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34
K.M. Sakthivel and Vidhya G RT&A, No 3 (74)
MIXTURE OF PROBABILITY MODELS Volume 18, September 2023
A NEW FINITE MIXTURE OF PROBABILITY MODELS
WITH APPLICATION
K.M. Sakthivel and Vidhya G •
Department of Statistics, Bharathiar University, Coimbatore 641046, Tamil Nadu, India [email protected],[email protected]
Abstract
In this research, we present an approach to model lifetime data by a weighted three-parameter probability distribution utilizing the exponential and gamma distributions. We have presented some of the essential characteristics such as the shapes ofpdf, cdf, moments, incomplete moments, survival function, hazard function, mean residual life, stochastic ordering, and order statistics of the proposed distribution. Furthermore, we also presented the Bonferroni index and Lorenz curve of the proposed distribution. The maximum likelihood approach is used to estimate the parameters of the distribution. Finally, the proposed probability distribution is compared to goodness of fit with Lindley, Akash, exponential, two-parameter Lindley, cubic transmuted Rayleigh, and Exponential-Gamma distributions for the real-time data set.
Keywords: Lifetime distribution, Hazard function, Mean residual life function, Order statistic, Maximum likelihood estimation.
1. Introduction
A scientific approach to the statistical modeling of a wide variety of random events has been made possible by finite mixture of probability models. Due to its adaptability in representing complicated data, finite mixture models have drawn significant interest recently, both from a theoretical and practical perspective. Karl Pearson [15] conducted one of the earliest significant analyses utilizing mixture models. He modeled a proportional combination of two normal probability density functions with varying means and variances. A variety of probability distributions were subsequently utilized by many authors to fit a combination of probability distributions. Similarly, Lindley [17] also modeled the 'Lindley distribution' which is a combination of an exponential distribution with a scale parameter of 9 and a gamma distribution having a shape parameter of 2 and a scale parameter of 9 with their corresponding mixing proportions, 9++1 and 9++1 respectively.
A probability density function (pdf) and cumulative distribution function (cdf) for the Lindley distribution were included below.
f (x) = 92 (1 + ^x; x > 0,9 > 0
9 +1
F(x) = 1-
1 +
9 x 9 +1
e-9x; x > 0,9 > 0
(1) (2)
Shanker [22] used the finite mixture model to propose the Akash distribution, which is
described by its pdf and cdf.
(X) = ; x > 0, * > 0
F(x) = 1-
1 +
02 + 2
0 x(0 x + 2) 92 + 2
e-0x; x > 0,0 > 0
Furthermore, the finite mixing model is
f (x) = wi gi (x) + W2 g2 (x)
(4)
(5)
Where Shanker [22] uses the mixing proportion for Akash distribution with weights as wi = 02+2 and w2 = 02+2. Here, g1(x) and g2 (x) denotes pdf of exponential (0) and gamma (3, 0) distribution respectively.
We make changes to the Akash distribution to make it more inclusive and adaptable. Shanker [22] used the term 0 to describe the parameters of an exponential and a gamma distribution. In this study, we presented a new probability distribution, which we called the Exp-Gamma distribution. The proposed distribution is more flexible and it performs like the Generalized version of the Akash distribution. We did this by employing the scale parameter A for the exponential distribution and shape parameter 3, and the scale parameter ft for the gamma distribution with the mixture proportion of and ^j+p^ respectively.
This paper is also arranged in the following manner. In section 2, we present the Exp-Gamma distribution. Section 3 contains the usual moments and their related measures for the Exp-Gamma distribution. Section 4 deals with reliability analysis. Log-odds rate is calculated in section 5. Section 6 discusses Entropy. Section 7 deals with stochastic ordering. The order statistics for the Exp-Gamma distribution are given in section 8. The Lorenz and Bonferroni curves are presented in Section 9. The section 10 Zenga index is derived. In section 11, it is discussed how to estimate the Exp-Gamma distribution's parameters using the maximum likelihood method. Finally, section 12's proposed distribution as an application makes use of real-time data.
2. Exponential-Gamma Distribution(Exp-Gamma)
The probability distribution of the Exp-Gamma distribution can be described by its probability
density function and cumulative distribution function.
f (x; 0, A, ft) = ^
02Ae-Ax + ft3 x2e-ftx
F(x)
02(1 - e-Ax) + 2 - e-ftx(x2ft2 + 2xft + 2)
02 + 2
(6) (7)
for, x > 0,0 > 0, A > 0, ft > 0.
