CC BY
26
Exponentiated Discrete Hypo Exponential Distribution
and its Generalizations
Krishnakumari.K1 and Dais George2
Research Scholar, St.Thomas College, Palai (SAS SNDP YOGAM College, Konni), Kerala, India. 2Catholicate College, Pathanamthitta, Kerala, India.
[email protected], [email protected]
Abstract
Generalizations of standard probability distributions is a thought-provoking concept in statistical literature and was inspired by many researchers in recent days. This is because the addition of parameters may increase the flexibility of the new models. Now a days various generalization techniques are available in literature. In this work, we proposed a generalization of discrete hypo exponential distribution and studied its various properties. A real data analysis is carried out and check the flexibility of the new model by comparing it with other standard distributions. Two generalizations of the proposed distribution are introduced.
Keywords: Discrete hypo exponential distribution, Estimation, Generalization, Moments, Stress-strength analysis.
1. Introduction
Over the last few decades, there has been growing interest in adding supplementary parameters to the baseline distributions to broaden generalized families of distributions. The addition of parameters may increase the flexibility of the new models. So generalization of the standard distributions are attracted by many researchers and are prominent in recent days. In literature there exists various generalization techniques and for a detailed review, see Tahir and Nadarajah[32]. These techniques resulted in the generalizations of various standard distributions. For details see, Gupta and Kundu [14], Eugene et.al [11], Zografos and Balakrishnan [33], Gomez-Deniz [13], Mahmoudi and Zakerzadeh [19], Cordeiro and Castro [4], Nadarajah [24], Nadarajah et.al [26], Cordeiro et al. [5], Ristic and Balakrishnan [30], Lemonte et.al [17], Liyanage and Pararai [18], Merovci and Elbatal [22], Merovci and Sharma [23], Nadarajah and Bakar [25], Ahmad and Ghazal [1], Sulami [2] etc. Recently exponentiated family of distributions due to Lehman [16] has got special attention and various standard distributions were generalized. The most prominent distribution introduced in the 20th century is the exponentiated exponential distribution and inspired by this many existing distributions were generalized and for details see, Pal et al. [28], Nekoukhou and Bidram [27], Morshedy et al. [8], El-Bassiouny et al. [7], Morshedy et al.[10], Morshedy et al. [9], Mashhadzadeh and Mirmostafaee [21] and Baharith and Alamoudi [3]. The layout of this article is in this way. In Section 2, we introduced exponentiated discrete hypo exponential distribution and studied its various properties. In Section 3 the parameters of the distribution is done through non linear maximization method. To evaluate the performance of the nlm estimator a simulation study is done in Section 4. A real data analysis is done in Section 5. In Section 6 some generalizations of the proposed distribution are introduced. Some concluding remarks are recorded in Section 7.
2. Exponentiated Discrete Hypo Exponential Distribution
Consider the discrete hypo exponential (DHE) distribution having model parameters 01, 02 > 0, _ 02 with the distribution function
F(x; 01,02)
- 01
(1
_ e-01x) _
- 01
(1
_ e_02x
)
(1)
By inserting (1) into the resilience parameter family of distributions, the distribution function of the resulting distribution is given by
where
G(x; 01,02, a) _ [F(x; 01,02 )]a _ V(x; 01,02, a) _ (02 _ 01 )a
V(x; 01,02, a) _ [02(1 _ e_01 x) _ 01(1 _ e_02x)]
(2)
(3)
We call such a random variable X, having distribution function (2), is an exponentiated DHE distribution with parameters 01, 02 > 0, 01 _ 02, a > 0 and denote it as EDHE (01,02, a). The probability mass function(pmf) of EDHE distribution is given by
P(X _ x) _ v(x; 01,02, a)
V(x + 1; 01,02, a) _ V(x; 01,02, a) _ (02 _ 01)a The plots of pmf of EDHE distribution is given in Figure 1.
;x _ 0,1,2,....
(4)
Figure 1: Plots of pmf of EDHE distribution
From Figure 1 it is understood that the EDHE distribution is unimodel. Since every log-concave density is unimodel, it is also inferred that EDHE distribution is log-concave.
a
2.1. Reliability characteristics
Survival function, S(x) = 1 — Vf1, f2'^ ; x = 0,1,2..., and
hazard rate, r(x)
(f 2 — f )Й V (x + 1; f 1, f 2, a) — V (x; f 1, f 2, a)
(f 2 — f1)a — V (x; f1, f 2, a) The plots of hazard rate of EDHE distribution is given in Figure 2.
