risks of the estimators of θ under GELF for real data set with fix M = 0.02 and δ = ±3. δ = −3 δ = +3 V RG (θˆM ) RG (θˆS ) RG (θˆG ) RG (θˆM ) RG (θˆS ) RG (θˆG ) 0.25
0.01712
0.01718
0.01695
0.01709
0.01704
0.01696
0.5
0.01679
0.01685
0.01667
0.01742
0.01736
0.01725
1
0.01876
0.01878
0.01875
0.0197
0.01959
0.01915
2
0.01834
0.01838
0.01829
0.01892
0.01883
0.01855
5
0.01708
0.01714
0.01697
0.0172
0.01714
0.01701
10
0.0174
0.01742
0.01739
0.01826
0.01817
0.01778
50
0.01618
0.01621
0.01614
0.01646
0.01638
0.01616
100
0.01894
0.01895
0.01894
0.02007
0.01995
0.01937
500
0.01725
0.0173
0.01714
0.01738
0.01732
0.01719
1000
0.01728
0.01732
0.01722
0.01754
0.01746
0.01725
Table 4: Risks of the estimators of θ under GELF for real data set with fixed M = 0.03 and δ = ±3.
7
RG (θˆM )
δ = −3 RG (θˆS )
0.25
0.01848
0.5
RG (θˆG )
δ = +3 RG (θˆM ) RG (θˆS )
RG (θˆG )
0.01854
0.01833
0.01853
0.01847
0.01834
0.01808
0.01808
0.01808
0.0191
0.01899
0.0184
1
0.01658
0.01661
0.01655
0.0171
0.01702
0.01678
2
0.01836
0.01841
0.01827
0.01864
0.01857
0.01837
5
0.0188
0.01886
0.01868
0.01906
0.01899
0.01881
10
0.01954
0.01958
0.01949
0.02001
0.01992
0.01959
50
0.01793
0.01797
0.01789
0.01844
0.01836
0.01808
100
0.01922
0.01925
0.01919
0.01999
0.01989
0.0195
500
0.01799
0.01805
0.0179
0.01831
0.01824
0.01806
1000
0.02024
0.02029
0.02018
0.02051
0.02042
0.0201
V
Conclusion
In this article, we propose a new lifetime distribution, capable of having increasing hazard rate. Here, also, we derived various statistical properties namely moments, conditional mo` and ments, MGF, CHF, CGF, mean deviation about mean and median, quantile function, Renyi Shannon entropies, distribution of rth order statistic. For the estimation of the parameter, we discussed the method of ML and Bayesian procedure. For the purpose of Bayesian inferences, we used MCMC technique under SELF and GELF. 100(1 − η)% asymptotic confidence interval and Fisher information has also been discussed. Further, we consider a real data set of yarn cycle failure data and seven other distributions namely exponential, DU SE , Lindley, inverse Lindley, generalised Lindley, gamma and Weibull distribution. It is shown that ETL fits the considered real data set very well, even best than other distributions. Hence, we can easily conclude that our proposed (ETL) distribution is more flexible and in this way be one may consider as a suitable model for lifetime data.
Acknowledgment The authors would like to thank the reviewer for constructive and pertinent comments, also thankful the referees for their valuable suggestions that improvised the original version of the manuscript. The corresponding author would like to express gratitude to Council of Scientific and Industrial Research (CSIR); New Delhi, India, for providing financial assistance to this work.
References Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. and Knuth, D. E. 1996. On the lambertw function, Advances in Computational mathematics 5(1): 329–359. Deniz, E. G. and Ojeda, E. C. 2011. The discrete lindley distribution: properties and applications, Journal of Statistical Computation and Simulation 81(11): 1405–1416. Elbatal, I., Merovci, F. and Elgarhy, M. 2013. A new generalized lindley distribution., Journal of Mathematical Theory and Modeling 3(13): 30–47. Gelmen, A., Carlin, J., Stern, H. and Rubin, D. 1995. Bayesian Data Analysis, Text in Statistical Science, Chapman & Hall. Ghitany, M. E. and Al-Mutairi, D. K. 2008. Size-biased poisson-lindley distribution and its application, Metron-International Journal of Statistics 66(3): 299–311. Ghitany, M. E., Al-Mutairi, D. K. and Nadarajah, S. 2008. Zero-truncated poisson–lindley distribution and its application, Mathematics and Computers in Simulation 79(3): 279–287. Ghitany, M. E., Atieh, B. and Nadarajah, S. 2008. Lindley distribution and its application, Mathematics and computers in simulation 78(4): 493–506. Glaser, R. E. 1980. Bathtub and related failure rate characterizations, Journal of the American Statistical Association 75: 667–672. Graham, R. L., Knuth, D. E. and Patashnik, O. 1989. Concrete mathematics. Hassan, A. S., Elbatal, I. and Hemeda, S. E. 2016. Weibull quasi lindley distribution and its statistical properties with applications to lifetime data, International Journal of Applied Mathematics and Statistics 55(3): 63–80. Hastings, W. K. 1970. Monte carlo sampling methods using markov chains and their applications, Biometrika 57(1): 97–109. Kaushik, A., Singh, U. and Singh, S. K. 2016. Bayesian inference for the parameters of weibull distribution under progressive type-i interval censored data with beta-binomial removals, Communications in Statistics - Simulation and Computation 0(ja): 0–0. Kumar, D., Singh, U. and Singh, S. K. 2015. A method of proposing new distribution and its application to bladder cancer patient data., Journal of Statistics Applications and Probability Letters 2: 235–245. Maurya, S. K., Kaushik, A., Singh, R. K., Singh, S. K. and Singh, U. 2016. A new method of proposing distribution and its application to real data, Imperial Journal of Interdisciplinary Research 2(6). Nadarajah, S., Bakouch, H. S. and Tahmasbi, R. 2011. A generalized lindley distribution, Sankhya B 73(2): 331–359.
Natzoufras, I. 2009. Bayesian Modeling Using WinBugs, Series B, Wiley. Picciotto, R. and Hersh, S. P. 1972. The tensile fatigue behavior of a warp yarn and its influence on weaving performance, Textile Research Journal 42(9): 512–522. Rashid, A. and Jan, T. 2016. A new family of lifetime distributions with real life applications, International Journal of Mathematics and Statistics 17(1): 23–38. Renyi, A. 1961. On measures of entropy and information, In Proceedings of the 4th Berkeley symposium on mathematical statistics and probability. Berkeley :University of California Press. 1: 547–561. Shanker, R., Hagos, F. and Sujatha, S. 2015. On modeling of lifetimes data using exponential and lindley distributions, Biometrics and Biostatistics International Journal 2(5): 1–9. Shannon, C. E. 1951. Prediction and entropy of printed english, The Bell System Technical Journal 30: 50–64. Singh, S. K., Singh, U. and Kumar, D. 2011. Estimation of parameters and reliability function of exponentiated exponential distribution: Bayesian approach under general entropy loss function, Pakistan journal of statistics and operation research 7(2): 119–216. Singh, S. K., Singh, U. and Kumar, D. 2013. Bayes estimators of the reliability function and parameter of inverted exponential distribution using informative and non-informative priors, Journal of Statistical Computation and Simulation 83(12): 2258–2269. Singh, S. K., Singh, U. and Kumar, D. 2014. Bayesian estimation of the parameters of generalized inverted exponential distribution, International Journal of Statistics & Economics 13(1): 57–69. Singh, S. K., Singh, U., Yadav, A. S. and Viswkarma, P. K. 2015. On the estimation of stress strength reliability parameter of inverted exponential distribution, International Journal of Scientific World 3(1): 98–112. Singh, S., Singh, U. and Yadav, A. S. 2014. Bayesian reliability estimation in extension of exponential distribution for progressive type ii censored data with binomial removals using different loss functions, International Journal of Statistics & Economics 13(1): 19–39.
