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A Generalization of Lindley Distribution: Characterizations, Methods of Estimation and Applications
A. Mohammed Shabeer1, Bindu Krishnan2, K. Jayakumar3
1,3 Department of Statistics, University of Calicut, India 2 Department of Data Science, CS & IT, Jain(Deemed-to-be-University), India md.shabeer.edv@gmail.com, bindusruthy@gmail.com, jkumar19@rediffmail.com
Abstract
Lindley distribution is a lifetime model with application in survival analysis and reliability theory problems often centred around its increasing hazard rate function and flexibility over exponential distribution. In this paper, we introduce a new generalization of the Lindley distribution referred to as Lindley Truncated Negative binomial (LTNB) distribution. The LTNB model has increasing, decreasing and upside-down bathtub(UBT) shapes for the hazard rate function. Various properties of the LTNB distribution are studied including moments, quantiles, and stochastic ordering. Characterizations of the new distribution are obtained. Maximum likelihood, Cramer-von-Mises, ordinary and weighted least squares methods of estimation are utilized to obtain the estimators of the model parameters. A simulation study is carried out to assess and compare the performance of different estimates. An autoregressive time series model with the LTNB as marginal is developed. The model is fitted to bladder cancer data set to show how the proposed model works in practice.
Keywords: Autoregressive Models, Characterizations, Lindley distribution, Maximum Likelihood, Stochastic Ordering.
1. Introduction
As a counter-example to fiducial statistics, the Lindley distribution, first put forth by [16], is one of the crucial lifetime distributions in the framework of Bayesian statistics. The Lindley distribution has an increasing hazard rate function; however, in real-life situations, the models exhibit non-monotone hazard rate shapes. For example, in the case of a serious illness condition such as cancer, the hazard rate increases and then decreases. Similarly, in the engineering field, the quality of production by untrained workers follows a similar pattern. As a result, many researchers have developed models with non-monotone hazard rates over time. Models with bathtub shapes can be found in [22]. Most often, upside-down bathtub (UBT) shape hazard rate models are used to model medical data, such as patient data for bladder cancer and lung cancer (see, [5] and [15]). In this paper, we introduce a generalized Lindley distribution with increasing, decreasing and UBT shapes for the hazard rate function. A positive random variable X is said to have Lindley distribution with the parameter a, denoted as L(a), if it has the probability density function (pdf)
f (x) = 0+1 (1 + X)e-ax;x > 0,a > (1)
The cumulative distribution function (cdf) is given by
F(x) = 1 - a + 1 + ax e-ax; x > 0. (2)
w a +1 w
A. Mohammed Shabeer, Bindu Krishnan, K. Jayakumar RT&A, No 1 (72)
LTNB distribution Volume 18, March 2023
Ghitany[8] studied the statistical characteristics of the Lindley distribution and demonstrated its flexibility in modelling lifetime data over exponential distribution. Generalized Lindley distribution was later developed by [20], who also studied its mathematical characteristics. Weighted Lindley distribution was introduced by [9]. An extended Lindley distribution was proposed and investigated by [3]. In 2013, [6] introduced a generalized Lindley distribution. Later Beta Lindley distribution was examined by [18]. The Lindley-Exponential distribution was first proposed by [4]. Inverse Lindley distribution was extended by [23]. The Odd Lindley Burr XII distribution was proposed by [14]. The generalized two-parameter Lindley distribution was introduced by [7]. Unit-Lindley distribution was studied in [2].
The rest of the paper is as follows: The Lindley Truncated Negative binomial (LTNB) distribution is introduced as a compounded distribution in Section 2. Section 3 discuss the statistical properties of the LTNB distribution such as moments, quantile function, skewness, and kurtosis. Characterizations of the LTNB distribution are obtained in section 4.In Section 5, we investigate the stochastic ordering property of LTNB. In section 6, we use the maximum likelihood method, least squares method, weighted least squares, and Cramer-von-Mises-estimator for estimating the parameters of the new distribution. In section 7 we carry out a simulation study to evaluate the performance of Maximum likelihood estimation and other estimation methods. The LTNB distribution is used to model remission times of bladder cancer patients in Section 8. It is demonstrated that the LTNB distribution fits this data set better than the other well-known competitors. In section 9, we develop a first-order autoregressive minification process with the LTNB distribution as the marginal distribution.
