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On gamma Lindley distribution: Properties and simulations
Accepted Manuscript On gamma Lindley distribution: Proprieties and simulations Sihem Nedjar, Halim Zeghdoudi PII: DOI: Reference:
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Journal of Computational and Applied Mathematics
Received date: 23 October 2015 Revised date: 15 November 2015 Please cite this article as: S. Nedjar, H. Zeghdoudi, On gamma Lindley distribution: Proprieties and simulations, Journal of Computational and Applied Mathematics (2015), http://dx.doi.org/10.1016/j.cam.2015.11.047 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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On Gamma Lindley Distribution : Proprieties and Simulations Sihem Nedjar1 and Halim Zeghdoudi1 LaPS laboratory, Badji-Mokhtar University, Box 12, Annaba, 23000,ALGERIA [email protected], [email protected] 1
summary A new distribution is recently proposed by Zeghdoudi and Nedjar, called as Gamma Lindley distribution (GaL). This paper proposes more properties and simulations which o¤ers a more ‡exible model for modelling lifetime data. Various statistical properties like the quantile function, Lorenz curve, moment method, maximum likelihood estimation, entropy, and limiting distribution of extreme order statistics are established. A simulation study is carried out to examine the bias and mean square error of the maximum likelihood estimators of the parameters. An application of the model to a real data set is presented …nally and compared with the …t attained by some other well-known two(three)-parameter distributions. Keywords : Lindley distribution, gamma lindley distribution, maximumlikelihood estimation Mathematics Subject Classi…cation: 60E05;62H10
1
Introduction
The real-life applications of contemporary numerical techniques in di¤erent …elds such as medicine, …nance, biological engineering sciences and statistics see (Taormina et al. (2015), Wang et al. (2015), Wu et al.(2009), Chau et al.(2010), S.W. Zhang et al.(2009) and J. Zhang et al.(2009)). To this end, statistics plays a crucial role in real life applications. Often by using the statistical analysis which strongly depends on the assumed probability model or distributions. However, several problems in statistics does not follow any of the classical or standard probability models. Let X be a random variable following the one-parameter distribution with
1
the density function f (x; ) =
8 < :
2
x
(1 + x)e 1+ 0;
x; > 0
(1)
otherwise
introduced by Lindley (1958). Sankaran (1970) used (1) as mixing distribution of Poisson parameter which it named Poisson- Lindley distribution. Recently, Asgharzadeh et al. (2013), Elbatal et al. (2013), Bakouch et al.(2012), Ghitany et al. (2008a) and (2008b) rediscovered and studied the new distribution bounded to (1), what they derived is known as Zero-truncated Poisson- Lindley and Pareto Poisson-Lindley distributions. Recently, Zeghdoudi and Nedjar (2015) introduced a new distribution, named gamma Lindley distribution, based on mixtures of gamma (2; ) and one-parameter Lindley distributions.Various properties of the distribution are examined including the density function (pdf), cumulative distribution (cdf), survival and hazard rate function, moment generating function (mgf), mean, variance and some results on stochastic orderings. Plots of pdf and cdf for some parameter values are also given. Moment estimates are discussed and two examples are presented to show the goodness of …t. The idea using a mixture of two known distribution to generate a new distribution is not new. For example, Shanker, Sharma and Shanker (2013) used a mixture of exponential ( ) and gamma (2; ) to create a two-parameter Lindley distribution. For another example, Zakerzadeh and Dolati (2009) used gamma ( ; ) and gamma ( + 1; ) to create a generalized Lindley distribution. However, it is the …rst time in the literature to use a mixture of gamma (2; ) and one-parameter Lindley distribution to generate Gamma Lindley distribution, where the density function of the random variable X is given by:
fGaL (x; ; ) =
8 < :
2
(( +
)x + 1)e (1 + )
0;
x
x; ;
>0
(2)
otherwise.
and the cumulative distribution function (c.d.f) is: FGaL (x) = 1
((
+
x
)( x + 1) + )e (1 + )
; x > 0;
> 0; > 0:
(3)
The mode, the mean and variance of GaL distribution are given by
mode(X) =
8 <
+ ( + : 0;
2 )
2
;
2 ;1 +1 otherwise.
2
(4)
E(X) = V ar(X) =
2 (1 + ) 6 ; E(X 2 ) = (1 + )
( 2
+ )2 + (2 + 6 ) 2 2
2
+6 2 +
2 ( 2
(1 + )
4
(5)
3
3
2
+ 2 2)
(6)
This paper proposes more properties and simulations of Gamma Lindley distribution. Various properties of the distribution are examined including the quantile function, Lorenz curve, probability density of the order statistics and entropies. It also provides more ‡exibility to analyze complex real data sets. We discuss maximum likelihood estimation of the distribution parameters. Two real data sets are analyzed using the new distribution, which show that the Gamma Lindley distribution can be used quite e¤ectively in analyzing real lifetime data. This work will add some value to the existing literature on modeling lifetime data and survival analysis The paper is organized as follows: in section 2 we give closed from expressions for the quantile functions of gamma Lindley distribution. Section 3 is devoted to introduce and give more properties of gamma Lindley distribution which we proposed the quantile function, Lorenz curve, probability density of the order statistics, entropies and moment method. Finally, we give an illustrative examples and simulations for compared gamma Lindley distribution with quasi Lindley, Two-Parameter Lindley, Generalized Lindley, Weibull and Lognormal distributions.
