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78
On an extension of the two-parameter Lindley distribution
Jiju Gillariose1 and Lishamol Tomy2 •
1 Department of Statistics, CHRIST (Deemed to be University), Bangalore, Karnataka- 560029, India
2 Department of Statistics, Deva Matha College, Kuravilangad, Kerala, 686633, India
[email protected] [email protected]
Abstract
AIM: Lindley distribution has been widely studied in statistical literature because it accommodates several interesting properties. In lifetime data analysis contexts, Lindley distribution gives a good description over exponential distribution. It has been used for analysing copious real data sets, specifically in applications of modeling stress-strength reliability. This paper proposes a new generalized two-parameter Lindley distribution and provides a comprehensive description of its statistical properties such as order statistics, limiting distributions of order statistics, Information theory measures, etc.
METHODS: We study shapes of the probability density and hazard rate functions, quantiles, moments, moment generating function, order statistic, limiting distributions of order statistics, information theory measures, and autoregressive models are among the key characteristics and properties discussed. The two-parameter Lindley distribution is then subjected to statistical analysis. The paper uses methods of maximum likelihood to estimate the parameters of the proposed distribution. The usefulness of the proposed distribution for modeling data is illustrated using a real data set by comparison with other generalizations of the exponential and Lindley distributions and is depicted graphically.
RESULTS/FINDINGS: This paper presents relevant characteristics of the proposed distribution and applications. Based on this study, we found that the proposed model can be used quite effectively to analyzing lifetime data.
CONCLUSIONS: In this article, we proffered a new customized Lindley distribution. The proposed distribution enfolds exponential and Lindley distributions as sub-models. Some properties of this distribution such as quantile function, moments, moment generating function, distributions of order statistics, limiting distributions of order statistics, entropy, and autoregressive time series models are studied. This distribution is found to be the most appropriate model to fit the carbon fibers data compared to other models. Consequently, we propose the MOTL distribution for sketching inscrutable lifetime data sets.
Keywords: Exponential distribution, Generalized family, Lindley distribution, Marshall-Olkin
extended distribution, Maximum likelihood estimation
1. Introduction
Lindley distribution [16, 17] has been proposed to describe a difference between fiducial distribution and posterior distribution. The works on Lindley distribution; see, for example, [11], [14], [8], [2], [28], [33], etc. In the last decades, a lot of attempts have been made to define new probability distributions based on Lindley model, for example, three parameters-Lindley distribution [37], generalized Poisson-Lindley distribution [18], generalized Lindley distribution [23], Marshall-Olkin Lindley distribution [38], power Lindley distribution [10], two-parameter Lindley distribution [29], quasi Lindley distribution [30], transmuted Lindley distribution [20], transmuted Lindley-geometric distribution [21], beta-Lindley distribution [22], discrete Harris extended Lindley distribution [35], etc. Moreover, [36] has provided a detailed review study on
the generalizations of the Lindley distribution.
[31] introduced a new distribution, called two-parameter Lindley distribution. A random variable X is said to have the two-parameter Lindley distribution with parameters a and ft if its survival function (sf) takes the form
F(x, a, ft) = (a e-ax, x > 0, a > 0, ft > —a (1)
and the corresponding probability density function (pdf) can be expressed as
f (x,a, ft)= a2(1 + ftx)e ax , x > 0,a > 0, ft > -a. (2)
J v a + ft
It can easily be seen that at ft = 1, the distribution in equation (2) reduces to the Lindley distribution and at ft = 0, equation (2) reduces to the exponential distribution. [12] has also studied this distribution as a new flexible form of exponential distribution is called flexible exponential distribution. Some generalizations and extensions of this flexible exponential distribution are proposed in [34] and [25].