2. Lindley Truncated Negative binomial distribution
We propose a new generalization for Lindley distribution with relatively simple expressions for the survival function, hazard rate function, and quantile's. The hazard rate function is more flexible than the previous generalizations and the proposed model has a closed-form for distribution function. In addition, the new distribution fits some real-world data sets better than existing Lindley distribution generalizations.
In 1997, [17] developed a method of adding a tilt parameter to a distribution to extend existing distributions. Many distributions were extended and published in the literature as a result of this technique. As a generalization of the above technique, [21] proposed a novel family of distributions through truncated negative binomial distribution. It's important to remember that these distributions arise as the distribution of random minimum or random maximum. The cdf is given by
G( z)
1 - y
-[(F(z) + jF(z))-n - 1];z e R, 7,0 > 0,
(3)
where F(.) is the cdf of baseline distribution . If F(x) follows Lindly with cdf (2) we have Lindley Truncated Negative binomial(LTNB) distribution, LTNB(a, 7,0), having pdf
, m a2(a + 1)0(1 - 7)0y0(1 + z)e-az
g(z; a, 7,0) —
(1 - 70) [(a + 1) - (1 - 7)(a + 1 + az)e-az]0+1 The cumulative distribution function of LTNB(a, 7,0) is given by
; z > 0 a, 7,0 > 0.
G(z; a, 7,0)
7°
1 - 70 1 - 70
a + 1
(a + 1) - (1 - 7)(a + 1 + az)e-
and hence the survival function is
G (z; a, 7,0) =
7
0
1 - 70
a + 1
(a + 1) - (1 - 7)(a + 1 + az)e-
1
(4)
(5)
(6)
Remark: The LTNB(a, 7,0) distribution reduces to the Lindley distribution when 0 = 1 and 7 1.
0
1
0
Figure 1: LTNB (a, у, в) pdf when у = 0.5, в = 0.5.
Figure 2: LTNB (a, у, в) pdf when a = 1.
The hazard rate function of LTNB distribution is
h(z; a, у, в) =
a2(a + 1)в(1 - у)в(1 + z)e-
[(a + 1) - (1 - y)(a + 1 + az)e-az] (a + 1)e - [(a + 1) - (1 - + 1 + az)e-azf and the reverse hazard rate function is
r(z; a, у, в) =
a2(a + 1)в(1 - у)в(1 + z)e-
[(a + 1) - (1 - y)(a + 1 + az)e-az] [(a + 1) - (1 - y)(a + 1 + az)e-az]в - ув(a + 1)
(7)
(8)
Figure 3 depicts hazard rate function graphs for various parameter values. The graphs showing decreasing, non-decreasing, and UBT shapes for hazard rate function.
3. Statistical Properties We discuss about moments, simulation, quantiles, skewness, and kurtosis in this section.
x
в
Figure 3: LTNB (a, 7,0) hazard rate function for various parameter values.
3.1. Moments
Let X ~ LTNB (a, 7,0), then its rth moment with respect to the origin is given by,
r a2(a + 1)0(1 - 7)070(1 + x)e-ax
E[Xr ]
-axl 0+1
/0 (1 - 70) [(a + 1) - (1 - 7)(a + 1 + ax)e-ax]
dx
= £ J^^i' r) r xr (1 + x)« + 1 + «x)>dx.
3.2. Simulation, Quantiles and Median
In order to generate random numbers from a LTNB(a, 7,0) distribution, we use
X = — - 1 - 1W-1 (-e-(a+1)^+1)(1 - 7((1 - 70)Y + 70)-1/0)) a a 1 - 7
where W-i(.) is the negative Limbert W function and Y ~ U(0,1). The qth quantile of the LTNB(a, 7,0) is
X = — - 1 - 1W1 (-e-(a+1)^+1)(1 - 7((1 - 70)q + 70)-1/0)), a a 1 - 7
and thus the median of LTNB(a, 7,0) is,
X = a - 1 - aW-1(-e-(a+1)(0)(1 - 7((1 - 70)2 + 70)-1/0)).