Figure 2: Plots of hazard rate of EDHE distribution.
From Figure 2, it is evident that for various model parameters, the hazard rate functions can be decreasing, increasing and increasing-decreasing, which makes the EDHE distribution more flexible and can model different types of data sets such as count data, failure time data etc.
2.2. Moments
Let X - EDHE(f 1, f2, a), then for n > 1,
V (x + 1; fi, f2, a) — V (x; fi, f2, a)
TO
E(Xn) = £ xn
x=0
In particular
and
E(X)= £ x
x=0
TO
E(X2) = £ x2
x=0
(f 2 — f1)a V (x + 1; f 1, f 2, a) — V(x; f 1, f2, a)
V (x + 1; f1, f2, a) — V (x; f1, f2, a)
The expression for V(X) can be obtained using he relation
V(X) = E(X2) - [E(X)]2
2.3. Infinite Divisibility
According to Steutel and Van Harn [31], a necessary condition for infinite divisibility of a discrete distribution Py is that Po > 0. For EDHE distribution this condition is satisfied for all values of the parameters. Hence it is infinitely divisible.
2.4. Theorem
If X follows an exponentiated hypo exponential distribution with parameters , 02 and a then the random variable W=[X] follows a exponentiated discrete hypo exponential distribution with parameters 01, 02 and a .
TO
Proof:
Consider w=0,1, 2,... then using Lemmal of Krishna and Pundir (2009), we have
P(W > w) = P([X] > w) = P(X > w)
1
02(1 - e-01x) - 01 (1 - e-02x)
02 - 01
which is the survival function of EDHE distribution and hence the theorem.
2.5. Order Statistics
Order statistics are sample values placed in ascending order. It is a very useful concept in statistical sciences. It has a far reaching applications especially in modeling auctions, car races, insurance policies and estimating parameters of distributions etc.
Let X(1:n) < X(2:n) < X(3:n) < ... < X(n.n) represents the order statistics obtained from the i.i.d. EDHE(01,02, a) distribution of size n. Then probability mass function of first order statistics is given by
fX(1:n)(x)
1
V(x - 1; 01,02, a)
(02 - 01 )a
1
V(x; 01,02, a) (02 - 01 )a
and the distribution function is
FX(1:n)(x) = 1 -
1
V (x; 01,02, a) (02 - 01)a
The probability mass function of nth order statistics is given by
fX(nn)(x) =
and the distribution function is
V(x; 01,02, a) (02 - 01)a
V(x - 1; 01,02, a)
(02 - 01)a
FX ( x)
(n:n)
V(x; 01,02, a) (02 - 01 )a
where V(x; 01,02, a) is given by (3).
2.6. Entropy
The Shannon's entropy of random variable X having probability mass function P(x) is given by
H(X) = E(-logP(x)).
For EDHE distribution, H(X) is obtained as
H(X) = - £
x=0
V(x + 1; 01,02, a) - V(x; 01,02, a) (02 - 01)a
log
V(x + 1; 01,02, a) - V(x; 01,02, a) (02 - 01)a
Renyi's entropy of order fi > 0 (fi = 1) is
Hp (p)
1
1- fil°g £ 1 fi x=0
V(x + 1; 01,02, a) - V(x; 01,02, a) (02 - 01 )a
a
n
n
n
n
n
n
fi
CO
2.7. Stress-strength Analysis
Stress-strength analysis is a mechanism and a topic used in reliability engineering. When the probability of stress exceeding the strength of an item, that item fails. Hence the expected reliability (R) is calculated as
TO
R = P (stress < strength) = £ fstress (x)Rstrength (x)
x=0
where the strength and stress are in the positive domain. When stress ~ EDHE(0i, 02, ai) and strength^ EDHE(03,04, a2), the expected reliability is
R = £ V(x + 1; 01, 02, «1) — V(x; 01,02, «2)
x=0 (f2 — f1)a1
' V (x; 0з, 04, «2 )
(04 - 03)k2
Tables 1-4 show the numerical values of R for different values of stress-strength parameters.