On the other hand, there is a vast amount of statistical literature on methods of introducing new family of distributions. Notable among them are Azzalinis skewed family of distributions [4], exponentiated family of distributions [13], gamma-generated family of distributions [39, 27], Kumaraswamy family of distributions [6], Weibull generalized family of distributions [5], logistic-generated family of distributions [Torabi and Montazari(2014)], Kumaraswamy Marshal-Olkin family of distributions [1] and Marshall-Olkin Kumaraswamy family of distributions [15]. Moreover, [19] has introduced a general method for adding parameter to a baseline distribution, the resulting distribution is called Marshall-Olkin family of distributions, its sf G(x) and pdf g(x) are given by the following formulae,
G(X' a) = I-fil) , x G R Y > 0 (3)
g(x,a) = (11%)2 , ^ e R,7 > 0 (4)
where F(x) is sf of the random variable X to be generated, 7 = 1 — 7 and 7 is a tilt parameter. If F(x) has the hazard rate function (hrf) r(x) then the hrf of MOE family is given by
r( x)
h(x, a) = -—v ' . , x e R, 7 > 0 K ' 1 — 7F (x)
The main object of this paper is to present an extension for the two-parameter Lindley distribution, that can be used as an alternative to the existing generalized exponential and Lindley distributions. The rest of this article is organized as follows: Section 2 introduces the Marshall-Olkin two-parameter Lindley distribution; its properties including quantile function, moments, moment generating function, distributions of order statistics, limiting distributions of order statistics, entropy and autoregressive time series models are presented in Section 3; Section 4 proposes parameter estimation of the proposed distribution by the method of maximum likelihood estimation; Section 5 deals with the application of the new distribution to a real data set; Section 6 presents the conclusion of the study.
2. Marshall-Olkin two-parameter Lindley distribution
If X is distributed according to equation (2), then the corresponding Marshall-Olkin (MO) generalized form of its sf and pdf using equations (3) and (4) is given by
aßx) e— ax
G^ß, y) = T-,a+Laßx)-VT, x > 0, ^ Y > 0, ß > -a (5)
1 ((a+ß+aßx) p-ax
1 a+ß p
and
g(x a ß 7) =
7a2 (1 + ßx)e_
a + ß [1 _ y ((g+fflx)e-ax
-, x > 0, a, y > 0, ß > -a
(6)
respectively. The new distribution given by the pdf equation (6) is called the Marshall-Olkin two-parameter Lindley (MOTL) distribution. In addition, hrf of the MOTL distribution is given by following equation
h(x, a, ß, y)
a2 (1 + ßx)
{1 _ Y
(a+ß+«ßx) _ax a+ß e
I (ß + a + aßx)
, x > 0, a, y > 0, ß > -a.
(7)
Notably, the classical exponential and one-parameter Lindley distributions are special cases of the MOTL distribution. Some distributions that are special cases of MOTL distribution are:
Exponential distribution : when 7=1 and j8=0 in equation (6) with pdf
g(x, 7) = ae-ax
One-parameter Lindley distribution : when 7=1 and j8=1 in equation (6) with pdf
g(x a Y)
a2(1 + x)e a +1
MO exponential distribution : when ß = 0 in equation (6) with pdf
g(x a Y)
Yae
[1 _ y(e_ax)]2
MO Lindley distribution: when ß = 1 in equation (6) with pdf
g(x a Y)
Ya2(1 + x)e
a+1
1 _ Y
(a+l+«x) _ax
a+1 e
The different shapes of the pdf and hrf of the MOTL distribution are displayed in Figure 1 and Figure 2 for selected parameter values. From figures it is clear that the pdf and hrf of MOTL distribution can be increasing, decreasing, upside-down bathtub (unimodal) depending on the values of its parameters.
3. Statistical Properties
In this section, we study the statistical properties for the MOTL distribution.
3.1. Quantiles
The quantile function of the MOTL distribution is given by
G-1(u)=_() _ a W_1
1
u1
ß V1 _ u + uy
(_ a+ß)
(a + ß)e( ß )
(8)
where G(x) = u and 0 < u < 1. W-1 denotes the negative branch of the Lambert W function. Table 1 represents the quantiles for selected values of the parameters of the MOTL distribution
using R programming language.