(9)
(10)
3.3. Skewness and Kurtosis
According to [13], the distribution skewness can be calculated using the below equation
Q (4) - 2Q( 1) + Q( 1)
S
Q( 4) - Q( 4)
and according to [19], the kurtosis of the LTNB distribution is as follows
K
Q (8) - Q(5) + Q(3) - Q( 1) Q(6) - Q(2) '
where Q(.) is the quantile function of X as defined by (1°). These metrics are less susceptible to outliers.
4. Characterizations of LTNB distribution
Many characterization results, established in various ways, can be found in the literature. In this section, the ratio of truncated moments and the hazard rate function are used to characterize the LTNB distribution.
4.1. Characterizations based on two truncated moments
This section discusses how to characterise the LTNB distribution using the ratio of two truncated moments. In our first characterization, we apply the Theorem 1 below established in [10].
Theorem 1. Let (Q, F, P) be a given probability space and let H = [d, e] be an interval for some d < e (d = — ro,e = romight as well be allowed). Let X : Q ^ H be a continuous random variable with the distribution function F and let q1 and q2 be two real functions on H such that
E[q2(X)|X > x] = E[q1(X)|X > x]Z(x),x e H,
is defined with some real function Z. Assume that q1, q2 e C1(H), Z e C2 (H) and F is twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation Zq1 = q2 has no real solution in the interior of H. Then F is uniquely determined by the functions q1, q2 and Z, particularly
F(x) = [C Z(u)q1(u()u— q2(u)lexp(—s(u))du,
where the function s is a solution of the differential equation s' = ^ —1q2 and C is the normalization constant, such that JHdF = 1.
Theorem 2. Let X be a positive real valued continuous random variable with pdf f (x).
1 + a + ax 1 + a + ax
Denote q1 (x) = (1 — (1 — y)--— e-ax)9+1 and q2(x) = q1 (x)--— e-ax for x > 0,a >
a + 1 a + 1
0,9 > 0, y > 0. Then, the random variable X has pdf (4) if and only if the function Z(■) defined in Theorem 1 is of the form
n \ 11 + a + ax _ax
Z (x) = -z-;— e
' 2 a +1
Proof. Suppose the random variable X has the pdf (4). Then
and
(1 — F(x))E [q1 (X)|X > x] = C1 +a+1 axe-ax, x > 0 (1 — F(x))E [q2(X)lX > x] = C ^ +aa++1 2e-2ax, x > 0
where C = 9—1),97^ .Further,
1 1 + a + ax Z(x)q1(x) — q2(x) = — ^q1(x) a + 1 e-ax < °, x > 0.
Conversely, if Z is of the above form, then
s' (x)= Z (x)q1(x) = a2(1 + x) , x > 0. ( ) Z(x)q1(x) — q2(x) 1 + a + ax' '
Thus
s(x) = —log(1 +aa++1 axe-ax), x > 0. Now from Theorem 1, result follows.
■
A. Mohammed Shabeer, Bindu Krishnan, K. Jayakumar RT&A, No 1 (72)
LTNB distribution Volume 18, March 2023
4.2. Characterization based on the hazard rate function
For 9 = 1, we characterise the LTNB distribution using the hazard rate function.
Theorem 3. Let X be a positive real valued continuous random variable with hazard rate function h(x). The random variable X has LTNB distribution, for 9 = 1, if and only if h(x) satisfies the differential equation
h(x)__h(x) = a4(1 - 7)(1 + x)2e-ax x > Q
(X) (1 + x)(1 + a + ax) (1 + a + ax)(1 - (1-7)(i+++ax)e-ax)2, x > .
Proof. If X has pdf (4), the above differential equation clearly holds. Now suppose the differential equation holds then,
d (h(x)(1 + a + ax)] = d \ 1 [
dx\ 1 + x ) dx \ i _ ((i-7)(i+«+«x)g-ax ) which implies,
m s a2(1 + x) 1
h(x) —
(1 + a + ax) i _ ((1_7)(1+"+"x)e-
1+a
which is the hazard rate function of LTNB distribution.