The following images Figure 1 and Figure 2 show a few potential pdf and cdf shapes for an Exp-Gamma distribution for various parameter values. The Akash and Gamma distributions are the special cases of the Exp-Gamma distribution when A = ft = 0 and 0 = 0 respectively. According to Figure 1, the Exp-Gamma distribution presents a variety of pdf patterns, including right-skewed and reversed-J shaped, pdf parameters that have fixed values.
Figure 1: The shape of the pdf of the Exp-Gamma distribution with varying parameter values.
Figure 2: The form of the Exp-Gamma distribution's cdf changes when the parameter values change.
3. Moments and related measures The rth moment (raw moments) has been obtained as
œ
r J
E(Xr )_ / xrf (x)dx
10
x
r 1 \al\n-Ax
!o 02 + 2 1
[62 Ae-Ax + ft3 x2 e-ftx ]dx (8)
92 r(r + 1) r(r + 3) Ar + ftr
e2 + 2
when r = 1,2,3,4 then the results follow.
The Exp-Gamma distribution's first four moments are:
Mean(p) = E(X) ^ + 6A
ftA(e2 + 2)
2 ) = 2ft2e2 + 24A2 E(X ) ft2A2(e2 + 2)
3) = 6ft3 e2 + 120A3
E(X ) ft3A3(e2 + 2)
4) = 24ft4e2 + 720A4 E(X ) ft4A4(e2 + 2)
As a result, the Exp-Gamma distribution's central moments are calculated as
T7 . 12A2 + 92(ft292 + 24A2 + 4ft2 - 12Aft) V2 _ Variance = - ft2 A2 (g2 + 2)2 --
_ 2 [13ft362 + 6ft3e4 + 60A3e4 + 24A3e2 - 36A2ftd4 - 36Aft2e2 - 54A2ftd2 - 84A3]
m _ ft3a3(e2 + 2)3
F4 _
24ft4 e2(o.375e6 + 3e4 + 8e2 + 8) - 48A4(93 + 114e2 + 51e4 - 1.5e6 )+ 72A2 ft2e2 (12 + 2e4 + e2 ) - 72ft3e2 (8A + 4Ae2 + e4 ) -192A3 fte2 (5.5 + e2 + 2.5e4)
1
ft4 a4 (e2 + 2)4
With the use of the aforementioned moments, closed-form formulas for the Exp-Gamma distribution's skewness, kurtosis, variation, and index of dispersion are produced. The variance-to-mean ratio is known as the index of dispersion (DI). The model is appropriate for datasets with low dispersion if the DI value is less than 1. The model works well with overly distributed datasets if the DI value is greater than 1.
, , E(X3) - 3E(X2+ 2}i3 skewness(x) = ——--^3—--—
= 2 [13ft3 e2 + 6ft3e4 + 60A3e4 + 24A3 e2 - 36A2 fte4 - 36Aft2 e2 - 54A2 fte2 - 84A3] = (12A2 + e2(ft2e2 + 24A2 + 4ft2 - 12Aft))2
Kurtosis = E(X4) - 4E(X^ + 6E(Xfr2 - V
a4
K.M. Sakthivel and Vidhya G RT&A, No 3 (74)
MIXTURE OF PROBABILITY MODELS Volume 18, September 2023
Kurtosis
24в4в2(0.375в6 + 3в4 + 8в2 + 8) - 48Л4(93 + 114в2 + 51в4 - 1.5в6) +
72Л2в2в2(12 + 2в4 + в2) - 72в3в2(8Л + 4Л62 + в4)
1
-192Л3 вв2 (5.5 + в2 + 2.5в4)
(12Л2 + в2(в2в2 + 24Л2 + 4в2 - 12Лв))2
COV = -
(12Л2 + в2(в2в2 + 24Л2 + 4в2 - 12Лв))2
а
DOI (Y) = -
вв2 + 6Л 12Л2 + в2 (в2 в2 + 24Л2 + '
- 12Лв)
(вЛ(в2 + 2))(вв2 + 6Л)
As seen in the table 1 to 5, the mean, variance, skewness, kurtosis, and index of dispersion are all expressed in quantitative terms.
From the tables, we can infer that the proposed distributions have the following features:
* The mean of the proposed function is a declining function of в, Л, and, в.
* The Exp-Gamma distribution is positively skewed for all the parameter values.
* Every positively skewed set of data can fits the suggested distribution.
* When the parameter values of the Exp-Gamma distribution are less than 1, then the Exp-Gamma distribution belongs to the light-tailed distribution, and when it exceeds the value of 1, then it belongs to the heavy-tailed distribution.
* The Exp-Gamma distribution is appropriate for both over- and under-dispersed datasets, as evidenced by the increasing and diminishing DI behavior.