2
2
x
(a, ß, Y) (a, ß, Y) (a, ß, Y) (a, ß, Y) (a, ß, Y)
u (0.5,0.5,1.5) (0.4,0.3,.05) (2.1,2.2,0.5) (3.5,4.2,2.5) (5,6,7)
0.1 0.5784 0.2331 0.0513 0.1360 0.1515
0.2 1.1374 0.5001 0.1087 0.2503 0.2596
0.3 1.7018 0.8107 0.1741 0.3563 0.3515
0.4 2.2923 1.1796 0.2504 0.4604 0.4360
0.5 2.9324 1.6295 0.3422 0.5676 0.5202
0.6 3.6556 2.1986 0.4565 0.6838 0.6078
0.7 4.5205 2.9585 0.6074 0.8170 0.7061
0.8 5.6539 4.0696 0.8255 0.9879 0.8282
0.9 7.4520 6.0382 1.2080 1.2505 1.0135
Figure 2: Graphs of pdf of the MOTL distribution for different values of a, ft and 7.
3.2. Moments
In statistical analysis and its applications, moments have received important role. It can be used to study the most eminent features and characteristics such as tendency, dispersion, skewness and kurtosis of a distribution. We now give simple expansions for the pdf of the MOTL distribution. We have following expansion
i=0
Put
(1 - z)-r = E ^ +1 ^*, |z| < 1,r > 0
= (a + ft + aftx) S(x) = a + ft '
ax
When 7 e (0,2)
(1 - (1 - 7)(1 - S(x))-2 = tt(i + 1)(1 - 7)l(i)s(x)> (9)
i j=0 j
Using the series expansion in equation (9) and the representation for the MOTL pdf in equation (6), we obtain
g« = 7 g +1)(1 - 7)i{1+aa+ft}'(1+fix^w
E
g0 a + ftv" ' ^ " y ' a + ft
7g£ dm(i+1)(1 - 7)i 0){ ^}'(1+ftx)e-i,+1]"
TO TO /i\ r 1
7 EE, (i + 1)(1 - 7)\ii (aft) \x'e-(l+1)ax + ftxi+1e-(l+1)ax] (10)
i=0 j=0 (a + ft) V//
We have
z*TO
E(Xr) = xrg(x, a, ft, 7)dx. (11)
0
Substituting equation (10) into the equation (11), we obtain the rth moment of MOTL distribution in the form
to TO a2 /i\ ¿'TO r
E(Xr) = 7 Eg + 1)(1 - 7)in (aft)j x^\x'e-('+1)ax + ftxj+1e-(i+1)*
i=0j=0 (a + ft) j J0
TO TO a2 . fi\
= 7j +1)(1 - 7)-{i)(j
dx
r(r + j + 1) r(r + j + 2)
Wi/r = r/. ^ , + ft TTT"
i jr
[(i + 1)a]r+j+1 ^[(i + 1)a]r+j+2
Similarly, when 7 > 1/2
g(x)
7a2 (1 + ftx)e-a + ft
1 - 7(1 - S(x))
TO TO i a2
g(x) = 7EEE (a + ft)j+1
(-1)l+k[ ! ^ - ^ j (aft) [xje-(i+1)ax + ftxj+1 e-(i+1)a
2
The rth moment of MOTL distribution is
e(x-) = , EEE jr (?)' (' - OO j
Table 3.2 lists the moments, standard deviation (SD), coefficient of variation (CV), coefficient of skewness (CS) and coefficient of kurtosis (CK) of the MOTL distribution for selected values of the parameters.