■
5. Stochastic Ordering
Let X and Y be two random variables with distribution functions of F and F2, respectively, with corresponding pdfs of /1 and /2. Then it is stated that X is less than Y in ,
i) stochastic order (denoted as X <st Y) if F1(x) > F2(x) for all x;
ii) hazard rate order (denoted as X <hr Y) if (1 _ F (x)/(1 _ F2(x)) is decreasing in x > 0; ii) likelihood ratio order (denoted as X <ir Y) if /1(x)//2(x) is decreasing in x > 0;
iv) reverse hazard rate order (denoted as X <rhr Y) if F1(x)/F2 (x) is decreasing in x > 0. The four orderings have a relationship with one another; for further information, see [24],
X <ir Y X <hr Y X <rhr Y X <st Y. (11)
For 72 > 71,
let X - LTNB(a, j1,9) and Y - LTNB(a,y2,9). Then
/x(y) _ 71 (1 _ 72)(1 _ 71 ) [(a + 1) _ (1 _ Y2)(a + 1 + az)e_ay]9+1
fY(y) y2(1 _ 71 )(1 _ 72) [(a + 1) _ (1 _ 71)(a + 1 + ay)e_ay]9+1
Since y2 > 71,
7?(1 _ y2)(1 _ 71 )a2e_ay(1 + y)(a + 1)(71 _ 72)
d_ dx
/x (y) L/y (y)J
(1 - 72)y2 (1 - 7?)
x [(a + 1) - (1 - 72)(a + 1 + ay)e-ay]9 [(a + 1) - (1 - 71 )(a + 1 + ay)e-ay]9+2' < 0.
fx (y) /fy (y) is decreasing in y. That is X <ir Y. From (11), the remaining ordering follows.
6. Different methods of estimation
In this section we use maximum likelihood estimation, least squares, weighted least squares and cramer-von-Mises estimation methods to estimate the parameters of the LTNB distribution.
6.1. Maximum likelihood Estimation
Let x1,x2, ---,xn be an observed random sample from LTNB distribution with unknown parameter vector v = (a, j, 9)T. Then the log-likelihood function is
n
l = 2nloga + nlogd + nlog(1 - y) + ndlogy + ndlog(a + 1) - nlog(1 - y9) - a £ Xi
i=1
n n
+ £ log(1 + Xi) - (9 + 1) £ log [(a + 1) - (1 - y)(a + 1 + aXi)e-aXi] (12)
i=1 i=1
The partial derivatives are given by dlog l _ 2n n9
_ + —__£;
da a a +1 =
n
-(9 +1) £ i=1
1 - (1 - y)e aXi (1 - aXi - ax2) (a + 1) - (1 - Y)(a + 1 + axi)e-axi
dlog l = -n n9 n9y9-1 dy 1 - y Y 1 - Y9
(a + 1 + axi )e
-(9 +1) £ i=1
(a + 1) - (1 - y)(a + 1 + axi)e
-axi
axi
(13)
(14)
dlog l , n , , „. ny9logy
-dT = nlogY + 9 + nlog(a +1)+ lTyr
n [ ]
- £ log [(a + 1) - (1 - Y)(a + 1 + axi)e-axi] (15)
i=1
Set the score vector to zero,
/dloge dloge Blogev U(V) V da ' dy ' 30 J
U(v) = 0, and solve them simultaneously to obtain the ML estimators a, j, and 9. These equations can be numerically solved using statistical software using iterative techniques like the Newton-Raphson algorithm as they cannot be solved analytically. All of the second order derivatives exist for the three-parameter LTNB distribution as well.
6.2. Least squares and weighted least squares estimators
Let t1 < t2 < t3 < ... < tn be the n ordered random sample from any distribution with cdf F(t). Then we have,
E[ F(ti)] = .
L W/J n + 1
The least squares method minimizes
Plse(a,Y,9) = t (f(U) — (16)
with respect to the unknown parameters. Here the least squares estimates are obtained by minimizing the following equation with respect to a, j, 9.