2 2.1
The quantile function of the Gamma Lindley distribution The Lambert W function
The Lambert W function has attracted a great deal of attention beginning with Lambert in 1758 and Euler in 1779.The name “Lambert W function”has become a standard after its implementation in the computer algebra system Maple in the 1980s and subsequent publication by Corless et al. (1996) of a comprehensive survey of the history, theory and applications of this function.The Lambert W function is a multivalued complex function de…ned as the solution of the equation: W (z) exp(W (z)) = z
3
(7)
where z is a complex number. If z is a real number such that z 1=e then W (z) becomes a real function and there are two possible real branches. The real branch taking on values in( 1; 1]is called the negative branch and denoted by W 1 . The real branch taking on values in [ 1; 1) is called the principal branch and denoted by W0 . We emphasize that Eq. (7) has two possible solutions if z 2 ( 1=e; 0] and a unique solution if z 0. For our results in this note, we shall use the negative branch W 1 , which satis…es the following elementary properties: W 1 ( 1=e) = 1, W 1 (z) is decreasing as z increases to 0 and W 1 (z) ! 1 as z ! 0. Lemme1 Let a,b and c complex numbers, The solution of the equation z + abz = c with respect to z 2 C is 1 W (abc log(b)) log(b)
z=c
(8)
where W denotes the lambert W function. For details of the proof. See P.Jodra (2010). The quantile function of X is QX (u) = FX 1 (u) ; 0 u 1: In the following result, we give an explicit expression for QX in terms of lambert W function. Theorem1 For any ; 0;the quantile function of the Gamma Lindley distribution X is
QX (u) =
(1 + ) ( (1 + )
1 )
W
(1 + ) (y (1 + )
1
1)
e
(1+ ) (1+ )
;
0
u
1;
(9) where W 1 denotes negative branche of lambert W function. Proof For any …xed ; 0;let u 2 (0; 1) :We have to solve the equation FX (x) = u with respect to x, for x 0:We have to solve the following equation: : [ [ (1 + ) ] x + (1 + )] e x = (1 + ) (1 u) : (10) Multiplying by
(1+ ) (1+ )
e
(1+ )
[ (1 + ) ]x + (1 + )
both sides of equation(10),we obtain:
(1 + )
e [
From equation(11), we see that tion of the real argument
[ (1+ ) ]x+ (1+ ) (1+ )
]=
[ (1+ ) ]x+ (1+ ) (1+ ) (1+ ) (1+ )
(1+ )(u 1) e (1+ )
4
(1 + ) (u (1 + )
1)
e
(1+ ) (1+ )
(11) is the Lambert W func-
:Then,we have
(1 + ) (u (1 + )
W
1)
e
Morever, for any ;
(1+ ) (1+ )
[ (1 + ) ]x + (1 + )
=
0 and x
0 it is immediate that
and it can also be checked that
[ (1+ ) ]x+ (1+ ) e (1+ )
(1 + )
;0 < u < 1 (12) 0 ) ] =
[ (1+ ) ]x+ (1+ ) (1+ )
[
[ (1+ ) ]x+ (1+ (1+ )
(1+ ) (1+ )
(1+ )(u 1) e (1+ )
2 e1 ; 0 since u 2 (0; 1). There for, by taking into account the proprietes of the negative branche of the Lambert W function ,equation (12) becomes
W
1
(1 + ) (u (1 + )
1)
e
(1+ ) (1+ )
=
[ (1 + ) ]x + (1 + )
(1 + )
(13)
Which the proof of theorem is complete. Further the …rst three quantiles we obtained by substituting u = 14 ; 12 ; 43 in equation (9)
Q1
= F
1
1 ; ; 4
(1 + ) ( (1 + )
Q2
= F
1
1 ; ; 2
(1 + ) ( (1 + )
Q3
= F
1
3 ; ; 4
(1 + ) ( (1 + )
= 1 )
LambertW
1;
(1 + ) ( (1 + )
)e
(1+ ) (1+ )
1 4
(1+ ) (1+ )
1 2
(1+ ) (1+ )
3 4
1
!
1
!
1
!
= 1 )
LambertW
1;
(1 + ) ( (1 + )
)e
= 1 )
LambertW
1;
(1 + ) ( (1 + )
)e
Table1 displays the M ode, M ean and M edian of GaL distribution for different choices of parameter and . Also for any choice of and it is observed that M ean > M edian > M ode. Table1. M ode, M ean and M edian of GaL distribution
5
Q1 M edian = Q2 Q3 M ean M ode
3
= 0:01; 86:284 157:74 259:05 190:10 89:011
= 0:1
= 0:1; 7:825 14:904 24:999 18:182 7:777
= 0:5
= 0:05; 0:068 0:164 0:325 39:048 19:0
=1
Lorenz curve
The Lorenz curve is often used to characterize income and wealth distributions. The Lorenz curve for a positive random variable X is de…ned as the graph of the ratio L(F (x)) =
E(XjX x)F (x) E(X)
(14)
against F (x) with the properties L(p) p; L(0) = 0 and L(1) = 1. If X represents annual income, L(p) is the proportion of total income that accrues to individuals having the 100 p% lowest incomes. If all individuals earn the same income then L(p) = p for all p. The area between the line L(p) = p and the Lorenz curve may be regarded as a measure of inequality of income, or more generally, of the variability of X. For the exponential distribution, it is well known that the Lorenz curve is given by L(p) = pfp + (1
p)log(1
p)g:
For the GaL distribution in (3),
E(XjX
x)F (x) = 2 (1 + ) (1 + )
e x ( + (1 + )
) x2 + 2x +
2
+ (x + 1) :
Thus, from (14) we obtain the Lorenz curve for the Gamma Lindley distribution as
L(p) = 1 where x = F
(1 1
p) (1 + )
) x2 + 2x + 2 + (x + 1) ) (x + 1) + ] [2 (1 + ) ]
( + [( +
(p) with F ( ) given by (4).
6
4
Extreme order statistics
If X1 ; :::; Xn is a random sample from (1) and if X =
X1 +
+ Xn
denotes pn n X E (X) p the sample mean then by the usual central limit theorem V ar (X) approaches the standard normal distribution as n ! 1. Sometimes one would be interested in the asymptotics of the extreme values Mn = max(X1 ; :::; Xn ) and mn = min(X1 ; :::; Xn ). For the c.d.f. in (2), it can be seen that lim
1
t!1
F (t + x) = exp( 1 F (t)
x)
and F (tx) =x t!0 F (t) lim
Thus, it follows from Theorem 1.6.2 in Leadbetter et al.(1987) that there must be norming constants an > 0; bn ; cn > 0 and dn such that P rfan (Mn
bn )
xg ! exp( exp(
x))
(15)
and P rfcn (mn
dn )
xg ! 1
exp ( x)
(16)
as n ! 1. The form of the norming constants can also be determined. For instance, using Corollary 1.6.3 in Leadbetter et al.(1987), one can see that an = 1 and bn = F 1 (1 1=n) with F ( ) given by (3).
