Int. J. Open Problems Compt. Math., Vol. 6, No. 2, June 2013 c ISSN 1998-6262; Copyright ICSRS Publication, 2013 www.i-csrs.org
Transmuted Lindley Distribution Faton Merovci Department of Mathematics University of Prishtina Prishtin¨e 10000 Republic of Kosovo e-mail:
[email protected]
Abstract In this article, we generalize the Lindley distribution using the quadratic rank transmutation map studied by Shaw et al. [8] to develop a transmuted Lindley distribution. We provide a comprehensive description of the mathematical properties of the subject distribution along with its reliability behavior. The usefulness of the transmuted Lindley distribution for modeling reliability data is illustrated using real data.
Keywords: Lindley Distribution, hazard rate function, reliability function, parameter estimation. 2010 Mathematics Subject Classification: 62N05, 90B25.
1
Introduction
In many applied sciences such as medicine, engineering and finance, amongst others, modeling and analyzing lifetime data are crucial. Several lifetime distributions have been used to model such kinds of data. The quality of the procedures used in a statistical analysis depends heavily on the assumed probability model or distributions. Because of this, considerable effort has been expended in the development of large classes of standard probability distributions along with revelent statistical methodologies. However, there still remain many important problems where the real data does not follow any of the classical or standard probability models. In this article we present a new generalization of Lindley distribution called the transmuted Lindley distribution.
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Definition 1.1 A random variable X is said to have transmuted distribution if its cumulative distribution function(cdf ) is given by F (x) = (1 + λ)G(x) − λG2 (x), |λ| ≤ 1.
(1)
where G(x) is the cdf of the base distribution. Observe that at λ = 0 we have the distribution of the base random variable. Aryal et al. [3] studied the the transmuted Gumbel distribution and it has been observed that transmuted Gumbel distribution can be used to model climate data. In the present study we will provide mathematical formulation of the transmuted Lindley distribution and some of its properties.
2
Transmuted Lindley Distribution
Definition 2.1 A random variable X is said to have the Lindley distribution with parameter θ if its probability density is defined as f (x, θ) =
θ2 (1 + x)e−θx , θ+1
x > 0, θ > 0.
(2)
The corresponding cumulative distribution function (c.d.f.) is: F (x) = 1 −
θ + 1 + θx −θx e , θ+1
x > 0, θ > 0
(3)
Now using (1) and (3) we have the cdf of a transmuted Lindley distribution θ + 1 + θx −θx θ + 1 + θx −θx F1 (x) = 1 − e 1+λ e θ+1 θ+1
(4)
Hence, the pdf of transmuted Lindley distribution with parameter λ is θ2 θ + 1 + θx −θx −θx f1 (x) = 1 − λ + 2λ (1 + x)e e θ+1 θ+1
(5)
Note that the transmuted Lindley distribution is an extended model to analyze more complex data and it generalizes some of the widely used distributios.The Lindley distribution is clearly a special case for λ = 0. Figure 1 illustrates some of the possible shapes of the pdf of a transmuted Lindley distribution for selected values of the parameters λ and θ. Figure 2 illustrates some of the possible shapes of the CDF of a transmuted Lindley distribution for selected values of the parameter λ by keeping θ = 1.
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1.5
Transmuted Lindley Distribution
θ=2 θ=2 θ = 0.5 θ = 0.5 θ=2
0.0
0.5
f(x)
1.0
λ=0 λ = −1 λ = − 0.5 λ = 0.5 λ=1
0.0
0.5
1.0
1.5
2.0
2.5
3.0
x
0.6 0.2
0.4
λ = −1 λ = − 0.8 λ = − 0.4 λ = 0.4 λ = 0.8 λ=1
0.0
F(x)− CDF
0.8
1.0
Figure 1: Pdf of Transmuted Lindley distribution
0
1
2
3
4
5
x
Figure 2: CDF of Transmuted Lindley distribution
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Faton Merovci
Moments
Now let us consider the different moments of the transmuted Lindley distribution. Suppose X denote the transmuted Lindley distribution random variable with parameter θand λ, then θ2 E(X k ) = θ+1
Z∞
k
−θx
x (1 + x)e
0
= Here we use
R∞
θ + 1 + θx −θx 1 − λ + 2λ e θ+1
k! λ·θ (1 − λ)(θ + k + 1) + (2θ + 3θ + k) θk (θ + 1) 2k−1 (θ + 1)
xk e−θx dx = k!θ−k−1 (see [9], 2.3.3.2 page 322) Therefore putting
0
k = 1, we obtain the mean as E(X) =
1 2λ · θ (1 − λ)(θ + 2) + (θ + 2) θ(θ + 1) (θ + 1)
(6)
and putting k = 2 we obtain the second moment as E(X 2 ) =
λ·θ 2 (1 − λ)(θ + 3) + (2θ + 5) θ2 (θ + 1) 2(θ + 1)
(7)
The maximum likelihood estimates, MLE’s, of the parameters that are inherent within the transmuted exponentiated Lindley probability distribution function is given by the following: n P
n
θxi Y − θ2n θ + 1 + θxi −θxi i=1 L= (1 + x ) · 1 − λ + 2λ e e i (θ + 1)n θ+1 i=1
ln L = 2n ln θ − n ln(θ + 1) − θ
n X
xi +
i=1
+
n X
ln(1 − λ + 2λ
i=1
Now setting
n X
ln(1 + xi )
i=1
θ + 1 + θxi −θxi e ) θ+1
∂ ln L = 0 and ∂θ
∂ ln L = 0, ∂λ
we have 0=
2n n − − θ θ+1
k X i=1
xi +
k X i=1
−θxi
2λxi e
h
1 (θ+1)2
−
θ+1+θxi θ+1