Table 2: Moments of the MOTL distribution for different values of the parameters a, fi, and 7
Fl F2 F3
F4 SD CV CS CK
(a, ß, Y) (a, ß, Y) (a, ß, Y) (a, ß, Y) (a, ß, Y)
(0.5,0.5,.5) (1.5,0.5,1.5) (2.5,0.005,3.5) (3,5,3.5) (5,6,7)
1.5273 1.0084 0.7029 0.8856 1.3452
3.0999 1.7819 0.7641 1.0953 2.5024
7.5709 4.3852 1.0910 1.6799 5.7702
21.4836 13.7554 1.9276 3.0565 15.7912
0.8760 0.8746 0.5196 0.5577 0.8326
0.5735 0.8676 0.7392 0.6297 0.6076
3811.034 749.2536 211.0792 548.7451 2284.693
3.8968 6.5565 5.3869 4.2850 4.3509
3.3. Moment Generating Function
Moment generating function is given by the following formula
œ ,r
tr
Mx(t) = E(etX) = E rrE(Xr) (12)
r=0 ''
The moment generating function of MOTL distribution is obtained by using equation (12). When 7 G (0,2), it has following form
m m m a2tr . /i\
M(t) = 7 E E E (OTj^(i +1)(1 - Y)i j (aft)jw j
Similarly, when 7 > 1/2
MX(t) = YE E E s (-1),+'( ^Y7) 'C - 0 j(j
where
3.4. Order Statistics
Let Xi, X2,..., X n be a random sample taken from the MOTL distribution and Xi:n, X2:n,.. ., Xn:n be the corresponding order statistics. The pdf gr:n (x, a, ft, 7) of the rth order statistics Xr:n is given by
gr:n(x,a,ß, j) = {r _ 1)(n _ r)!g(x,a,ß, j)G(x,a,ß,j)r_1 [1 - G(x,a,ß, j)]n_r (13)
where g(x), G(x) are pdf and cdf of MOTL distribution by equations (5) and (6). We can use the binomial expansion of [1 _ G(x)]n_l given as follows
[1 _ G(x)]n_i = E (" _ ^ (_1)'G(x,a,ß, j) (14)
i=0 \ 1 /
substitute equation (15) into the equation (14), we get
gr:n(x,a,ß, j) = {r _"1^n1)_r)!g(x,a,ß, j)G(a,ß, j)^1 (15)
We get pdf of rth order statistics for MOTL distribution from equation (15), by using above equations (5) and (6). We can express the kth ordinary moment of the rth order statistics Xr:n (E(Xk:n)) as a liner combination of the kth moments of the MOTL distribution with different shape parameters. The rth order statistic for MOTL distribution can be expressed as
a2 (1+ßx)e_ax
n!(_1)i Y ( a+ß
2
n_r n _ r n!(_1)i
gr:n (x, a, ß, j) = E . ?--tt
1=0 V 1 J (r _ 1)(n _ r)!
1 a+ß _(ß+a+aß x) 1 ' a+ß e
ß+a-a+ß
a+ß
a+ß_(ß+a+aßx) ^ l+r_1
j +(1 _ j) a+ß_(ß++aßx) e_ax
(16)
When r=1 and r=n in equation (16), we get the equations of pdf of the smallest and largest order statistics, respectively.
3.5. Limiting distributions of order statistics
Theorem: 1. IfX1:n be the minimum and Xn:n be the maximum of a random sample x = (x1,...., xn) from MOTL distribution, then
(a) lim p( Xl:n - Pn < x) = 1 - e-
n^M y qn
Xn:n pn
, qn
where pn = 0, qn = G-1(n), p*n = G-1(1 - n), q*n = 1 and G-1 (•) is given in (8).
(b) lim p(Xn:n pn < t] = exp(_e_t)
n^™ V qn < j
Proof
For MOTL distribution, by applying L' Hospital rule, we obtain
G(ex) xg(ex)
lim ' / = lim , . ' = x e^0+ G(e) e^0+ g(e)
Hence, the minimal domain of attraction of the MOTL distribution can be the standard exponential
distribution (see Theorem 8.3.6 [3]).
Subsequently, for the MOTL distribution, we can express
xfe dx iyh(xy) 0
Therefore, the maximal domain of attraction of MOLT distribution can be the standard Gumbel distribution (see Theorem 8.3.3 [3]).
