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53
A New Reliability Model and Applications
Manoharan M. and Kavya P. •
University of Calicut [email protected], [email protected]
Abstract
The Lomax or Pareto Type II distribution has a wide range of applications in many areas including reliability and life testing. In this paper, we modify the Lomax distribution using KM transformation to enhance the applicability of the Lomax distribution. The distribution introduced using KM transformation is parsimonious in parameter. Substituting the cumulative distribution function (cdf) of the Lomax distribution in KM transformation provides a new modified Lomax distribution. The behavior of hazard rate function is studied graphically and also theoretically using Glacer method. Its analytical properties are derived and parameters are estimated using maximum likelihood estimation method. We consider two real data sets to show the flexibility of the proposed model. The model proposed in this paper provides a better fit to the data sets compared to other well-known distributions given in this study.
Keywords: Parsimonious model Lifetime KM transformation Lomax distribution Decreasing failure rate.
1. Introduction
The Lomax distribution has wide applications in many fields like economics, actuarial science, and so on. The Lomax distribution is also called Pareto Type II distribution. The distribution was introduced by Lomax [13] and it is a heavy-tailed distribution. It has also been useful in reliability and life testing problems in engineering and survival analysis as an alternative distribution [[9], [11]]. The Lomax distribution shows decreasing failure rate. Modified and extended versions of the Lomax distribution have been studied; examples include the weighted Lomax distribution [11], exponential Lomax distribution [7], exponentiated Lomax distribution [19], gamma Lomax distribution [5], transmuted Lomax distribution [3], Poisson Lomax distribution [2], McDonald Lomax distribution [12], Weibull Lomax distribution [21], power Lomax distribution [18], Kumaraswamy-Generalized Lomax distribution [20], Gompertz-Lomax distribution [16], and DUS-Lomax distribution [6]. Besides, estimation of the parameters of Lomax distribution under general progressive censoring has been considered by Al-Zahrani and Al-Sobhi [1].
The principal objective of the study is to introduce a modified Lomax distribution which is parsimonious in parameter and enhance the application of the Lomax distribution in reliability theory and survival analysis. We try to improve the properties of the Lomax distribution as a useful lifetime model.
We organize the paper as follows: In Section 2, we introduce a new life distribution using the Lomax distribution as the baseline distribution in the KM transformation. We then discuss the analytical characteristics of the new distribution in Section 3. In Section 4, we establish the ordering of the new distribution. In section 5, the parametric estimation for the new distribution is studied. We carry out an analysis using a real-life data set to illustrate the model's flexibility in Section 6. In section 7, we summarize the conclusions and outline our future works.
- a=0.5, 0=1.0
---- a=1.0, 0=0.5
a=2.0, 0=1.0
.0 0.5 1.0 1.5 2.0 2.5 3.0
x
Figure 1: Probability density plot
2. The modified Lqmax model
In this paper we have modified the Lomax distribution with cumulative distribution function
(cdf)
G(x) = 1 - (1 + ¡x)-a, x > 0, a, 5 > 0,
(1)
using KM (Kavya and Manoharan) transformation introduced by Kavya and Manoharan [10]. Let X be a random variable with cdf G(x) and probability density function (pdf) g(x) of some baseline distribution. Then the cdf F(x) of new distribution is defined as,
F(x)
e- 1
[1 - e-G(x)
(2)
Here we introduce a new distribution by substituting the cdf of Lomax distribution (1) in (2). The cdf and pdf of the new distribution are respectively obtained as
F(x)
f (x)
e1
[1 - e-(1-(1+5x)-a)], x > 0, a, 5 > 0,
xp(1 + 5x)-(a+1) e(1+5x)-
-, x > 0, a, 5 > 0,
(3)
(4)
(5)
The graphical representation of pdf is given in Fig. 1 for different values of parameters. In the whole paper we used the software MATHEMATICA [23] for plotting the graphs.
3. Hazard rate function of the model
The hazard function is defined as
h(x) =
f (x) 1 - F(x)
The hazard function of the proposed model is obtained as
h(x) =
xp(1 + px)-(a+1) e(1+5x)-
e(1+5x)-a -1
The shape of the hazard rate function is given in Fig. 2.