5
Entropies
It is well known that entropy and information can be considered as measures of uncertainty of probability distribution. However, there are many relationships established on the basis of the properties of entropy. An entropy of a random variable X is a measure of variation of the uncertainty. Rényi entropy is de…ned by J( )=
where
> 0 and
1 1
log
Z
f (x) dx
= 1. For the GaL distribution in (1), note that
7
(17)
Z
2
f (x) dx =
E
e + E (1 + ) =
+
Z1
(18)
+ u e
u
+
du
0
From (18), one obtains the Rényi entropy as given by J( )= if
6
1 1
log
"
2
e + E (1 + )
#
+
(19)
! 1 ,we …nd shannon entropy.
Maximum Likelihood Estimates (ML)
In this section we shall discuss the point estimation on the parameters that index the GaL( ; ). Let the log-likelihood function of single observation(say xi ) for the vector of parameter ( ; ) can be written as
ln l(x; ; ) == 2 ln
ln
ln( + 1) + ln(( +
The derivatives of ln l(xi ; ; ) with respect to @ ln l(xi ; ; ) @ @ ln l(xi ; ; ) @
= =
2
1 1+
x+
( 8 <
: E b =
^= 1 x b= 1 1+x
E ^ = e ( +1)
2 (1+ ) (1+ )
2 +
8
x
are: (
1)x )x + 1
( + 1 (1 + )x + ( + )x + 1
The maximum likelihood estimator ^ of non linear equation(20) and (21) we give
and
and
)x + 1)
and ^ of
(20) (21)
is obtained by solving
(22)
=m 2
+
2
(23)
7
Moments estimates
Using the …rst moment m and m2 second moment about GaL distribution, we have 8 2 (1 + ) > > < m= (1 + ) (24) 6 +6 4 > > m = ; : 2 2 + 3
where m2 = s2 + m2 and s2 is the variance. We solve this non linear system we …nd the couple ^; ^ , where ^; ^ > 0 for all s > 0; m > 0: The solving of nonlinear system (24) gives m2
2
4m + 2 = 0 and
=
(1 + ) (2
m)
:
It is easy to check that the solution of m2
2
4m + 2 = 0
is :
1 if m 6= 0 ^ m2 = 0 2m
; if m = 0 ^ m2 = 0; and p p 1 m2 + 2m2 ; 2m + 2 m2
1 m2
2m +
since,
^=
8
p p 1 2 s2 + m2 2m + s2 + m2
p p 2 m2 + 2m2
and ^ =
if m2 6= 0
^ (1 + ^) 2
^m
:
Examples and Simulations
In this section, we give some examples and simulations. Example 1 Table2 shows some quantiles of the Gamma Lindley distribution, which have been calculated from the closed-form expression for QX given in theorem1.
9
Table2. y 0:01 0:05 0:1 0:15 0:2 0:25 0:3 0:35 0:40 0; 45 0:5 0:55 0:60 0:65 0:7 0:75 0:8 0:85 0:9 0:95 0:99
Quantiles of the GaL distribution = 0:1; = 0:1 = 0:1; = 0:5 0:110 55 0:506 76 0:564 08 2:095 1:158 4 3:6952 1:786 3 5:129 6 2:451 9 6:491 4 3:16 7: 825 3 3:916 6 9:159 6 4:728 7 10:516 5:605 3 11:912 6:557 4 13:368 7:599 5 14: 904 8:750 5 16:545 10:036 18: 322 11:492 20:278 13:171 22:473 15:154 24: 999 17: 577 28:008 20:695 31:781 25:080 36:946 32:549 45:477 49: 785 5 64:409
= 3; = 1 0:00446 0:02273 0:04656 0:07161 0:09802 0:125 99 0:155 71 0:187 44 0:221 51 0:258 31 0:298 36 0:342 35 0:391 19 0:446 16 0:509 13 0:583 01 0:672 63 0:787 0:946 31 1:214 4 1:822 2
= 5; = 1 0:00241 0:01229 0:02523 0:03888 0:05332 0:06866 0:08502 0:10256 0:12145 0:14192 0:16429 0:188 95 0:21642 0:24746 0:283 17 0:325 22 0:376 46 0:442 16 0:534 16 0:690 00 1:046 4
In this subsection, we investigate the behavior of the ML estimators for a nite sample size (n). A simulation study consisting of following steps is being carried out for each triplet ( ; ; n), where = 0:1; 0:5; 1; 3; = 0:1; 0:5; 0:75; 1; 6 and n = 10; 30; 50. - Choose the initial values of 0 ; 0 for the corresponding elements of the parameter vector = ( ; ) to specify GaL distribution; - choose sample size n; - generate N independent samples of size n from GaL( ; ); - compute the ML estimate ^ n of 0 for each of the N samples; - compute the mean of the obtained estimators over all N samples, average bais ( ) = and the average square error M SE ( ) =
N 1 P ^i N i=1
N 1 P ^i N i=1
10
0
2 0
Table3. Average bias of the simulated estimates = 1; = 6 = 1; = 0:1 bais ( ) bais ( ) bais ( ) n = 10 0:091 0:328 0:400 n = 30 0:030 0:109 0:133 n = 50 0:018 0:065 0:080 = 0:1; = 1 = 0:5; = 1 bais ( ) bais ( ) bais ( ) n = 10 1:899 0:100 0:283 n = 30 0:633 0:033 0:094 n = 50 0:379 0:020 0:056 = 3; = 0:5 = 0:5; = 0:5 bais ( ) bais ( ) bais ( ) n = 10 0:283 3:967 0:216 n = 30 0:094 1:322 0:072 n = 50 0:056 0:793 0:043 Table4. Average M SE of = 1; = 6 M SE ( ) n = 10 0:084 n = 30 0:028 n = 50 0:016 = 0:1; = 1 M SE ( ) n = 10 36:065 n = 30 12:022 n = 50 7:213 = 3; = 0:5 M SE ( ) n = 10 0:802 n = 30 0:267 n = 50 0:160
the simulated estimates = 1; = 0:1 M SE ( ) M SE ( ) 1:077 1:600 0:359 0:533 0:215 0:320 = 0:5; = 1 M SE ( ) M SE ( ) 0:101 0:802 0:033 0:267 0:0203 0:160 = 0:5; = 0:5 M SE ( ) M SE ( ) 15:380 0:469 5:460 0:156 3:476 0:093
bais ( ) 0:261 0:087 0:052 bais ( ) 0:119 0:039 0:023 bais ( ) 0:142 0:047 0:028
= 1; = 0:75 bais ( ) 0:028 0:009 0:005 = 3; = 1 bais ( ) 0:258 0:086 0:051 = 0:1; = 0:5 bais ( ) 1:808 0:602 0:361
M SE ( ) 0:685 0:228 0:137 M SE ( ) 0:143 0:047 0:028 M SE ( ) 0:202 0:067 0:040