_ / 1 j9 ' ■ X2
Plse (a, y, 9) = £
=1 \ l - y9 1 - y9
a + 1
(a + 1) - (1 - y)(a + 1 + ax)e-
i ^ (17)
n +1
Weighted least squares estimates of a, 7, 9 are obtained by minimizing the following equation with respect to a, 7, 9.
v i m n (" +1)2 (" +2)/ 1
Pwlse (a, Y, Ö) = 1
¿=1 i(n — i + 1) y 1 — Ye 1 — y'
a +1
(a + 1) — (1 — Y)(a + 1 + ax)e-
n +1
(18)
2
e
e
Y
6.3. Cramer-von-Mises-estimator(CME)
CME is obtained by minimizing the following equation with respect to a, 7,9. Here F(.) is the distribution function of LTNB distribution given by (5) .
1 n ( 2i — 1 \ 2 Pcme(a b,7,9) = + L(m) - . (19)
7. Simulation
Monte Carlo simulation is used to evaluate the performance of the maximum likelihood estimation procedure for estimating the LTNB parameters. From the LTNB model we generates samples of sizes n = 100,200,300,400, and 500 for various combinations of a, 7, and 9. We ran the simulation 1000 times to determine the MLE's and MSE's of the parameter estimates. Table 1 displays the results. We can observe that the ML estimates are consistent. To investigate the efficiency of least squares estimators, we took samples from the LTNB distribution with n = 100,200,300,400, and 500. We repeated the simulation 1000 times and computed the estimates and corresponding MSE's for the same set of parameter values using three methods. Table 2 displays the results. Table 2 shows that least squares and CME estimators perform similarly and both methods are giving smaller MSE's compared to weighted least squares estimates.
8. Data Analysis
Table 3 contains data on the remission times (in months) of a random sample of 128 bladder cancer patients provided by [15]. Table 4 displays the descriptive statistics for the data.
We will now look at the Total Time on Test (TTT) plot, a graphical method for determining the shape of the data's hazard rate function. The empirical TTT plot is defined as,
(r \ n
E x(1) +(n — r)xM / E x(i), r =12.., n
i=1 J i=1
where x(i) denote the ith order statistic of the sample. The shape of the hazard rate function would be increasing, decreasing, bathtub-shaped, and UBT if the TTT transform is concave, convex, convex then concave, and concave then concave (see [1]). Figure 3 depicts the data's TTT plot. The hazard rate function is clearly UBT. The Akaike Information Criterion (AIC), Corrected Akaike Information Criterion (AICC), Bayesian Information Criterion (BIC), and Hannan-Quinn Information Criterion(HQIC) are s the goodness-of-fit metrics that we take into consideration,
AIC = 2k — 2log l,
AICC = 2kn , — 2log I, n — k— 1 —
BIC = klog n — 2log I, HQIC = 2klog(log n) — 2log I,
Table 1: MLE's and their corresponding MSE's of LTNB distribution parameters.
a y 0 n a( MSE(a)) Y( MSE(Y)) 0( MSE( 0))
.8 .2 1.5 100 .9196(.1598) .2055(.0266) 1.5251(.7527)
200 .8551(.0888) .1989(.0158) 1.5342(.7232)
300 .8813(.1184) .2001(.0193) 1.5428(.6812)
400 .8347(.0506) .2015(.0091) 1.5034(.6691)
500 .8171(.0382) .1986(.0064) 1.5098(.5061)
.5 .3 2 100 .5698(.0539) .2824(.0449) 1.9985(1.2211)
200 .5298(.0313) .2726(.0309) 1.9952(.9892)
300 .5202(.0230) .2836(.0245) 2.0922(.7610)
400 .5122(.0192) .2856(.0222) 2.0962(.5948)
500 .5087(.0142) .2942(.0164) 2.0962(.5484)
.3 .2 1.5 100 .3403(.0229) .2075(.0309) 1.4718(.6438)
200 .3179(.0111) .2088(.0201) 1.6118(.4881)
300 .3141(.0087) .2047(.0149) 1.5339(.2959)
400 .3109(.0065) .2026(.0112) 1.4966(.2363)
500 .3078(.0049) .2073(.0092) 1.5387(.1401)
where 1 is the likelihood function evaluated at the maximum likelihood estimates, the sample size is n, and k is the number of parameters.