Table 1: Values of R for 0i = 0.1, 02 = 0.3, 03 = 0.3, 04 = 0.6 and different values of ai and a2
ai
a1 0.2 0.6 1 1.5 2 2.5
0.2 0.5185 0.6342 0.6962 0.7409 0.7685 0.7874
0.6 0.1707 0.2952 0.3708 0.4319 0.4735 0.5040
1 0.0735 0.1588 0.2166 0.2676 0.3050 0.3338
1.5 0.0336 0.0834 0.1207 0.1564 0.1843 0.2068
2 0.0181 0.0479 0.0718 0.0960 0.1157 0.1321
Table 2: Values of R for 0i = 0.3, 02 = 0.5, 03 = 0.6, 04 = 0.8 and different values of ai and a2
a2
«1 0.2 0.6 1 1.5 2 2.5
0.2 0.6338 0.7240 0.7777 0.8185 0.8441 0.8617
0.6 0.2799 0.4162 0.5027 0.5733 0.6210 0.6557
1 0.1430 0.2657 0.3487 0.4210 0.4730 0.5127
1.5 0.0753 0.1690 0.2373 0.3011 0.3499 0.3890
2 0.0469 0.1169 0.1712 0.2249 0.2680 0.3037
Table 3: Values of R for 0i = 0.5, 02 = 0.8, 03 = 0.5, 04 = 0.8 and different values of ai and a2
a2
«1 0.2 0.6 1 1.5 2 2.5
0.2 0.7372 0.8258 0.8748 0.9092 0.9293 0.9421
0.6 0.4272 0.5967 0.6952 0.7686 0.8138 0.8442
1 0.2720 0.4598 0.5747 0.6649 0.7235 0.7646
1.5 0.1759 0.3550 0.4713 0.5685 0.6352 0.6838
2 0.1277 0.2886 0.3991 0.4962 0..5658 0.6183
Table 4: Values of R for «i=«2 = 0.6, and different values of 0i, 02, 03, 04
02 = 04
01 = 03 0.6 0.7 0.8 1 1.5
0.1 0.4940 0.4974 0.50000 0.5037 0.5091
0.2 0.5437 0.5470 0.54972 0.5540 0.5611
0.3 0.5627 0.5668 0.5704 0.5761 0.5863
0.4 0.5751 0.5803 0.5848 0.5923 0.6058
0.5 0.5850 0.5912 0.5967 0.6059 0.6226
From tables 1-3 it is clear that for fixed values of 01, 02, 03, 04 and a\ reliability increases as a2 tends to infinity. But the reliability decreases with a1 tends to infinity for fixed values of 01, 02, 03, 04 and a2 . Table 4 shows that reliability increases with increasing values of 01, 03 for fixed values of 02, 04, «1 and «2. Also for fixed values of 01, 03, «1 and «2, reliability increases with increasing values of 02 , 04.
2.8. Estimation
In this section we estimate the parameters 01,02 and a of EDHE distribution using the method of maximum likelihood. Let us take a random sampleX1, X2...Xn of size n from EDHE distribution. Then the logarithm of likelihood function is
logL = E log
x=0
V(x + 1; 0i, 02, a) - V(x; 0i, 02, a)
(02 - 01 )a
The maximum likelihood estimators of 01, 02 and a are obtained by solving the equations
dlogL _ dlogL
d0i
0,-
0,
dlogL da
0.
But these equations cannot be solved analytically. So we use Non Linear Maximization (nlm) method for estimating the parameters 01, 02 and a.
3. Simulation Study
In this section, we use Monte-Carlo simulation method to illustrate the performance of the nlm
estimator of the parameters 01 and 02 and a. We generate 5000 random samples of sizes n=20, 30, 75 and 100 from the HE(01,02) distribution for some true values of the parameter set (01,02) = (15,18), (15,21), (16,18) and (16,21). We discretize the generated data and find out 5000 estimates of 01 and 02 and a using (4)for each sample sizes. The estimate of the parameter, average bias and mean square error of the estimate (MSE) are computed and it is given in Table 5 to Table 12.
Table 5: Values of estimates, average bias and average MSE for 01=15 and different values of, 02, a and n=20.