bais ( ) 0:904 0:301 0:180 bais ( ) 0:141 0:047 0:028
= 1; = 0:75 M SE ( ) 0:011 0:003 0:002 = 3; = 1 M SE ( ) 0:667 0:222 0:133 = 0:1; = 0:5 M SE ( ) 3:695 1:898 0:653
In this subection, we illustrate, the applicability of GaL distribution by considering two di¤erent datasets used by di¤erent researchers(lifetime data). We also …t GaL, quasi Lindley (Shanker and Mishra (2013)), Two-Parameter Lindley (Shanker et al. (2013)), Generalized Lindley (Zakerzadah and Dolati (2013)), Weibull and Lognormal distributions. In each of these distributions, the parameters are estimated by using the moment method because it is simple, easy to handle, exact, and for comparison we use negative log-likelihood values ( LL), the Akaike information criterion (AIC) and Bayesian information criterion (BIC) which are de…ned by 2LL +
11
bais ( ) 0:196 0:065 0:039
M SE ( ) 0:387 0:129 0:077 M SE ( ) 8:177 2:725 1:635 M SE ( ) 0:201 0:067 0:040
2q and 2LL + qlog(n), respectively, where q is the number of parameters estimated and n is the sample size. Further K S (Kolmogorov-Smirnov) test statistic de…ned as K S = supx jFn (x) F (x)j, where Fn (x) is empirical distribution function and F (x) is cumulative distribution function is calculated and shown for all the datasets. Example 2 We consider from Lawless (2003),pp 204 and 263 two series of real data. The …rst one, represents the failure times (mm) for a sample of …fteen electronic components in an acceleration life test : 1.4, 5.1, 6.3, 10.8, 12.1, 18.5, 19.7, 22.2, 23, 30.6, 37.3, 46.3, 53.9, 59.8, 66.2. The second set of data, are the number of cycles to failure for 25 100-cm specimens of yarn, tested at a particular strain level : 15, 20, 38, 42, 61, 76, 86, 98, 121, 146, 149, 157, 175, 176, 180, 180, 198, 220, 224, 251, 264, 282, 321, 325, 653. In these examples we compared several two and three parameters distributions which might attract wider sets of applications in lifetime data reliability analysis and actuarial sciences. The results for these data are presented in the follwing table and hence best ts the data among all the models considered. Table5. Comparison between distributions Data Distribution LL K S Serie1 1:203 0:064 0:083 64:080 0:095 Generalized Lindley GaL 1:129 0:684 64:015 0:094 n=15 QLD 4:016 0:99 1504 0:93 m=27:546 T woP LD 0:0704 1:110 0:196 s=20:059 Gamma 1:442 0:052 64:197 0:102 W eibull 1:306 0:034 64:026 0:450 Lognormal 1:061 2:931 65:626 0:163 Serie2 1:505 0:012 0:018 152:369 0:137 Generalized Lindley GaL 1:086 0:010 152:132 0:129 n=25 QLD 0:0107 8:514 1045:9 0:94 m=178:32 s = 131:097 T woP LD 0:0107 0; 125 0:232 Gamma 1:794 0:010 152:371 0:135 W eibull 1:414 0:005 152:440 0:697 Lognormal 0:891 4:880 154:092 0:155 For table 5, the negative log-likelihood values con…rm that the GaL distribution is a much better …tted distribution than quasi Lindley ,Two-Parameter Lindley, Generalized Lindley Distribution, Weibull distribution and Lognormal distributions for …tting of lifetime data. According to the BIC and the AIC values the best model for prediction and most plausibly generated the data
12
AIC 134:16 132:03 3012
BIC 136:28 133:45 3013:4
132:39 132:05 135:25 310:74 308:26 2131:8
133:81 133:47 136:67 314:39 310:7 2156:2
308:74 308:88 312:18
311:18 310:7 314:62
is GaL distribution. Also, we can observe that Gamma Lindley distribution provide smallest k S as compare to quasi Lindley ,Two-Parameter Lindley, Generalized Lindley , Weibull and Lognormal distributions. Conclusion We discussed more statistical properties of the distribution, including the quantile function, Lorenz curve, probability density of the order statistics and entropies. The maximum likelihood estimates of the two parameters index to the new distribution are discussed. The distribution includes the Lindley and the exponential distributions as special cases.Two real data sets are analyzed using the new distribution and it is compared with six immediate sub-models mentioned above in addition to another distributions (quasi Lindley ,Two-Parameter Lindley, Generalized Lindley, Weibull and Lognormal distributions). The results of the comparisons showed that the new distribution provides a better …t than those three mentioned distributions to the three data sets. We hope our new distribution might attract wider sets of applications in lifetime data reliability analysis and actuarial sciences. For future studies, we can explain the derivation of posterior distributions for the Gamma Lindley distribution under Linex loss functions and squared error using non-informative and informative priors(the extension of Je¤reys and Inverted Gamma priors) respectively. Acknowledgment The authors are grateful for the comments and suggestions by the referee and the Editor. Their comments and suggestions greatly improved the article.