We fitted the LTNB distribution to the data and compared it to the Lindley distribution with pdf (1) and the New Generalized Lindley distribution (NGLD) with pdf
e-0x (0a+V-1
f (x) = 1 + e{ ~rW~ +
Beta Lindely(BL) distribution having pdf,
c. , 02(0 + 1 + 0x)?-1 (1 + x)e-0^x f(x) = —
m
1 _ 0 + 1 + 0x e-0x
a-1
B(a, p)(0 + 1)P V 0 +1
new generalized two-parameter Lindley distribution (NG2PLD) distribution having pdf,
f(x)
02
1+
0a-2xa-1
0 +1 ' r(a)
transmuted Lindley distribution(TLD) having pdf,
O-0 x
f (x) = ^^^ (1 - A + 2A0-±l+xe-"
(0 + 1)
and exponential distribution(ED) having pdf,
f (x) = ae-
Table 2: LSE, WLSE, CVME and corresponding MSEs of parameters of LTNB distribution.
a Y 0 n a( MSE(a)) y( MSE(Y)) (MSE( ))
Least squares 100 .6262(.0015) .2178(.0007) 1.5386(.0043)
.6 .2 1.5 200 .6206(.0009) .2151(.0005) 1.5408(.0045)
300 .6173(.0005) .2141(.0004) 1.5408(.0038)
400 .6162(.0005) .2136(.0003) 1.5437(.0039)
500 .6150(.0004) .2122(.0002) 1.5441(.0039)
Weighted Least squares 100 .6194(.1305) .1308(.0288) 1.3082(2.3691))
.6 .2 1.5 200 .6062(.0893) .1474(.0186) 1.3082(1.7783)
300 .5902(.0641) .1595(.0134) 1.3271(.7774)
400 .5870(.0514) .1660(.0102) 1.3618(.6277)
500 .5938(.0478) .1729(.0089) 1.3942(.5434)
CVM 100 .6261(.0015) .2178(.0008) 1.5386(.0042)
.6 .2 1.5 200 .6206(.0009) .2151(.0006) 1.5440(.0046)
300 .6172(.0005) .2141(.0004) 1.5408(.0038)
400 .6162(.0005) .2135(.0004) 1.5434(.0041)
500 .6150(.0004) .2122(.0002) 1.5441(.0039)
The table 5 lists the parameter estimates and goodness of fit statistics for bladder cancer patient data.
The LTNB distribution is more suitable for these data since the values of -logl, AIC, AICC, BIC, and HQIC for the LTNB distribution are lower than those of the other competing models. Figure 4 displays the fitted densities.
9. Time series Models with LTNB marginals
Several researchers have developed and studied time series models with non-Gaussian marginals (see, for example, [11], [12], and [25]). The following definition is required in order to create the time series model with LTNB marginal distribution
Definition 9.1. A positive real valued random variable X is said to have Marshall-Olkin Lindley
Truncated Negative binomial distribution and write X == MOLTNB(#,a, y,0) if it has the survival function
Fx (x)
1
1 + 1
1 + «
((a+1)-(1-7)(a+1+ax)e-ax)0 -je (a+11)0 Y0 [(a+1)0 — ((a+1) — (1—y)(a+1+ax)e-ax)0]
(21)
Theorem 4. The first order auto regressive (AR(1)) process given by
Xn
£ n w.p 5
min(Xn-1,£n) w.p 1 - 5
(22)
Table 3: Remission times of Bladder Cancer Patient Data.