Table 1: Mean values of the model
в в
Л
0.5
1
1.5
2
2.5
3
0.5 4.6667 4.3333 4.2222 4.1667 4.1333 4.1111
1 2.6667 2.3333 2.2222 2.1667 2.1333 2.1111
1 1.5 2.0000 1.6667 1.5556 1.5000 1.4667 1.4444
2 1.6667 1.3333 1.2222 1.1667 1.1333 1.1111
2.5 1.4667 1.1333 1.0222 0.9667 0.9333 0.9111
3 1.3333 1.0000 0.8889 0.8333 0.8000 0.7778
0.5 3.3333 2.6667 2.4444 2.3333 2.2667 2.2222
1 2.3333 1.6667 1.4444 1.3333 1.2667 1.2222
2 1.5 2.0000 1.3333 1.1111 1.0000 0.9333 0.8889
2 1.8333 1.1667 0.9444 0.8333 0.7667 0.7222
2.5 1.7333 1.0667 0.8444 0.7333 0.6667 0.6222
3 1.6667 1.0000 0.7778 0.6667 0.6000 0.5556
0.5 2.7273 1.9091 1.6364 1.5000 1.4182 1.3636
1 2.1818 1.3636 1.0909 0.9545 0.8727 0.8182
3 1.5 2.0000 1.1818 0.9091 0.7727 0.6909 0.6364
2 1.9091 1.0909 0.8182 0.6818 0.6000 0.5455
2.5 1.8545 1.0364 0.7636 0.6273 0.5455 0.4909
1.8182 1.0000 0.7272 0.5909 0.5091 0.4545
3
Table 2: The variance of the model
9 ß
A
0.5
1
1.5
2
2.5
3
0.5 12.8889 13.8889 14.4691 14.8056 15.0222 15.1728
1 3.5556 3.2222 3.3580 3.4722 3.5556 3.6173
1.5 2.2222 1.4444 1.4321 1.4722 1.5111 1.5432
2 1.8889 0.8889 0.8025 0.8056 0.8222 0.8395
2.5 1.7956 0.6622 0.5314 0.5122 0.5156 0.5240
3 1.7778 0.5556 0.3951 0.3611 0.3556 0.3580
0.5 10.2222 10.2222 10.6173 10.8889 11.0756 11.2099
1 3.8889 2.5556 2.5062 2.5556 2.6089 2.6543
1.5 3.1111 1.3333 1.1358 1.1111 1.1200 1.1358
2 2.9722 0.9722 0.7006 0.6389 0.6256 0.6265
2.5 2.9689 0.8356 0.5195 0.4356 0.4089 0.4010
3 3.0000 0.7778 0.4321 0.3333 0.2978 0.2840
0.5 7.8347 6.7190 6.7769 6.8864 6.9779 7.0496
1 3.9669 1.9587 1.7190 1.6798 1.6820 1.6942
1.5 3.5152 1.2094 0.8705 0.7817 0.7542 0.7466
2 3.4463 0.9917 0.6033 0.4897 0.4473 0.4298
2.5 3.4552 0.9114 0.4932 0.3647 0.3134 0.2899
3 3.4821 0.8788 0.4408 0.3023 0.2451 0.2176
1
2
3
Table 3: Skewness of the model
9 ß
A
0.5 1_1.5 2_2.5 3
~Ö5 0.0252 0.0224 0.0205 0.0194 0.0187 0.0182
1 0.1912 0.2015 0.1910 0.1794 0.1705 0.1639
1 1.5 0.5940 0.6636 0.6800 0.6587 0.6306 0.6054
2 1.0277 1.5293 1.5954 1.6119 1.5773 1.5279
2.5 1.2964 2.9363 3.0324 3.1345 3.1482 3.0980
3 1.4238 4.7520 5.1614 5.3091 5.4316 5.4400
0.5 0.0494 0.0592 0.0574 0.0554 0.0539 0.0528
1 0.2053 0.3950 0.4645 0.4738 0.4680 0.4593
2 1.5 0.3739 0.9375 1.3331 1.5255 1.5893 1.5990
2 0.4567 1.6424 2.4939 3.1600 3.5401 3.7158
2.5 0.4828 2.3873 3.9009 5.1726 6.1719 6.8048
3 0.4856 2.9913 5.5430 7.5000 9.2689 10.6651
0.5 0.0877 0.1541 0.1641 0.1639 0.1620 0.1601
1 0.2215 0.7013 1.0708 1.2328 1.2935 1.3129
3 1.5 0.3088 1.2455 2.3669 3.2923 3.8550 4.1606
2 0.3406 1.7724 3.5959 5.6103 7.3395 8.5668
2.5 0.3483 2.1934 4.8130 7.8061 10.9577 13.7362
0.3475 2.4707 5.9817 9.9640 14.4031 18.9349
3
Table 4: Kurtosis of the model
в ß A
0.5 1 1.5 2 2.5 3
0.5 0.0291 0.0239 0.0212 0.0198 0.0189 0.0183