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[17] K.W. Chau, C.L. Wu (2010), "A Hybrid Model Coupled with Singular Spectrum Analysis for Daily Rainfall Prediction, Journal of Hydroinformatics 12 (4): 458-473. [18] H. Zakerzadah, A. Dolati (2010). Generalized Lindley distribution. J. Math. Ext. 3(2), pp. 13-25. [19] H. Zeghdoudi, S. Nedjar (2015). Gamma Lindley distribution and its application. Journal of Applied Probability and Statistics Vol. 10, N 2. [20] S. Zhang, K.W. Chau (2009), "Dimension Reduction Using SemiSupervised Locally Linear Embedding for Plant Leaf Classi…cation, Lecture Notes in Computer Science 5754: 948-955. [21] J. Zhang, K.W. Chau (2009), "Multilayer Ensemble Pruning via Novel Multi-sub-swarm Particle Swarm Optimization, Journal of Universal Computer Science 15 (4): 840-858.
15
S0377-0427(15)00618-4 http://dx.doi.org/10.1016/j.cam.2015.11.047 CAM 10404
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Journal of Computational and Applied Mathematics
Received date: 23 October 2015 Revised date: 15 November 2015 Please cite this article as: S. Nedjar, H. Zeghdoudi, On gamma Lindley distribution: Proprieties and simulations, Journal of Computational and Applied Mathematics (2015), http://dx.doi.org/10.1016/j.cam.2015.11.047 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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On Gamma Lindley Distribution : Proprieties and Simulations Sihem Nedjar1 and Halim Zeghdoudi1 LaPS laboratory, Badji-Mokhtar University, Box 12, Annaba, 23000,ALGERIA [email protected], [email protected] 1
summary A new distribution is recently proposed by Zeghdoudi and Nedjar, called as Gamma Lindley distribution (GaL). This paper proposes more properties and simulations which o¤ers a more ‡exible model for modelling lifetime data. Various statistical properties like the quantile function, Lorenz curve, moment method, maximum likelihood estimation, entropy, and limiting distribution of extreme order statistics are established. A simulation study is carried out to examine the bias and mean square error of the maximum likelihood estimators of the parameters. An application of the model to a real data set is presented …nally and compared with the …t attained by some other well-known two(three)-parameter distributions. Keywords : Lindley distribution, gamma lindley distribution, maximumlikelihood estimation Mathematics Subject Classi…cation: 60E05;62H10
1
Introduction
The real-life applications of contemporary numerical techniques in di¤erent …elds such as medicine, …nance, biological engineering sciences and statistics see (Taormina et al. (2015), Wang et al. (2015), Wu et al.(2009), Chau et al.(2010), S.W. Zhang et al.(2009) and J. Zhang et al.(2009)). To this end, statistics plays a crucial role in real life applications. Often by using the statistical analysis which strongly depends on the assumed probability model or distributions. However, several problems in statistics does not follow any of the classical or standard probability models. Let X be a random variable following the one-parameter distribution with
1
the density function f (x; ) =
8 < :
2
x
(1 + x)e 1+ 0;
x; > 0
(1)
otherwise
introduced by Lindley (1958). Sankaran (1970) used (1) as mixing distribution of Poisson parameter which it named Poisson- Lindley distribution. Recently, Asgharzadeh et al. (2013), Elbatal et al. (2013), Bakouch et al.(2012), Ghitany et al. (2008a) and (2008b) rediscovered and studied the new distribution bounded to (1), what they derived is known as Zero-truncated Poisson- Lindley and Pareto Poisson-Lindley distributions. Recently, Zeghdoudi and Nedjar (2015) introduced a new distribution, named gamma Lindley distribution, based on mixtures of gamma (2; ) and one-parameter Lindley distributions.Various properties of the distribution are examined including the density function (pdf), cumulative distribution (cdf), survival and hazard rate function, moment generating function (mgf), mean, variance and some results on stochastic orderings. Plots of pdf and cdf for some parameter values are also given. Moment estimates are discussed and two examples are presented to show the goodness of …t. The idea using a mixture of two known distribution to generate a new distribution is not new. For example, Shanker, Sharma and Shanker (2013) used a mixture of exponential ( ) and gamma (2; ) to create a two-parameter Lindley distribution. For another example, Zakerzadeh and Dolati (2009) used gamma ( ; ) and gamma ( + 1; ) to create a generalized Lindley distribution. However, it is the …rst time in the literature to use a mixture of gamma (2; ) and one-parameter Lindley distribution to generate Gamma Lindley distribution, where the density function of the random variable X is given by:
fGaL (x; ; ) =
8 < :
2
(( +
)x + 1)e (1 + )
0;
x
x; ;
>0
(2)
otherwise.
and the cumulative distribution function (c.d.f) is: FGaL (x) = 1
((
+
x
)( x + 1) + )e (1 + )
; x > 0;
> 0; > 0:
(3)
The mode, the mean and variance of GaL distribution are given by
mode(X) =
8 <
+ ( + : 0;
2 )
2
;
2 ;1 +1 otherwise.
2
(4)
E(X) = V ar(X) =
2 (1 + ) 6 ; E(X 2 ) = (1 + )
( 2
+ )2 + (2 + 6 ) 2 2
2
+6 2 +
2 ( 2
(1 + )
4
(5)
3
3
2
+ 2 2)
(6)
This paper proposes more properties and simulations of Gamma Lindley distribution. Various properties of the distribution are examined including the quantile function, Lorenz curve, probability density of the order statistics and entropies. It also provides more ‡exibility to analyze complex real data sets. We discuss maximum likelihood estimation of the distribution parameters. Two real data sets are analyzed using the new distribution, which show that the Gamma Lindley distribution can be used quite e¤ectively in analyzing real lifetime data. This work will add some value to the existing literature on modeling lifetime data and survival analysis The paper is organized as follows: in section 2 we give closed from expressions for the quantile functions of gamma Lindley distribution. Section 3 is devoted to introduce and give more properties of gamma Lindley distribution which we proposed the quantile function, Lorenz curve, probability density of the order statistics, entropies and moment method. Finally, we give an illustrative examples and simulations for compared gamma Lindley distribution with quasi Lindley, Two-Parameter Lindley, Generalized Lindley, Weibull and Lognormal distributions.