3.6. Information Theory Measures
The concept of entropy has played a vital role in information theory. The entropy of a random variable can be defined in terms of its probability distribution and it has been used in distinct fields in science as a measure of variation of the uncertainty. Numerous measures of entropy have been mentioned and compared in the literature. The entropy and mutual information concepts have been formalized by Shannon (1948). [26] proposed a other measure of entropy namely, Rényi entropy as a generalization of Shannon entropy. The Rényi entropy is defined as (?) = rb log fRg5(x)dx, 5 > 0 and 5 = 1. Rényi entropy of order 1 is Shannon entropy. We consider first g5(x) given by,
g5 (x)
7'
5 a25(1+fix)5e-a5x 5 a+fi
ii ^ a+fi-(fi+a+afix) e—ax \
I1 7 a+fi e }
25
Suppose that j > Using the series expansion
' a + fi — (fi + a + afix) ax'
a + fi 6
25
7
—25 f r(25 + k) / 1 k=0 r(25)k! ' 1
a + fi — (fi + a + afix) e—ax
7
a + fi
Thus we obtain that in the case 7 > 1/2, the Renyi entropy is
IR (5) =
1 Y5a25
1 log{ ?a
E
r(25 + k)
1-1
1 — 5 6L(a + fi)5 i,y,kï=0 r(25)k!(a + fi)j V 5 'A /i + k\ fl
i
5 . (—1) (afi)jfi5 J x(5+j)e—a(5+k)x
dx}
1 1 ; 5a25 f log{ , fi)5 E
r(25 + k)
1-1
1 —5 6L(a + fi)5 ijk-r=0 r(25)k!(a + fi)j V 5
k)(i+k)(5 )(—1)'- (-fi) fi5 -amj
Similarly, we can show that in the case 0 < 7 < 2 and by using the series expansion
1 _ a + fi — (fi + a + afix)e — 7 0+fi
25
f r(25 + k) (1_ 7)k ¿0 r(25)k! (1 7)
(fi + a + afix)e a + fi
nk
Corresponding Rényi entropy is
Ir (5)
7—5a25 f
15
r(25 + k) (1— 5)k
log{ (a + fi)5 ¿0 r(25)k!(a + fi)
k) (5) (afi)fi IR x(i+j)e—a(5+k)xdx}
r(25 + k)
1 l , 7—5a25 f 1 —5 og{ (a + fi)5 ^ r(25)k!(a + fi)
i)(j) (afi)ifi r(i + j + 1) } k) \5J (afi) fi [a(5 + k)]i+j+1 }
(1 5)k
k
k
k
k
k
1
3.7. Autoregressive Time Series Modeling
Autoregressive models are types of random process, has utilized to model and predict various type of natural phenomena. In other words, autoregressive models are group of linear prediction formulas which try to predict an output of a system based on the past observations. In the following Subsections we construct and explore different autoregressive models of order 1 (AR(1)), that is, MIN AR(1) model I, MIN AR(1) model II, MAX-MIN AR(1) model I and MAX-MIN AR(1) model II with MOTL as marginals.
3.7.1 MIN AR (1) Model-1 with MOTL Marginal Distribution
The first AR (1) structure is given by
X f e„, with probability q
1 min(xn_i, en), with probability 1 — q
where {en } is a sequence of independently and identically distributed (iid) random variables independent of {Xn} and q € (0,1). Hence the process is stationary Markovian with MOTL distribution as marginal.
Theorem: 2. {Xn } is stationary Markovian with MOTL distribution with parameters q, a, ft ^^ {en } is districted as two-parameter Lindley distribntion, nnder in an AR (2) process defined in (27).
Proof. Let en follows two-parameter Lindley distribution with parameters a and ft. Using equation (17), we can express
Fxn (x) = QFen (x) + (1 — Q)Fxn _ 1(x)
While under stationary equilibrium,
F (x) = QjPe(x)
Fx (x) = 1 — (1 — q)F (x)
and therefore
Fe (x) - FX (X)
Q + (1_ Q)FFx(x)
When en, it follows two-parameter Lindley with parameters a and ft
F (x) _ (a + ft + aftx) e _ax
F (x)= a + ft '
Hence
FFx (x) = r-
Q(a+ft+ aftx) e _ ax
1 — q( ^O+ft1 e _ ax)
a+ft
It is easy to see that this is the sf of the MOTL(a, ft, q). Conversily, if
Q(a+ft+ aftx) e ax
Fx(x) ~ P
1 _ q(ia+Mx) e _axj
a+ft
then Fen (x) is distributed as two-parameter Lindley distribution with parameters a and ft, and
the process is stationary.