01 02 a 01 Bias(01 ) MSE(01 ) 02 Bias(02) MSE(02) a Bias(a) MSE(a)
15 18 0.4 14.5668 -0.4331 0.1876 17.8307 -0.1693 0.0287 0.3775 -0.0224 0.00005
0.8 14.4120 -0.5879 0.3457 17.7375 -0.2624 0.0689 0.7133 -0.0866 0.0075
15 21 0.4 14.8996 -0.1003 0.0101 20.3668 -0.6331 0.4009 0.3370 -0.0629 0.004
0.8 14.3655 -0.6344 0.4025 20.6269 -0.3730 0.1392 0.6763 -0.1236 0.0153
Table 6: Values of estimates, average bias and average MSE for 01=15 and different values of 02, a and n=30.
01 02 a 01 Bias(01 ) MSE(01 ) 02 Bias(02) MSE(02) a Bias(a) MSE(a)
15 18 0.4 14.5689 -0.4310 0.1858 17.8316 -0.1683 0.0283 0.3807 -0.0192 0.00004
0.8 14.5509 -0.4490 0.2016 17.5699 -0.4300 0.185 0.7356 -0.0643 0.0041
15 21 0.4 14.9042 -0.0957 0.0092 20.6514 -0.3485 0.1215 0.3487 -0.0512 0.0026
0.8 14.5340 -0.4659 0.2171 20.6291 -0.3708 0.1375 0.6883 -0.1116 0.0125
Table 7: Values of estimates, average bias and average MSEfor p\=15 and different values of p2, a and n=75.
$1 $2 a $1 Bias($ ) MSE($ ) $2 Bias($? ) MSE($?) a Bias(a) MSE(a)
15 18 0.4 0.8 14.6128 14.4120 -0.3871 -0.5879 0.1499 0.3457 17.8372 17.7375 -0.1627 -0.2624 0.0265 0.0689 0.4090 0.7133 0.0099 -0.0866 0.00001 0.0075
15 21 0.4 0.8 14.9614 14.6453 -0.0380 -0.3546 0.0014 0.1258 20.8440 20.6317 -0.1550 -0.3683 0.0241 0.1356 0.3502 0.6979 -0.0497 -0.1020 0.0024 0.0104
Table 8: Values of estimates, average bias and average MSEfor p\=15 and different values of p2, a and n=100.
$1 $2 a $1 Bias(^1 ) MSE$ ) $2 Bias(^2) MSE(^2) a Bias(a) MSE(a)
15 18 0.4 0.8 14.6154 14.7504 -0.3845 -0.2494 0.1478 0.0622 17.8376 17.8349 -0.1623 -0.2494 0.0263 0.0273 0.4089 0.7988 0.0089 -0.0011 0.00001 0.0000
15 21 0.4 0.8 15.0249 15.0320 0.0249 0.0320 0.00006 0.0010 20.8483 20.65213 -0.1516 -0.3478 0.0230 0.1210 0.37770 0.7445 -0.0226 -0.0554 0.00005 0.0031
Table 9: Values of estimates, average bias and average MSEfor pi=16 and different values of p2, a and n=20.
$1 $2 a $1 Bias($ ) MSE($ ) $2 Bias($2) MSE($2) a Bias(a) MSE(a)
16 18 0.4 0.8 15.4568 15.6653 -0.5431 -0.3346 0.2950 0.1120 17.8664 17.4611 -0.1335 -0.5388 0.0178 0.2903 0.3409 0.6452 -0.0590 -0.1547 0.0035 0.0240
16 21 0.4 0.8 15.1495 15.7852 -0.8504 -0.2147 0.7233 0.0461 20.4267 20.4893 -0.5732 -0.5101 0.3286 0.2603 0.3003 0.6690 -0.0996 -0.1309 0.0099 0.0172
Table 10: Values of estimates, average bias and average MSEfor p\=16 and different values of p2, a and n=30.