2 2.1
The quantile function of the Gamma Lindley distribution The Lambert W function
The Lambert W function has attracted a great deal of attention beginning with Lambert in 1758 and Euler in 1779.The name “Lambert W function”has become a standard after its implementation in the computer algebra system Maple in the 1980s and subsequent publication by Corless et al. (1996) of a comprehensive survey of the history, theory and applications of this function.The Lambert W function is a multivalued complex function de…ned as the solution of the equation: W (z) exp(W (z)) = z
3
(7)
where z is a complex number. If z is a real number such that z 1=e then W (z) becomes a real function and there are two possible real branches. The real branch taking on values in( 1; 1]is called the negative branch and denoted by W 1 . The real branch taking on values in [ 1; 1) is called the principal branch and denoted by W0 . We emphasize that Eq. (7) has two possible solutions if z 2 ( 1=e; 0] and a unique solution if z 0. For our results in this note, we shall use the negative branch W 1 , which satis…es the following elementary properties: W 1 ( 1=e) = 1, W 1 (z) is decreasing as z increases to 0 and W 1 (z) ! 1 as z ! 0. Lemme1 Let a,b and c complex numbers, The solution of the equation z + abz = c with respect to z 2 C is 1 W (abc log(b)) log(b)
z=c
(8)
where W denotes the lambert W function. For details of the proof. See P.Jodra (2010). The quantile function of X is QX (u) = FX 1 (u) ; 0 u 1: In the following result, we give an explicit expression for QX in terms of lambert W function. Theorem1 For any ; 0;the quantile function of the Gamma Lindley distribution X is
QX (u) =
(1 + ) ( (1 + )
1 )
W
(1 + ) (y (1 + )
1
1)
e
(1+ ) (1+ )
;
0
u
1;
(9) where W 1 denotes negative branche of lambert W function. Proof For any …xed ; 0;let u 2 (0; 1) :We have to solve the equation FX (x) = u with respect to x, for x 0:We have to solve the following equation: : [ [ (1 + ) ] x + (1 + )] e x = (1 + ) (1 u) : (10) Multiplying by
(1+ ) (1+ )
e
(1+ )
[ (1 + ) ]x + (1 + )
both sides of equation(10),we obtain:
(1 + )
e [
From equation(11), we see that tion of the real argument
[ (1+ ) ]x+ (1+ ) (1+ )
]=
[ (1+ ) ]x+ (1+ ) (1+ ) (1+ ) (1+ )
(1+ )(u 1) e (1+ )
4
(1 + ) (u (1 + )
1)
e
(1+ ) (1+ )
(11) is the Lambert W func-
:Then,we have
(1 + ) (u (1 + )
W
1)
e
Morever, for any ;
(1+ ) (1+ )
[ (1 + ) ]x + (1 + )
=
0 and x
0 it is immediate that
and it can also be checked that
[ (1+ ) ]x+ (1+ ) e (1+ )
(1 + )
;0 < u < 1 (12) 0 ) ] =
[ (1+ ) ]x+ (1+ ) (1+ )
[
[ (1+ ) ]x+ (1+ (1+ )
(1+ ) (1+ )
(1+ )(u 1) e (1+ )
2 e1 ; 0 since u 2 (0; 1). There for, by taking into account the proprietes of the negative branche of the Lambert W function ,equation (12) becomes
W
1
(1 + ) (u (1 + )
1)
e
(1+ ) (1+ )
=
[ (1 + ) ]x + (1 + )
(1 + )
(13)
Which the proof of theorem is complete. Further the …rst three quantiles we obtained by substituting u = 14 ; 12 ; 43 in equation (9)
Q1
= F
1
1 ; ; 4
(1 + ) ( (1 + )
Q2
= F
1
1 ; ; 2
(1 + ) ( (1 + )
Q3
= F
1
3 ; ; 4
(1 + ) ( (1 + )
= 1 )
LambertW
1;
(1 + ) ( (1 + )
)e
(1+ ) (1+ )
1 4
(1+ ) (1+ )
1 2
(1+ ) (1+ )
3 4
1
!
1
!
1
!
= 1 )
LambertW
1;
(1 + ) ( (1 + )
)e
= 1 )
LambertW
1;
(1 + ) ( (1 + )
)e
Table1 displays the M ode, M ean and M edian of GaL distribution for different choices of parameter and . Also for any choice of and it is observed that M ean > M edian > M ode. Table1. M ode, M ean and M edian of GaL distribution
5
Q1 M edian = Q2 Q3 M ean M ode
3
= 0:01; 86:284 157:74 259:05 190:10 89:011
= 0:1
= 0:1; 7:825 14:904 24:999 18:182 7:777
= 0:5
= 0:05; 0:068 0:164 0:325 39:048 19:0
=1
Lorenz curve
The Lorenz curve is often used to characterize income and wealth distributions. The Lorenz curve for a positive random variable X is de…ned as the graph of the ratio L(F (x)) =
E(XjX x)F (x) E(X)
(14)
against F (x) with the properties L(p) p; L(0) = 0 and L(1) = 1. If X represents annual income, L(p) is the proportion of total income that accrues to individuals having the 100 p% lowest incomes. If all individuals earn the same income then L(p) = p for all p. The area between the line L(p) = p and the Lorenz curve may be regarded as a measure of inequality of income, or more generally, of the variability of X. For the exponential distribution, it is well known that the Lorenz curve is given by L(p) = pfp + (1
p)log(1
p)g:
For the GaL distribution in (3),
E(XjX
x)F (x) = 2 (1 + ) (1 + )
e x ( + (1 + )
) x2 + 2x +
2
+ (x + 1) :
Thus, from (14) we obtain the Lorenz curve for the Gamma Lindley distribution as
L(p) = 1 where x = F
(1 1
p) (1 + )
) x2 + 2x + 2 + (x + 1) ) (x + 1) + ] [2 (1 + ) ]
( + [( +
(p) with F ( ) given by (4).
6
4
Extreme order statistics
If X1 ; :::; Xn is a random sample from (1) and if X =
X1 +
+ Xn
denotes pn n X E (X) p the sample mean then by the usual central limit theorem V ar (X) approaches the standard normal distribution as n ! 1. Sometimes one would be interested in the asymptotics of the extreme values Mn = max(X1 ; :::; Xn ) and mn = min(X1 ; :::; Xn ). For the c.d.f. in (2), it can be seen that lim
1
t!1
F (t + x) = exp( 1 F (t)
x)
and F (tx) =x t!0 F (t) lim
Thus, it follows from Theorem 1.6.2 in Leadbetter et al.(1987) that there must be norming constants an > 0; bn ; cn > 0 and dn such that P rfan (Mn
bn )
xg ! exp( exp(
x))
(15)
and P rfcn (mn
dn )
xg ! 1
exp ( x)
(16)
as n ! 1. The form of the norming constants can also be determined. For instance, using Corollary 1.6.3 in Leadbetter et al.(1987), one can see that an = 1 and bn = F 1 (1 1=n) with F ( ) given by (3).