0.08 2.09 3.48 4.87 6.94 8.66 13.11 23.63 0.20 2.23 3.52 4.98
6.97 9.02 13.29 0.40 2.26 3.57 5.06 7.09 9.22 13.80 25.74 0.50
2.46 3.64 5.09 7.26 9.47 14.24 25.82 0.51 2.54 3.70 5.17 7.28
9.74 14.76 26.31 0.81 2.62 3.82 5.32 7.32 10.06 14.77 32.15 2.64
3.88 5.32 7.39 10.34 14.83 34.26 0.90 2.69 4.18 5.34 7.59 10.66
15.96 36.66 1.05 2.69 4.23 5.41 7.62 10.75 16.62 43.01 1.19 2.75
4.26 5.41 7.63 17.12 46.12 1.26 2.83 4.33 5.49 7.66 11.25 17.14
79.05 1.35 2.87 5.62 7.87 11.64 17.36 1.40 3.02 4.34 5.71 7.93
11.79 18.10 1.46 4.40 5.85 8.26 11.98 19.13 1.76 3.25 4.50 6.25
8.37 12.02 2.02 3.31 4.51 6.54 8.53 12.03 20.28 2.02 3.36 6.76
12.07 21.73 2.07 3.36 6.93 8.65 12.63 22.69
Table 4: Descriptive Statistics of Cancer data.
Min. Qi Median Mean Q3 Max. Var.
0.08 3.348 6.395 9.366 11.838 79.05 110.425
where 0 < 5 < 1; n > 1, defines a stationary AR(1) minification process with LTNB(a, 7,9) as marginal distribution if and only if £n's are i.i.d MOLTNB(5-1, a, 7,9) with X0 = CL(a, 7,9). Proof. If {Xn } is stationary with LTNB(a, 7,9) marginals, then
F n (x)
Fx (x)
S + (1 - S)Fx(x)
•JÖ
7
1-7»
a+1
(a+1)-(1-7)(a+1+ax)e-ax
1
s + (1 - S)
a+1
(a+1)-(1-7)(a+1+ax)e-1
1
1+S
((x+1)-(1-7)(x+1+xx)e-ax )»-7» (a+1)» 7» [(a+1)»-((x+1)-(1-7)(x+1+xx)e-ax )» ]_
That is, £n's are i.i.d MOLTNB(5-1, a, 7,9).
Conversely, if £ n's are i.i.d MOLTNB(5-1, a, 7,9) with X0 = CL(a, 7,9), then,
Fx (x) = 5F^ (x) + (1 - 5)F£1 (x)Fx0 (x)
1
= S
1+S
(1 - S)
((x+1)-(1-7)(x+1+xx)e-ax )»-7» (a+1)» 7» [(a+1)»-((a+1)-(1-7)(a+1+ax)e-ax )» ]_
1
+
1 + S
((a+1)-(1-7)(a+1+ax)e-ax )»-7» (a+1)» 7» [(a+1)»-((a+1)-(1-7)(a+1+ax)e-ax )» ]_
1
1+
e
7
1 - 71
9
((a+1)-(1-7)(a+1+ax)e-ax )»-7» (a+1)» 7» [(a+1)»-((a+1)-(1-7)(a+1+ax)e-ax )» ]_
a + 1
(a + 1) - (1 - 7)(a + 1 + ax)e-ax_
(23)
0
0
cq o
CD
_ o c
o
0.0 0.2 0.4 0.6 0.8 1.0
i/n
Figure 4: The empirical TTT plot of the data.
Table 5: Parameter estimates and goodness of fit statistics for various models fitted to the data.
Model Estimates -l°gi AIC AICC BIC HQIC
LTNB a = = 0.0617, Y = 0.1218, 409.23 824.47 824.67 833.03 827.95
§ = 1.5407
NGLD a = 4.679, ji = 1.324, 412.75 831.50 831.69 840.06 834.98
§ = 0.180
BL a = 1.340, j = 0.065, 412.80 831.60 831.80 840.16 835.08
§ = 1.861
NG2PLD a = = 0.8303, § = 0.2942 413.37 830.73 830.83 836.44 833.05
TLD § = = 0.156, X = -0.617 415.15 834.31 834.41 840.01 836.62
LD a = 0.1960 419.53 841.06 841.09 843.89 842.20
ED a = 0.097 426.76 855.53 855.56 858.37 856.68
0 20 40 60 80
x
Figure 5: Estimated pdf
That is, Xi == LTNB (a, y, 9).
If we assume that Xn-1 == LTNB(a,y,9), then by induction, we can establish that Xn == LTNB(a, y, 9) . Hence the process {Xn} is stationary with LTNB marginals.
FUNDING
The first author wishes to thank the University Grants Commission, New Delhi for the financial assistance in the form of Junior Research Fellowship.
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