We have to prove its stationarity, so we take that Xn-1^ MOTL(a, ft, q) and en follows two-parameter Lindley distribution with parameters a and ft, then
Q(a+ft+ aftx) e—ax
r r \ a+ft
fx (x)- F
1 - q^la+J+Me-axj
It is easy to see that Xn is distributed as MOTL(a, ft, q).
3.7.2 MIN AR (1) Model-II with MOTL Marginal Distribution
The second AR (1) structure is given by
{Xn-1, with probability Qi
en, with probability q2 (18)
min(Xn-1, en), with probability 1 - Q1 - q2
where Q1, q2 > 0, Q1 + q2 < 1 and {en} is a sequence of iid random variables independent of {Xn }. This structure allows probabilistic selection of process values, innovations and combinations of both. Then the process is stationary with MOTL as marginal.
Theorem: 3. {Xn} is stationary Markovian with MOTL distribution with parameters 7, a and ft ^^ {en} is distributed as two-parameter Lindley distribution with parameters a and ft, where 7 = j-^, under in an AR (1) process with structure defined in equation (18).
Proof.
Let en follows two-parameter Lindley with parameters a and ft. By using equation (18),
Fxn (x) = Q1 [1 - (1 - FXn-1 (x))(1 - Fen (x))] + Q2F^1 (x)Fen (x) + (1 - Q1 - Q2^1 (x)
Under stationary equilibrium it becomes,
F (x) = YFe(x) FX(x) = 1 - (1 - Y)Fe(x)
where 7 = 1-Q-, it is evident that it has Marshall-Olkin form. Next, we assume that {Xn} ~ MOTL(a, ft, 7). By using equation (18), under stationarity, we can write
F (x) = (1 - Q1 )FX(x) ' (x) Q2 + (1 - Q1 - Q2)Fx (x)
Next by using Xn as MOTL(a, ft, 7), it can be obtained as
F (x) = (a + ft + aftx) e-ax
Fe (%)= a + ft ' which is the sf of two-parameter Lindley distribution with parameters a and ft.
3.7.3 MAX-MIN AR(1) model-I with MOTL Marginal Distribution
Consider now the third model with AR(1) structure
{max(Xn-1, en), with probability Q1
min(Xn-1, en), with probability q2 (19)
Xn-1, with probability 1 - Q1 - q2
where 0 < Q1, q2 < 1, q2 < Q1, Q1 + q2 < 1 and {en} is a sequence of iid random variables independent of {Xn}. Then the process is stationary Markovian with MOTL distribution as marginal.
Theorem: 4. {xn } is stationary Markovian AR (2) max-MIN process with MOTL distribution with parameters a, ft and 7 ^^ {en} is distributed as two-parameter Lindley distribution with parameters a and ft, where 7 = O^, under an AR (2) MAx-MIN process with structure (29).
Proof. Let en follows two-parameter Lindley with parameters a and ft. It is obvious from equation (19),
Fxn (x) = Q1 [1 — (1 — Fxn _! (x))(1 — Fen (x))] + Q2 FXn _ 1 (x)Fen (x) + (1 — Q1 — Q2 )Fxn _! (x) Under stationary equilibrium,
Fxn (x)- 7F (x)
1 — (1 _ 7)FFe(x)
where 7 = Q1 and Fxn (x) has Marshall-Olkin form. Let xn ~ MOTL(a, ft, 7). Then by using (19), under stationarity,
Fe (x)- ^ (x)
Q1 + (Q2_ Q1)FFxn(x) Thus, after simplification it can be written as
F (x) = (a + ft + aftx) e _ax
Fe (x)= a + ft ' Consequently, which is the sf of two-parameter Lindley with parameters a and ft.
3.7.4 MAX-MIN AR(1) model-II with MOTL Marginal Distribution
Finally, we consider the more general max-min process that includes minimum, maximum innovations and the process. The relating model with AR(1) structure being of the form
xn =
max(xn_ 1, en), with probability Q1
min(xn _ 1, en), with probability Q2 (20)
en, with probability q3
xn _ 1, with probability 1 — Q1 — q2 — q3
where 0 < Q1, q2, q3 < 1, Q1 + q2 + q3 < 1 and {en} is a sequence of iid random variables independent of {xn}. Then the process is stationary Markovian with MOTL distribution as marginal.