i −θxi 1 − λ + 2λ θ+1+θx e θ+1
i
(8)
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Transmuted Lindley Distribution
0=
n X i=1
i −θxi e −1 2 θ+1+θx θ+1 i −θxi e 1 − λ + 2λ θ+1+θx θ+1
ˆ λ) ˆ ′ of β = (θ, λ)′ is obtained by The maximum likelihood estimator βˆ = (θ, solving this nonlinear system of equations. It is usually more convenient to use nonlinear optimization algorithms such as quasi-Newton algorithm to numerically maximize the log-likelihood function given in (8).In order to compute the standard error and asymptotic confidence interval we use the usual large sample approximation in which the maximum likelihood estimators of θ can be treated as being approximately trivariate normal. Hence as n → ∞ the ˆ λ) ˆ is given by, see Zaindin et al. [10], asymptotic distribution of the MLE (θ, θˆ θ Vˆ11 Vˆ12 , ˆ ∼N λ Vˆ21 Vˆ22 λ where, Vij = Vij|θ=θˆ and
V11 V12 V21 V22
=
A11 A12 A21 A22
−1
is the approximate variance covariance matrix with its elements obtained from A11 = −
∂ 2 ln L , ∂α2
A12 = −
∂ 2 ln L , ∂α∂β
∂ 2 ln L . ∂β 2 where ln L is the log-likelihood function given in (??). An approximate 100(1− α)% two sided confidence intervals for α, β and λ are, respectively, given by q q ˆ ˆ ˆ θ ± zα/2 V11 , and λ ± zα/2 Vˆ22 A22 = −
where zα is the upper α−th percentiles of the standard normal distribution. Using R we can easily compute the Hessian matrix and its inverse and hence the values of the standard error and asymptotic confidence intervals.
4
Reliability Analysis
The reliability function R(x), which is the probability of an item not failing prior to some time t, is defined by R(x) = 1 − F (x). The reliability function of a transmuted Lindley distribution is given by θ + 1 + θx −θx θ + 1 + θx −θx R(x) = 1−λ+λ (9) e e θ+1 θ+1
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Faton Merovci
0.6 0.4 0.0
0.2
R(t)− Reliability function
0.8
λ = −1 λ = − 0.8 λ = − 0.4 λ = 0.4 λ = 0.8 λ=1
0
1
2
3
4
5
t
Figure 3: The reliability function of a transmuted Lindley distribution The other characteristic of interest of a random variable is the hazard rate function defined by f (x) h(x) = 1 − F (x) which is an important quantity characterizing life phenomenon. It can be loosely interpreted as the conditional probability of failure, given it has survived to the time t. The hazard rate function for a transmuted Lindley random variable is given by h(x) =
θ+1+θx θ2 (1 + x) 1 − λ + 2λ θ+1 e−θx θ + 1 + θx λ − 1 − λ θ+1+θx e−θx θ+1
(10)
Figure 2 illustrates the reliability behavior behavior of a transmuted Lindley distribution as the value of the parameter λ varies from −1 to 1, keeping θ = 1.
The hazard rate function h(x) of a transmuted Lindley distribution has the following properties:
Transmuted Lindley Distribution
69
• if λ = 1, the failure rate is decreasing . 2
θ • h(x) tends to − θ+1 (1+λ) and −θ where x → 0 and x → ∞, respectively.
Proof. For λ = 1, we have θ2 (1 + x) h(x) = −2 θ + 1 + θx The derivation of h(x) with respect to x is given by 2 ∂h(x) θ = −2 0, and parameters θ > 0 and 0 < p < 1. Is it transmuted Lindley geometric distribution better than transmuted Lindley distribution for fitting data?
References [1] Akaike, H. A New Look at the Statistical Model Identification, I.E.E.E. Transactions on Automatic Control, AC (1974) 19 , 716- 723. [2] Akaike, H. Canonical Correlation Analysis of Time Series and the Use of an Information Criterion, in R. K. Mehra and D. G. Lainotis (eds.), System Identification: Advances and Case Studies, Academic Press, New York,(1976) 52- 107. [3] Aryal G.R. and Chris P. Tsokos. On the transmuted extreme value distribution with application. Nonlinear Analysis: Theory, Methods and Applications, (2009) 71: 1401 1407. [4] Aryal G.R., Tsokos Ch. P., Transmuted Weibull Distribution: A Generalization of the Weibull probability distribution,Europian journal of pure and applied mathemtics (2011) Vol. 4, No. 2, 89- 102.
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[5] Ghitany M.E., Atieh B. and Nadarajah S., Lindley distribution and its application,Mathematics and Computers in Simulation 78 (2008) 493506 [6] Lindley D.V. , Fiducial distributions and Bayes theorem, Journal of the Royal Statistical Society, Series B 20 (1958) 102107. [7] Lindley D.V. , Introduction to Probability and Statistics from a Bayesian Viewpoint, Part II: Inference, Cambridge University Press, New York, 1965. [8] Shaw W., Buckley I., The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map.Research report, (2007). [9] Prudnikov, A. P., Brychkov, Y. A., Marichev, O. I. Integrals and series. Volumes 1, Gordon and Breach Science Publishers, Amsterdam (1986). [10] Zaindin M. and Sarhan A. M., Parameters Estimation of the Modified Weibull Distribution. Applied Mathematical Sciences, (2009) 3, 11, 541 550. [11] Zakerzadeh H.and Mahmoudi E.,A new two parameter lifetime distribution: model and properties, Arxiv, (2012). [12] Lee E.T. and Wang J.W., Statistical Methods for Survival Data Analysis, 3rd ed.,Wiley, NewYork, 2003.