5
Entropies
It is well known that entropy and information can be considered as measures of uncertainty of probability distribution. However, there are many relationships established on the basis of the properties of entropy. An entropy of a random variable X is a measure of variation of the uncertainty. Rényi entropy is de…ned by J( )=
where
> 0 and
1 1
log
Z
f (x) dx
= 1. For the GaL distribution in (1), note that
7
(17)
Z
2
f (x) dx =
E
e + E (1 + ) =
+
Z1
(18)
+ u e
u
+
du
0
From (18), one obtains the Rényi entropy as given by J( )= if
6
1 1
log
"
2
e + E (1 + )
#
+
(19)
! 1 ,we …nd shannon entropy.
Maximum Likelihood Estimates (ML)
In this section we shall discuss the point estimation on the parameters that index the GaL( ; ). Let the log-likelihood function of single observation(say xi ) for the vector of parameter ( ; ) can be written as
ln l(x; ; ) == 2 ln
ln
ln( + 1) + ln(( +
The derivatives of ln l(xi ; ; ) with respect to @ ln l(xi ; ; ) @ @ ln l(xi ; ; ) @
= =
2
1 1+
x+
( 8 <
: E b =
^= 1 x b= 1 1+x
E ^ = e ( +1)
2 (1+ ) (1+ )
2 +
8
x
are: (
1)x )x + 1
( + 1 (1 + )x + ( + )x + 1
The maximum likelihood estimator ^ of non linear equation(20) and (21) we give
and
and
)x + 1)
and ^ of
(20) (21)
is obtained by solving
(22)
=m 2
+
2
(23)
7
Moments estimates
Using the …rst moment m and m2 second moment about GaL distribution, we have 8 2 (1 + ) > > < m= (1 + ) (24) 6 +6 4 > > m = ; : 2 2 + 3
where m2 = s2 + m2 and s2 is the variance. We solve this non linear system we …nd the couple ^; ^ , where ^; ^ > 0 for all s > 0; m > 0: The solving of nonlinear system (24) gives m2
2
4m + 2 = 0 and
=
(1 + ) (2
m)
:
It is easy to check that the solution of m2
2
4m + 2 = 0
is :
1 if m 6= 0 ^ m2 = 0 2m
; if m = 0 ^ m2 = 0; and p p 1 m2 + 2m2 ; 2m + 2 m2
1 m2
2m +
since,
^=
8
p p 1 2 s2 + m2 2m + s2 + m2
p p 2 m2 + 2m2
and ^ =
if m2 6= 0
^ (1 + ^) 2
^m
:
Examples and Simulations
In this section, we give some examples and simulations. Example 1 Table2 shows some quantiles of the Gamma Lindley distribution, which have been calculated from the closed-form expression for QX given in theorem1.
9
Table2. y 0:01 0:05 0:1 0:15 0:2 0:25 0:3 0:35 0:40 0; 45 0:5 0:55 0:60 0:65 0:7 0:75 0:8 0:85 0:9 0:95 0:99
Quantiles of the GaL distribution = 0:1; = 0:1 = 0:1; = 0:5 0:110 55 0:506 76 0:564 08 2:095 1:158 4 3:6952 1:786 3 5:129 6 2:451 9 6:491 4 3:16 7: 825 3 3:916 6 9:159 6 4:728 7 10:516 5:605 3 11:912 6:557 4 13:368 7:599 5 14: 904 8:750 5 16:545 10:036 18: 322 11:492 20:278 13:171 22:473 15:154 24: 999 17: 577 28:008 20:695 31:781 25:080 36:946 32:549 45:477 49: 785 5 64:409
= 3; = 1 0:00446 0:02273 0:04656 0:07161 0:09802 0:125 99 0:155 71 0:187 44 0:221 51 0:258 31 0:298 36 0:342 35 0:391 19 0:446 16 0:509 13 0:583 01 0:672 63 0:787 0:946 31 1:214 4 1:822 2
= 5; = 1 0:00241 0:01229 0:02523 0:03888 0:05332 0:06866 0:08502 0:10256 0:12145 0:14192 0:16429 0:188 95 0:21642 0:24746 0:283 17 0:325 22 0:376 46 0:442 16 0:534 16 0:690 00 1:046 4
In this subsection, we investigate the behavior of the ML estimators for a nite sample size (n). A simulation study consisting of following steps is being carried out for each triplet ( ; ; n), where = 0:1; 0:5; 1; 3; = 0:1; 0:5; 0:75; 1; 6 and n = 10; 30; 50. - Choose the initial values of 0 ; 0 for the corresponding elements of the parameter vector = ( ; ) to specify GaL distribution; - choose sample size n; - generate N independent samples of size n from GaL( ; ); - compute the ML estimate ^ n of 0 for each of the N samples; - compute the mean of the obtained estimators over all N samples, average bais ( ) = and the average square error M SE ( ) =
N 1 P ^i N i=1
N 1 P ^i N i=1
10
0
2 0
Table3. Average bias of the simulated estimates = 1; = 6 = 1; = 0:1 bais ( ) bais ( ) bais ( ) n = 10 0:091 0:328 0:400 n = 30 0:030 0:109 0:133 n = 50 0:018 0:065 0:080 = 0:1; = 1 = 0:5; = 1 bais ( ) bais ( ) bais ( ) n = 10 1:899 0:100 0:283 n = 30 0:633 0:033 0:094 n = 50 0:379 0:020 0:056 = 3; = 0:5 = 0:5; = 0:5 bais ( ) bais ( ) bais ( ) n = 10 0:283 3:967 0:216 n = 30 0:094 1:322 0:072 n = 50 0:056 0:793 0:043 Table4. Average M SE of = 1; = 6 M SE ( ) n = 10 0:084 n = 30 0:028 n = 50 0:016 = 0:1; = 1 M SE ( ) n = 10 36:065 n = 30 12:022 n = 50 7:213 = 3; = 0:5 M SE ( ) n = 10 0:802 n = 30 0:267 n = 50 0:160
the simulated estimates = 1; = 0:1 M SE ( ) M SE ( ) 1:077 1:600 0:359 0:533 0:215 0:320 = 0:5; = 1 M SE ( ) M SE ( ) 0:101 0:802 0:033 0:267 0:0203 0:160 = 0:5; = 0:5 M SE ( ) M SE ( ) 15:380 0:469 5:460 0:156 3:476 0:093
bais ( ) 0:261 0:087 0:052 bais ( ) 0:119 0:039 0:023 bais ( ) 0:142 0:047 0:028
= 1; = 0:75 bais ( ) 0:028 0:009 0:005 = 3; = 1 bais ( ) 0:258 0:086 0:051 = 0:1; = 0:5 bais ( ) 1:808 0:602 0:361
M SE ( ) 0:685 0:228 0:137 M SE ( ) 0:143 0:047 0:028 M SE ( ) 0:202 0:067 0:040
bais ( ) 0:904 0:301 0:180 bais ( ) 0:141 0:047 0:028