Theorem: 5. AR (2) MAx-MIN process {xn } with structure (20) is a stationary Markovian AR (2) MAx-MIN process with MOTL distribution (a, ft, 7) ^^ {en } is distributed as two-parameter Lindley distribution with parameters a and ft where 7 = Q|+|
Proof. Let en follows two-parameter Lindley with parameters a and ft. It is clear from equation (20),
Fxn(x) = Q1 [1 _ (1_ Fxn_ 1 (x))(1_ Fen (x))]+ Q2Fxn_ 1 (x)Fen (x) + Q3Fen (x) + (1 — Q1 _ Q2_ Q3)Fxn_ 1(x)
Under stationary equilibrium it gives,
7Fe (x)
Fxn(x)
1 — (1 — 7)Fe (x)
Figure 3: Graphs of sample path ofAR(1) Minification model I for different values of q=0.3,0.5,0.8, a=30 and ft = 0.02
Figure 4: Graphs of sample path of AR(1) Minification model II for different sets of (q1, q2)=(0.1,0.4), (0.4,0.1), (0.4,0.4), a=0.2 and ft =0.3
_ Q1+Q3 Q2+Q3
we obtain
where 7 = Q1+Q3, which has Marshall-Olkin form. Now let Xn ~ MOTL(a, ft, 7) and from (20),
(Q2 + Q3) FX(x)
Fe (x) =
(Q1 + Q3) + (Q2 - Q1 )Fx(x)
Thus, after simplification it reduces to
F (x) = (a + ft + aftx) e-ax
Fe (x)= a + ft 6
which is the sf of two-parameter Lindley with parameters a and ft. The sample path properties of the four AR(1) models developed in this section are displayed in Figure 3-6 and it shows how these measures vary with different values of parameters.
Figure 5: Graphs of sample path of AR(1) Minification model II for different sets of (Q1, Q2)=(0.3,0.5), (0.5,0.3), (0.4,0.4), a=1.7 and ft =1.4
Figure 6: Graphs of sample paih of AR(1) Min-max model Ilfor dzffereni seis of (qi, q2, q3 )=(0.1,0.3,0.4), (0.2,0.2,0.4), (0.1,0.1,0.1), a=0.02 and ft =0.02
4. Estimation of Parameters
In this section, we consider maximum likelihood estimation (MLE) for a given sample of size xi, x2,..., xn from MOTL(a, ft, 7), then the log likelihood function is given by
n n
l(g) = n(log7 + 2loga) + J]log(1 + ftx,) -a J^xf-nlog(a + ft)
i=1 i=1
- 2^lo^1-(1-7)
z'=1
(a + ft + aftx/ )e a + ft
The partial derivative of the log likelihood functions with respect to the parameters are
ft(g) da
2n n
--L xi -
i=1
a
—^ - 2 L(1 - 7)
a + ft i=1( ^ 1 - (1 - 7)
((a2 + (a + ft)aft%i ^"^i)
(a+ft+aftx,' )e-a+ft
(a + ft)2
dl (g) 3ft
E
+2
(1 - 7)a2X'e
2V „—ax,'
11 + ftx' a + ft i=1{1 - (1 - 7)
(a+ft+aftx, )e
a+ft
(a + ft)2
dl (g) d7
(a+ft+aftx, )e-a+ft
2 L_
7 ,=1 (1 - (1 - 7) (a+ft+aftx,)e-
a+ft
The MLE g=(a, ft, 7 )T of g =(a, ft, 7)T can be numerically obtained by solving the equations = 0, ^ft = 0, = 0. For this purpose, we can use functions like nlm, fitdist or optimize from the programming language R.