1 0.4635 0.4655 0.4200 0.3824 0.3567 0.3388
1 1.5 2.2113 2.3624 2.3568 2.2050 2.0551 1.9358
2 4.5271 7.4158 7.5377 7.4486 7.0975 6.7211
2.5 5.9951 18.3288 18.0734 18.4501 18.1850 17.5152
3 6.6248 35.3808 37.5423 37.7987 38.2654 37.7084
0.5 0.0612 0.0709 0.0657 0.0617 0.0590 0.0571
1 0.4662 0.9786 1.1455 1.1349 1.0926 1.0512
2 1.5 1.0564 3.3125 4.9543 5.6862 5.8242 5.7456
2 1.3369 7.4593 11.9326 15.6581 17.6348 18.3282
2.5 1.4021 12.5799 22.7055 31.2223 38.2278 42.3995
3 1.3871 16.9017 37.7625 53.0000 67.5222 79.2691
0.5 0.1318 0.2697 0.2848 0.2798 0.2730 0.2670
1 0.4966 2.1087 3.6540 4.3159 4.5235 4.5574
3 1.5 0.7823 4.7057 10.6753 16.4434 20.0144 21.8491
2 0.8732 7.9452 18.9774 33.7393 48.0182 58.4636
2.5 0.8850 10.7268 29.1432 52.9549 82.3714 110.9024
0.8726 12.5162 40.2223 75.2908 119.4368 170.8054
3
Table 5: Index of dispersion of the model
в ß
A
0.5
1
1.5
2
2.5
3
0.5 2.7619 3.2051 3.4269 3.5533 3.6344 3.6907
1 1.3333 1.3809 1.5111 1.6026 1.6667 1.7135
1 1.5 1.1111 0.8667 0.9206 0.9815 1.0303 1.0684
2 1.1333 0.6667 0.6566 0.6905 0.7255 0.7556
2.5 1.2224 0.5843 0.5198 0.5299 0.5524 0.5751
3 1.3333 0.5556 0.4444 0.4333 0.4444 0.4603
0.5 3.0667 3.8333 4.3434 4.6667 4.8863 5.0444
1 1.6667 1.5333 1.7350 1.9167 2.0596 2.1717
2 1.5 1.5556 1.0000 1.0222 1.1111 1.2000 1.2778
2 1.6212 0.8333 0.7418 0.7667 0.8159 0.8675
2.5 1.7128 0.7833 0.6152 0.5939 0.6133 0.6444
3 1.8000 0.7778 0.5556 0.5000 0.4963 0.5111
0.5 2.8727 3.5195 4.1414 4.5909 4.9203 5.1697
1 1.8182 1.4364 1.5758 1.7597 1.9273 2.0707
3 1.5 1.7576 1.0233 0.9576 1.0116 1.0915 1.1732
2 1.8052 0.9091 0.7374 0.7182 0.7455 0.7879
2.5 1.8631 0.8794 0.6459 0.5814 0.5745 0.5906
3 1.9151 0.8788 0.6061 0.5117 0.4814 0.4788
The rth Incomplete moment for Exp-Gamma distribution has been obtained as
<r(x) = J xrf (x)dx
ft 1 xr
!о в2 + 2
[в2Ae-Ax + ß3 x2 e-ßx ]dx
1
в2 + 2
d2y(r + 1, At) Y(r + 3, ßt)
Ar + ß
when r = 1 the first incomplete moment of the Exp-Gamma distribution is
<1(x)
1
в2 + 2
ße2 y(2, At) + Ay(4, ßt) Aß
The related Exp-Gamma distribution moment-generating function is
f TO
Mx (t) = E(etX ) = j etXf (x)dx
E1
L-i л
1
i=0
i! V в2 + 2
в2 Г(г + 1) + Г(г + 3)
Аг
ß*
The corresponding characteristic function of the Exp-Gamma distribution is
<x (t) = E(e ) = J e f (x)dx
TO itk
E k!
1
¿=0
k! в2 + 2
в2 Г(к + 1) + Г(к + 3)
Ak
ßk
The Exp-Gamma distribution's associated cumulant-generating function is
(10)
(11)
Kx (t) = loge Mx (t)
tl
П
¿=0
1
¿! в2 + 2
в2 Г (г + 1) + Г (г + 3)
A1
ßl
(12)
Probability-weighted moments are derived using a different method for statistical distributions whose inverse form is difficult to define. The corresponding probability-weighted moment for the Exp-Gamma distribution can be found using the formula below.