= 1; = 0:75 M SE ( ) 0:011 0:003 0:002 = 3; = 1 M SE ( ) 0:667 0:222 0:133 = 0:1; = 0:5 M SE ( ) 3:695 1:898 0:653
In this subection, we illustrate, the applicability of GaL distribution by considering two di¤erent datasets used by di¤erent researchers(lifetime data). We also …t GaL, quasi Lindley (Shanker and Mishra (2013)), Two-Parameter Lindley (Shanker et al. (2013)), Generalized Lindley (Zakerzadah and Dolati (2013)), Weibull and Lognormal distributions. In each of these distributions, the parameters are estimated by using the moment method because it is simple, easy to handle, exact, and for comparison we use negative log-likelihood values ( LL), the Akaike information criterion (AIC) and Bayesian information criterion (BIC) which are de…ned by 2LL +
11
bais ( ) 0:196 0:065 0:039
M SE ( ) 0:387 0:129 0:077 M SE ( ) 8:177 2:725 1:635 M SE ( ) 0:201 0:067 0:040
2q and 2LL + qlog(n), respectively, where q is the number of parameters estimated and n is the sample size. Further K S (Kolmogorov-Smirnov) test statistic de…ned as K S = supx jFn (x) F (x)j, where Fn (x) is empirical distribution function and F (x) is cumulative distribution function is calculated and shown for all the datasets. Example 2 We consider from Lawless (2003),pp 204 and 263 two series of real data. The …rst one, represents the failure times (mm) for a sample of …fteen electronic components in an acceleration life test : 1.4, 5.1, 6.3, 10.8, 12.1, 18.5, 19.7, 22.2, 23, 30.6, 37.3, 46.3, 53.9, 59.8, 66.2. The second set of data, are the number of cycles to failure for 25 100-cm specimens of yarn, tested at a particular strain level : 15, 20, 38, 42, 61, 76, 86, 98, 121, 146, 149, 157, 175, 176, 180, 180, 198, 220, 224, 251, 264, 282, 321, 325, 653. In these examples we compared several two and three parameters distributions which might attract wider sets of applications in lifetime data reliability analysis and actuarial sciences. The results for these data are presented in the follwing table and hence best ts the data among all the models considered. Table5. Comparison between distributions Data Distribution LL K S Serie1 1:203 0:064 0:083 64:080 0:095 Generalized Lindley GaL 1:129 0:684 64:015 0:094 n=15 QLD 4:016 0:99 1504 0:93 m=27:546 T woP LD 0:0704 1:110 0:196 s=20:059 Gamma 1:442 0:052 64:197 0:102 W eibull 1:306 0:034 64:026 0:450 Lognormal 1:061 2:931 65:626 0:163 Serie2 1:505 0:012 0:018 152:369 0:137 Generalized Lindley GaL 1:086 0:010 152:132 0:129 n=25 QLD 0:0107 8:514 1045:9 0:94 m=178:32 s = 131:097 T woP LD 0:0107 0; 125 0:232 Gamma 1:794 0:010 152:371 0:135 W eibull 1:414 0:005 152:440 0:697 Lognormal 0:891 4:880 154:092 0:155 For table 5, the negative log-likelihood values con…rm that the GaL distribution is a much better …tted distribution than quasi Lindley ,Two-Parameter Lindley, Generalized Lindley Distribution, Weibull distribution and Lognormal distributions for …tting of lifetime data. According to the BIC and the AIC values the best model for prediction and most plausibly generated the data
12
AIC 134:16 132:03 3012
BIC 136:28 133:45 3013:4
132:39 132:05 135:25 310:74 308:26 2131:8
133:81 133:47 136:67 314:39 310:7 2156:2
308:74 308:88 312:18
311:18 310:7 314:62
is GaL distribution. Also, we can observe that Gamma Lindley distribution provide smallest k S as compare to quasi Lindley ,Two-Parameter Lindley, Generalized Lindley , Weibull and Lognormal distributions. Conclusion We discussed more statistical properties of the distribution, including the quantile function, Lorenz curve, probability density of the order statistics and entropies. The maximum likelihood estimates of the two parameters index to the new distribution are discussed. The distribution includes the Lindley and the exponential distributions as special cases.Two real data sets are analyzed using the new distribution and it is compared with six immediate sub-models mentioned above in addition to another distributions (quasi Lindley ,Two-Parameter Lindley, Generalized Lindley, Weibull and Lognormal distributions). The results of the comparisons showed that the new distribution provides a better …t than those three mentioned distributions to the three data sets. We hope our new distribution might attract wider sets of applications in lifetime data reliability analysis and actuarial sciences. For future studies, we can explain the derivation of posterior distributions for the Gamma Lindley distribution under Linex loss functions and squared error using non-informative and informative priors(the extension of Je¤reys and Inverted Gamma priors) respectively. Acknowledgment The authors are grateful for the comments and suggestions by the referee and the Editor. Their comments and suggestions greatly improved the article.
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