n
x
n
n
5. Application
Now we use a real data set to show that the MOTL distribution can be a better model than the some other generalized exponential and Lindley distributions. The distributions are given below:
1. beta generalized exponential distribution (BGE) [9]
2. exponentiated exponential distribution (EE) [7]
3. gamma generalized exponential distribution (GGE) [39]
4. Kumaraswamy generalized exponential distribution (KGE) [6]
motl
"1-T
2 3
x
Figure 7: pdffor fitted distributions of the breaking stress of carbon fibers data
0
2
3
4
5
0
4
5
x
5. Weibull generalized exponential distribution (WGE) [1]
6. two-parameter Lindley distribution (TL) [31]
7. one-parameter Lindley distribution (OL) [16,17]
8. classical exponential distribution (E)
The data set represents the breaking stress of carbon fibers of 50mm in length (n=66) and it has been given by [24]. The data set is given as:
3.70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11,3.56, 4.42, 2.41, 3.19, 3.22,1.96, 3.28, 3.09, 1.87, 3.15, 4.90, 1.57, 2.67, 2.93, 3.22, 3.39, 2.81, 4.20,3.33, 2.55, 3.31, 3.31, 2.85,1.25, 4.38,1.84, 0.39, 3.68, 2.48, 0.85,1.61, 2.79, 4.70, 2.03, 1.89, 2.88,2.82, 2.05, 3.65, 3.75, 2.43, 2.95, 2.97, 3.39,2.96, 2.35, 2.55, 2.59, 2.03, 1.61, 2.12, 3.15, 1.08, 2.56,1.80, 2.53
The distribution of this data set is unimodal and slightly left skewed (skewness = 0.131 and kurtosis = 3.223). For each distribution, we estimated the unknown parameters (by the maximum likelihood method), the values of the -log-likelihood (—logL), AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), the values of the Kolmogorov-Smirnov (K-S) statistic and the corresponding p-values.
All the computations were done through the use of the R programming language. The results for these data are listed in Table 3. From the table, we can observe that MOTL distribution provides smallest -logL, AIC, BIC and K-S statistics values and highest p-value as compare to other distributions. This indicates that the MOTL distribution provides a better fit than other distributions. Plots of the histogram with fitted density functions and comparison of the cumulative distribution function for the each models with the empirical distribution function are displayed in Figure ?? and Figure ??. From figures, one can easily identify the suitability behavior of the MOTL distribution. Therefore, the new model may be an interesting alternative to the other available generalized exponential models in the literature.
Table 3: Estimated values, -logL, AIC, BIC, K-S statistics and p-valuefor data set.
Distribution Estimates -logL AIC BIC K-S p-value
MOTL(a, /,Y) a = 2.2964 84.5821 175.1641 181.7331 0.0599 0.9717
/= 11012.6970
Y= 79.8819
//= 4.323281e+06
BEG( a,b,/) a=7.5387 90.9961 187.9922 194.5611 0.1309 0.2082
b=20.1066
/3=0.1176
EE( a,/) a=9.2945 95.2187 194.4373 198.8166 0.1527 0.0921
/3=1.0092
GGE( a,/) a=7.5710 90.9359 185.8719 190.2512 0.1303 0.2124
/3=2.7395
KGE(a,b,/) a =4.3710 86.5579 179.1158 185.6848 0.0899 0.6603
b= 53.0724
//= 0.1696
WGE(a,b,/) a3 = 3.4608 85.7900 177.58 184.149 0.0799 0.7925
b= 5.0392
//= 1.6439
Y=50.4204
TL(a, /) a= 2.2410 112.0511 228.1022 232.4815 0.2510 -
//=174.011
OL(/) /3=0.5895 181.7535 246.8953 249.085 0.3017 -
E(/) //= 0.3618 133.0921 268.1842 270.3739 0.35764 -
x x
Figure 8: Estimated cumulative distribution function for the data set
6. Conclusions
The quality of the methods used in statistical analysis is eminently dependent on the underlying statistical distributions. In this article, we proffered a new customized Lindley distribution. The proposed distribution enfolds exponential and Lindley distributions as sub-models. Some properties of this distribution such as quantile function, moments, moment generating function, distributions of order statistics, limiting distributions of order statistics, entropy and autoregressive time series models are studied. This distribution is found to be the most appropriate model to fit the carbon fibers data compared to other models. Consequently, we propose the MOTL distribution for sketching inscrutable lifetime data sets.
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