$1 a $1 Bias($ ) MSE($ ) Bias($2 ) MSE($?) a Bias(a) MSE(a)
16 18 0.4 0.8 15.4593 15.8042 -0.5406 -0.1957 0.2923 0.0383 17.8662 17.4674 -0.1337 -0.5325 0.0179 0.2836 0.3452 0.6708 -0.0547 -0.1291 0.0030 0.0167
16 21 0.4 0.8 15.1507 15.8184 -0.8492 -0.1815 0.7212 0.0330 20.4282 20.4918 -0.5717 -0.5081 0.3269 0.2583 0.3008 0.7341 -0.0991 -0.0658 0.0098 0.0043
Table 11: Values of estimates, average bias and average MSEfor pi=16 and different values of p2, a and n=75.
$1 $2 a $1 Bias($1 ) MSE($1 ) $2 Bias($2) MSE($2) a Bias(a) MSE(a)
16 18 0.4 0.8 15.4822 15.8212 -0.5177 -0.1787 0.2681 0.0319 17.8708 17.5036 -0.1291 -0.4963 0.0167 0.2463 0.3791 0.7450 -0.0208 -0.0549 0.00004 0.0030
16 21 0.4 0.8 15.1631 15.9054 -0.8368 -0.0945 0.7002 0.0089 20.4289 20.5039 -0.5710 -0.4961 0.3261 0.2461 0.3171 0.7946 -0.08281 0.0053 0.0069 0.0012
Table 12: Values of estimates, average bias and average MSEfor pi=16 and different values of p2, a and n=100.
$1 $2 a $1 Bias($1 ) MSE($1 ) $2 Bias($2) MSE($2) a Bias(a) MSE(a)
16 18 0.4 0.8 15.4951 15.9119 -0.5048 -0.0880 0.2548 0.0078 17.8731 17.9695 -0.1268 -0.0304 0.0161 0.00009 0.3989 0.7942 -0.0010 -0.0057 0.0000 0.0000
16 21 0.4 0.8 15.1645 15.9066 -0.8354 -0.0933 0.6979 0.0087 20.4290 20.5044 -0.5710 -0.4956 0.3261 0.2456 0.3183 0.8010 -0.0816 0.0010 0.0067 0.0000
From tables 5-12, it is clear that as sample size increases, the average bias and average MSE becomes very small for different choices of the values of the parameters. This indicates the consistency of the estimators.
4. Real Data Analysis
For studying the efficiency of EDHE distribution we consider the data set used by Krishna and Pundir [15] and it represents the total number of carious teeth among the four deciduous molars in a sample of 100 children 10 and 11 years of old. The data are given in Table 13.
Table 13: Observed data
X 0 1 2 3 4
f 64 17 10 6 3
Figure 3 shows the observed data.
Figure 3: Observed data.
We fit the EDHE distribution using the empirical data set and the embeded figure is given in Figure 4.
Figure 4: Embeded figure.
In order to assess the suitability of the proposed model, we use chi-square test of goodness of fit. Also, we compare the EDHE distribution with discrete Lindley (DL) distribution discrete Pareto (DP) distribution and the values of Log-likelihood, AIC, BIC are computed and is shown in Table 14.
Table 14: MLE's, Chi-square value, -Log-likelihood value, AIC values, BIC values and P values for the observed data.
Distribution fitted estimators Chi-square -LL value AIC value BIC value P value
DLD d = 0.275 6.637 113.68 229.36 229.36 0.036
DPD P = 0.1837 3.226 116.83 235.66 235.66 0.199
EDHED 01 = 0.9824779 (p2 = 0.9824794 a = 0.3346 1.2611 111.54 229.08 229.08 0.8679
From Table 14, it is inferred that the EDHE distribution is a better fit than discrete Lindley
and discrete Pareto distributions.
5. Generalizations
5.1. Transmuted exponentiated discrete hypo exponential (TEDHE) distribution
Many transmuted distributions are proposed and studied in literature. For details see Rahman et al. [29] and Dey et al. [6]. In this section we present a generalization of (4) called the transmuted exponentiated discrete hypo exponential distribution. A random variable X is said to have transmuted distribution if its distribution function and probability mass functions are respectively given by
and
F(x) = G(x)[1 + fi - fiG(x)]; №< 1
P(X = x) = g(x)[1 + fi - 2fiG(x)]
(5)
(6)
where G(x), g(x) are the distribution function and probability mass function of the baseline distribution. Also if fi = 0, we will get the baseline distribution. By using equations (5) and (6), the distribution function and probability mass function of the TEDHE distribution is obtained as
F(x; 01,02, a, fi)
V (x; 01,02, a) (02 - 01 )a
1 + fi - fi
V(x; 01,02, a) (02 - 01)a
and
f (x; 01,02, a, fi)
V(x + 1; 01,02, a) - V(x; 01,02, a) (02 - 01 )a
1 + fi - 2fi
V(x; 01,02, a) (02 - 01)a
(7)
(8)
The plot of pmf of TEDHE distribution is given in Figure 5.