^r,s = E(XrF(x)s )
f to
= J xrf (x)[F(x)]sdx
1 {• to
1 1 „ПО 2
(13)
(в2 +
рто
2y+1Jo xr ^Ae-^ + ß3x2e
- в2e-Ax + 2 - e-ßx(x2ß2 + 2xß + 2)]sdx
The corresponding n conditional moment of the Exp-Gamma distribution is defined as
E[Xn/X > x]
S(x)
xnf (x)dx
TO
to
1
dx
E[Xn/X > x] =
62 + 2 JX° xn^ ¿2+2 62Ae-Ax + ß3x2e-ßx])
(62 + 2) - 62(1 - e-Ax) + [2 - e-ßx(x2ß2 + 2xß + 2)] _—r(n + 1, Ax)62ßn - r(n + 3, ßx)An_
Anßn( (62 + 2) - 62(1 - e-Ax) + [2 - e-ßx(x2ß2 + 2xß + 2)]
4. Reliability Analysis
4.1. Survival Function The odds that an item won't fail before x is specified is the survival function S(x).
S(x) = P(X > x) = 1 - F(x)
02(1 - e-Ax) + [2 - e-£x(x2£2 + 2x£ + 2)]
1
62 + 2
(62 + 2) - 62(1 - e-Ax) + [2 - e-ßx(x2ß2 + 2xß + 2)] 62 + 2
(15)
Survival Function
i\Y\ V-x \ \\\W \ U\.\ \ — e,a.,p = 0.2,1,1 -- e,a.,p = 0.4,0.5,1.5 •••• = 0.5,1,2 --- e,A.,ß = 0.8,1.5,2.5 — e,a.,p = 1,2,3 -— e,A.,p = 1.5,2.5,3.5
1 \ • \ \ \ H \v \ n \\ \ \\\ \ \ \
\\\ \ \ \ \ \ \ *. * \ \ \ \ \ \ \
\ * \ \ \ V '•- N
Figure 3: The different shapes of the sfofan Exp-Gamma distribution for different parameter values.
4.2. Hazard Rate Function
Assume that X is a continuous random variable with pdf f (x) and cdf F(x). The hazard function of X is
h(x)
f (x) 1 - F(x)
[62\e-Xx + ß3 x2 e-ßx ]
(16)
(02 + 2) - 02(1 - e-A-x) + [2 - e-ftx(x2ft2 + 2xft + 2)] 4.3. Mean Residual Life Function
Assume that X is a continuous random variable with pdf f (x) and cdf F(x). According to X, the mean residual life function is
m(x) = E[X - x/X > x]
[1 - F(t)]dt
1 - F(x) Jx -Xx + A(x2 ß2 + 4x ß + 6)e-ßx
(17)
(62 + 2) - d2(1 - e-A-x) + [2 - e-ßx(x2ß2 + 2xß + 2)]
The Exp-Gamma distribution's hazard function can take three different shapes: decreasing HF, unimodal HF, increasing HF, and decreasing-increasing HF. A declining function is also a property of the mean residual life function.
Hazard Function
— e,ä.,ß = 0,1,2 ~~ e,A.,p = 1,2,0 ■■•■ e,i,ß = 1,0,2
/ --9,^,ß = 0.5,1,2.5 -— 9,A.,ß = 1.5,2,3.5
/ i
W'/ — \
p: -
Figure 4: Hazard function of the Exp-Gamma distribution for different parameter values. The shape of the hazard function changes as the parameter values are varied.
1
Figure 5: The various forms of an Exp-Gamma distribution's mean residual life function for various parameter values.
4.4. Mean Inactivity Time
The mean inactive time is the amount of time that has passed after an item's failure based on the premise that it failed in (0, t).