Figure 5: Plot of pmf of TEDHE distribution.
The survival function and hazard rate functions are given by the expressions
S(x) = 1 -
V(x; 01,02, a) (02 - 01 )a
1 + fi - fi
V(x; 01,02, a) (02 - 01 )a
and
h(x)
V(x+1;01,02,a)-V(x;01,02,a)
(02-01 )a
1+fi - 2fi(
1 _ V(x;01,02,a)
1 (02-01 )a
1+fi - fii
The hazard plots of TEDHE distribution is given in Figure 6.
Figure 6: Plot of hazard function ofTEDHE distribution.
From Figure 6, it is understood that for different model parameters, the hazard rate function can be decreasing, increasing and increasing-decreasing, which makes the TEDHE distribution more flexible and can model different types of data sets.
5.2. Marshall-Olkin exponentiated discrete hypo exponential (MOEDHE)
distribution
Marshall and Olkin [20] introduced a new method for adding a parameter d(> 0) to the baseline distribution in order to generalize it. Using this method many generalized distributions are proposed and for a detailed review see Gillariose et al. [12]. If F(x) is the survival function of a distribution, then, by Marshall-Olkin method, another survival function G(x) is obtained as
G(x, 9)
9 F(x)
1 - (1 - d)F(x)
=—; -to < X < TO, 9 > 0.
The corresponding distribution function, probability mass function and hazard rate is obtained as
F(x)
G(x, 9)
g(x,9)
h(x)
1 - (1 - 9)F(x) G(x,9) - G(x - 1,9)
9f (x)
[1 - (1 - 9)F(x)][1 - (1 - 9)F(x - 1)]
g_(x) G(x).
where f(x) is the probability mass function corresponding to the distribution function F(x). Using Marshall-Olkin method the survival function of MOEDHE distribution is
G(x, 9)
9(i Yixxj$1j$ial) 9(1 to-^r )
1 - [(1 - 9)(1 - V^-^l)]
The corresponding distribution function, probability mass function and hazard rate are respectively given by
G(x, 9)
g(x,9)
and
h(x)
V(x;fa,4>2A) to
1 - [(1 - 9)(1 - V^-f^)]
9 V(x+l;<p1,<p2,K)-V(x;<p1,<p2,K) 9 (<p2-<h )a
(1 - [(1 - 9)(1 - Vx-ffi)])(1 - [(1 - 9)(1 - V%-1f)22A))])
V(x+1;<ft1,<fe,a)-V (x;<fr,<fe,a) (02-$1)x
(1 - [(1 - 9)(1 - V(X--$A) )])(1 -
The plot of probability mass function of MOEDHE distribution is given in Figure 7.
Figure 7: Plot of pmf of MOEDHE distribution.
The hazard plots are given in Figure 8.
Figure 8: Hazard plots of MOEDHE distribution.
Figure 8 shows different shapes of hazard rate functions and so we can conclude that the MOEDHE distribution is a flexible model in modeling different types of data sets.
6. Summary
Recently, there has been thriving interest in developing new families of distributions by adding one or more additional parameters to the baseline distributions. The existence of various generalization techniques were attracted by many researchers and using one among them we proposed and studied a new distribution called exponentiated discrete hypo exponential distribution. Various distributional and structural properties of this distribution are studied. Also stress-strength analysis is carried out. To evaluate the performance of the nlm estimator, we conducted a simulation study and found that the nlm estimator is consistent. A real data application is carried out and inferred that our proposed distribution is better model than discrete Lindley and discrete Pareto distribution. Two generalizations of the proposed distribution namely transmuted exponentiated discrete hypo exponential distribution and Marshall-Olkin exponentiated discrete hypo exponential distribution are introduced.
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