(t) = E(X - t/X < t)
t -
fr(t) F(t)
t
ßd2 y(2, At) + Ay(4, ßt)
Aßid2(1 - e-At) + [2 - e-ßt(t2ß2 + 2tß + 2)]
(18)
4.5. Cumulative Hazard
The cumulative hazard function is
H(x) = - log(1 - F(x))
a2 + 2) - logl (d2 + 2) - d2(1 - e-Ax) + [2 - e-ßx(x2ß2 + 2xß + 2)]
4.6. Reversed Hazard Rate
The Reversed Hazard Rate is
T(x) = fM
T(x) = F(x)
(20)
__[62 Ae-Ax + ft3 x2 e-ftx ]__v '
5. Log-odds Rate
The log-odds rate was used by Wang et al. (2003) to propose a model for time to failure as well as some definition of failure time distributions. By simulating the failure process in terms of the log odds rate, the model may be used to analyze the distribution of time until failure. The odds function is given by
^o (x) - F(x)
S(x)
e2(1 - e-Ax) + [2 - e-ftx(x2ft2 + 2xft + 2)]
(21)
(e2 + 2) - e2(1 - e-xx) + [2 - e-ftx(x2ft2 + 2xft + 2)]
The log-odds function is given by
LO(x) = log ^-Fl) = (log(02(1 - e^Ax) + [2 - e-ftx(x2ft2 + 2xft + 2)]) - log((e2 + 2)-
e2(1 - e-Ax) + [2 - e-ftx(x2ft2 + 2xft + 2)]))
The log-odds rate is defined as
LOR(x)(x) = F^
F(x) (23)
=_[e2Ae-Ax + ft3 x2 e-ftx ](e2 + 2)_
= (e2 + 2) - e2(1 - e-Ax) + [2 - e-ftx(x2ft2 + 2xft + 2)]
6. Entropy
Entropy is a metric for describing the degree of uncertainty in a random variable (X) for the
probability density function obtained from the lifetime distribution.
6.1. Renyi Entropy
Renyi entropy of a random variable X ~ Exp - Gamma(d, A, ft) with pdf is defined as
1 rM
Ir(n) = log y0 /n(x)dx; n > 0, n = 1
1
1-n 'o^0 52+2[e2 Ae-Ax + ft3 x2 e-ftx] dx (24)
1 - n "*(why f[e2Ae-Ax+ft3x2e-ft']")dx
7
1
6.2. Shannon Entropy
The Shannon Entropy of X ~ Exp - Gamma(d, A,£) is given by
E[- log f (X)] = - f f (x) log f (x)dx J X
E
- log[
1
e2 + 2
[e2Ae-Ax + £3 x2e-£x ]
+ 2) - E
log[e2 Ae-Ax + £3 x2 e-£x ]
e2 + 2
e2 Ae-Ax + £3 x2 e-£x
(25)
log
e2 + 2
[e2Ae-Ax + £3x2e-£x] dx
6.3. Generalized Entropy
The Generalized Entropy of X ~ Exp - Gamma(e, A, £) is given by
GE(w, 5)
5(5 - 1)^5
x5 f (x)dx
1
5(5 -d(
r xs( 1
e2 + 2
[e2Ae-Ax + £3x2e-£x] dx
1
(26)
£5e2r(5 + 1) + A5r(5 + 3) j \£A(e2 + 2) (e2 + 2)A5£5(5(5 - 1)(£e2 + 6A)5)
1
7. Stochastic ordering
Stochastic ordering can be used to assess the relative performance of positive continuous random variables. The size of random variable X is less than that of random variable Y.
• Stochastic order ( X <st Y ) if FX (x) > Fy (y) for all x.
• Hazard rate order ( X <hr Y ) if hX(x) > hY(y) for all x.
• Mean residual life order ( X <mrlY ) if mX (x) > mY (y) for all x.
• Likelihood ratio order (X <y Y ) if fX^y) decreases in x.
The stochastic ordering of distributions was created by Shaked and Shanthi Kumar (1994) using the results.
The Exp-Gamma distribution is sorted according to the strongest 'likelihood ratio'. Let X ~ Exp - Gamma(e1, Ai,£1) and Y ~ Exp - Gamma(e2, A2,£2). If, £1 > £2, then X <lr Y and hence X <hr Y, X <mlr Y and X <st Y.we have
fX(x) _ (e2 + 2)[e2A1 e-A1x + x2e-£1x] fY(x) _ (e2 + 2)[e2A2e-A2x + £2x2e-£2x]
1 fX (x) ,
log T^T _ log
'fr (x)
(e22 + 2) [e12A1 e-A1x + £13x2e-£1x] (e12 + 2) [e22A2e-A2x + £23x2e-£2x]
_ log (e22 + 2) + log k2A1 e-A1 x + £13x2e-£1 x] - log (e2 + 2) - log |e22A2e-A2x + £23x2e-£2x
1
1
X
1
1
5
d fx(x) _ 022A22e-A2x - ß23 (2xe-ß2x - x2ße-ß2x) e12A12e-A1x + ß3 (2xe-ß1x - x2ße-ß1x) log"
dx fY(x)
[022 A2 e-A2x + ß23 x2 e-ß2x]
[032 A1 e-Aix + ß13 x2 e-ßix]
(27)
Now if 01 = e2 = e, Ai = A2 = A, fti > ft2, then it implies log f^ < 0. This means that
X <lr Y and hence X <hr Y, X <mlr Y and X <st Y.
8. Order Statistics
If X(1) < X(2) < ... < X(n) denotes the order statistic of a random sample X1, X2,..., Xn from a continuous population with cdf FX(x) and pdf fX(x) then the pdf X(r) is given by
fX(r)(x)
(r - 1)! (n - r)!
fx(x) [Fx(x)](r-1) [1 - FX(x)]
(n-r)
For, r = 1,2,... n. The pdf of the rth order statistic for the Exp-Gamma distribution is calculated, and the pdf of the largest order statistic X(n) and smallest order statistic X(1) are given below.
nth order statistics
fx(n)(x) = nfx(x) [Fx(x)](n-1)
e2 + 2
1st order statistics
e2Ae-Ax + ß3 x2e-ßx
e2 (1 - e-A^ + [2 - e-ßx (x2ß2 + 2xß + 2)]
(n-1)
(28)
fx(1)(x) = nfx(x) [1 - Fx(x)](n-1)
e2 + 2
e2 Ae-Ax + ß3 x2 e-ßx
(e2 + 2) - e2 (1 - e-Ax) + [2 - e-ßx (x2ß2 + 2xß + 2)]
e2 + 2
(n-1)
(29)
The pdf of a median of order statistic is given as
fm+1:n(x) = ^m+^fx(x) [Fx(x)f [1 - Fx(x)]m
m!m! (2m + 1) f 1
e2 Ae-Ax + ß3 x2 e-ßx
e2 (1 - e-Ax) + [2 - e-ßx (x2ß2 + 2xß + 2)]
m!m! e2 + 2 (e2 + 2) - e2 (1 - e-Ax) + [2 - e-ßx (x2ß2 + 2xß + 2)]
e^T2
e2 + 2
(30)
9. Lorenz and Bonferroni Curves
The Bonferroni and Lorenz curves (Bonferroni, 1930) are used in a variety of sectors, including economics, demography, insurance, and medicine. An Exp-Gamma distribution's Bonferroni and Lorenz curves are calculated as follows:
n
n
n
m
m
Bo (x)
t'xf (x)dx _
pF(x) Jo
£e2j(l, At) + Ay(4, £t)
A£v (e2 (1 - e-Ax) + [2 - e-£x (x2£2 + 2x£ + 2)])
Lo (x)_ -Ixf (x)dx _ Kg _ [£e2y(2, At) + Ay(4, £t)]
A£v (e2 + 2)
10. Zenga index
The Gini index is commonly used to account for the extent of income inequality in a population. The Zenga index (Zenga, 2007) is a relatively new metric and a novel alternative to the Gini index and other current inequality measurements and curves, and the Zenga index is denoted by z.
z _ 1 -
V+x)
where,
1 f x
V-x) _ F« Lxf (x)dx
£e2 y(2, Ax) + Ay(4, £x)
+
1
V(x)
x) 1 - F(x) 0
x f(x) dx
A£ (e2 (1 - e-Ax) + [2 - e-£x (x2£2 + 2x£ + 2)])
+ 6A
z_1
£A ((e2 + 2) - e2 (1 - e-Ax) + [2 - e-£x (x2£2 + 2x£ + 2)])
£e2y(2, Ax) + Ay(4,£x) (£A ((e2 + 2) - e2 (1 - e-Ax) + [2 - e-£x (x2£2 + 2x£ + 2)])) + 6A) A£ (e2 (1 - e-Ax) + [2 - e-£x (x2£2 + 2x£ + 2)])
11. Estimation of Parameters
In this section, the MLE approach is used to estimate the parameters e, A, and £. Consider a sample drawn at random from the Exp-Gamma distribution. Then the log-likelihood function is provided by
1
g(x)
e2 + 2
e2 Ae-Ax + £3 x2 e-£x
L (xi, e, A, £) _ ng (xi, e, A,t i_1
L (xi, e, A, £) _ n
1
|V e2 + 2
Ae2 e-Axi + £3 x2e-£xi
Ve2+2
The respective sample log-likelihood function is
n Ae2e-Axi + £3x2e-£xi
log L(xi, e, A, £) _ log n - log(e2 + 2) + £ log[Ae2e-Axi + £3x2e-£xi]
i_1
Now that we have differentiating w.r.t. e, A, and £, we can write
n
n
n
d log L d0
d log L
dX
and d log L dft
The MLEs are obtained by solving this system of nonlinear equations. The sample likelihood function can be quantitatively improved by using nonlinear optimization techniques, which are frequently more practical. R programming can be used to solve these equations numerically.
12. Application
Biomedical science lifespan data sets have been fitted with Exp-Gamma distribution. This section compares the goodness of fit of the Exp-Gamma model to the one-parameter Akash [22], Lindley [17], Exponential, two-parameter Lindley [26], Cubic transmuted Rayleigh, and Exponential-Gamma [18] distributions on a real-life data set. A density comparison diagram is also included in this section.
The data, according to Gross and Clark (1975, P.105), represents the lifetime data on the minutes of pain alleviation experienced by 20 people who received an analgesic. The details are as follows:
1.1, 1.4,1.3, 1.7,1.9,1.8, 1.6, 2.2,1.7, 2.7, 4.1,1.8, 1.5,1.2,1.4, 3.0,1.7, 2.3, 1.6, 2.0
For a real lifetime dataset, the -2lnL, AIC, AICC, BIC, K - S, CVM, andAD statistics have been calculated and shown in Table 7 to compare the goodness of fit of the Exp-Gamma, Akash, Lindley, Exponential, Cubic transmuted Rayleigh, Two parameter Lindley, Exponential-Gamma distributions.
Table 6: Estimated parameter values of the distributions for the dataset
Model Parameter Estimate Log-Lik
Exp-Gamma 0 = 5.3520e-05 , X = 0.29 1 4
ft = 1.5789 -22.8873
Akash 0 = 1.1569 -29.7613
Lindley 0 = 0.8161 -30.2496
Exponential X = 0.5263 -32.8371
Cubic transmuted Rayleigh cr = 2.63597
X = 2.5971 -24.9371
Two parameter Lindley 0 = 1.48 a = -0.2914 -25.8862
Exponential-Gamma X = 0.7361 a = 1.7971 -62.2516
-29 " 20Xe-Xxi
(02 + 2) ¿1 [02Xe-Xxi + ft3x2e-ftxi] ^ 02 (e-Xxi - Xxie-Xxi) _n
1 [02Xe-Xxi + ft3x2e-ftxi] "
" x2 (3ft2e-ftxi - ft3xte-ftxi] ¿1 [ 02 Xe-Xxi + ft3 x2e-ftxi]
0
0
The variance-covariance matrix of the MLEs is computed as
m-
1.1362e-01 -9.5914e-06 8.3587e-07
-9.5914e-06 8.3587e-07 8.0969e-10 -8.1641e-11 —8.1641e-11 4.1550e-02
The variances of the MLEs of the parameters of Exp-Gamma 6, A and £ are var(6) = 0.1136, var(A) = 8.0969e-10 and var(£) = 0.0415. And 95% confidence intervals of 6, A and £ are [-6.60597,6.60704], [0.29136,0.29147] and [1.1794,1.9785] respectively.
Table 7: Criteria for comparison
Model -2lnL AIC AICC BIC AD K-S statistic CVM
Exp-Gamma 45.7745 51.7745 53.2747 54.7617 1.9324 (0.097) 0.2587 (0.1007) 0.3508 (0.1375)
Akash 59.5226 61.5226 61.7471 62.5206 3.3554 (0.0185) 0.3705 (0.0082) 0.6555 (0.0154)
Lindley 60.4991 62.4991 62.7213 63.4948 3.7504 (0.0118) 0.3911 (0.0044) 0.7550 (0.0086)
Exponential 65.6742 67.6742 67.8964 68.6699 4.6035 (0.0046) 0.4395 (0.0009) 0.9630 (0.0026)
Cubic transmuted Rayleigh 49.8742 53.8742 54.5801 55.8657 2.216 (0.0707) 0.26534 (0.1196) 0.3873 (0.0772)
Two parameter Lindley 51.7724 55.4375 54.7785 55.8564 3.7822 (0.0085) 0.4102 (0.0075) 0.5275 (0.0058)
Exponential -Gamma 124.503 128.5032 130.4946 129.2091 41.855 (0.0000) 1.000 (0.0000) 5.4779 (0.0000)
Figure 6: Comparison of model fit for the distributions.
The Exp-Gamma distribution fits the dataset better than the Akash, Lindley, exponential, two-parameter Lindley, Cubic transmuted Rayleigh, and Exponential-Gamma distributions as
observed from Table 7.
1
13. Conclusion
A weighted three-parameter probability distribution is developed in this study for modelling skewed lifetime data. We derive expansions of important statistical measures like mean, variance, moments, and moment generating function, etc., as well as maximum likelihood estimation is used to estimate the Exp-Gamma distribution's parameters and hazard and reliability functions are used to examine the distribution's properties. The proposed distribution was fitted using real-time data.
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