(6)
e
e
a
1......
. 1 1
1 ; \ \ \ \ » \ \ ■ ■ ■ ■ a=2.0, 0=1.0
..........
0
1
2 3
x
4
5
Figure 2: Hazard rate plot
3.1. Theoretical explanation of the shape of the hazard rate function
We follow Glaser [8] for theoretical explanation of the shape of the hazard rate. Suppose f (t) is the pdf of some distribution and f'(t) is the first derivative of f (t). Then
v(t)
- f'(t) f (t)
For our proposed distribution
v(x) = fi la + (a + 1)(1 + fix)-1
and
V (x) = -fi2 (a + 1)(1 + fix)-
(8)
based on Glaser [8] we get a result from Equation (8):
v'(x) < 0 for all x > 0 when a > 0. Then the distribution has decreasing failure rate (DFR).
4. Some analytical characteristics Here we discuss some of the analytical characteristics of our proposed distribution.
4.1. Moments
The moments of a random variable, if they exist, are useful for estimating measures of central tendency, dispersion, and shapes. The rth raw moments of the proposed distribution is
afi
E(Xr) = xr(1 + fix)-(a+1)e(1+fix)~"dx.
After transformation, we get,
fir (e -
applying binomial expansion, then
1 TO
fir (e-1)
1 r1 1 r
E(Xr)= „ . ' (1 - ua)ru-aeudu,
v ' fir (e -1) J0y '
E(X' ) = wieh £(-1)(01:u'' "-r) '"du-
2
Expanding exponential term, and we get the rth raw moment as
E(Xr) = 1 f tH (r) a
( ) /r(e - 1) f j! \ij aj + a - r + i'
4.2. Moment generating function
The moment generating function of the proposed distribution is
M a f f 1 m (na + a - 2)!
Mx(t)=(e - 1) ¿=0 nf0 m! n! /m (n + a)! (9)
4.3. Characteristic function
The characteristic function of the proposed distribution is obtained as
( a f f 1 (it)m (na + a - 2)!
<PX(t) = (e-1) m=0nf0 m!^! (n + a)! ( 0)
where i = \f-1.
4.4. Quantile function
The quantile function is useful when generating random observations from a distribution. It can also be utilized in estimating measures of shapes (skewness and kurtosis) when the moments of the random variable do not exist. The pth quantile function of the proposed distribution is obtained as
1
Q( P) = ß
1 + log(1 - - 1
(11)
We can easily find the first, second and third quartile functions after substituting p = 4, 2, and 4 in Equation (11).
4.5. Order statistic
Order statistics are important for estimating summary statistics such as the minimum, maximum, and range of a data set. They are also used in quality control testing and reliability to forecast failure of future items based on the times of few early failures. Let X1, X2, ■ ■ ■ , Xn be a random sample of size n from the proposed distribution and X(X), X(2), ■ ■ ■ , X(n) denote the corresponding order statistics. The pdf of the rth order statistic fr(x) is given by
n!
fr(x) = (r - 1)!(n - r)!F (x)[1 - F(x)]n-rf
and the pdf of the rth order statistic of our proposed model is obtained as
n! a/3 ~ (-1)l+j ((r -1)
fr(X (r - 1)!(n - r)! (e - 1)n£L E jlkl { j
[i(1 - (1 + ßx)-a)]j(1 + ßx)-(xk+x+1) (e(1+ßx)-* - 1^n-r (12)
Substitute r = 1 and r = n in Equation (12), we get the pdf of the smallest and the largest order statistics respectively.
The cdf of the rth order statistic is
Fr(x) = t(ni)Fi(x)[1 - F(x)]n-i,
j=r ^ ' '
and the cdf Fr (x) of rth order statistic of the new distribution is obtained by using the Equation (3) as,
Fr (x) = t (n) (j E E ^ (0 i(1 - (1 + ex)-)f - l)n-i. (13)
The cdf of X(!) and X(n) are obtained by putting r = 1 and r = n respectively in Equation (13).
5. Ordering
Stochastic ordering of positive continuous random variables is an important tool for judging the comparative behavior. There are different types of stochastic orderings that are useful in ordering random variables in terms of different properties. Here we consider four different stochastic orders, namely, the usual, the hazard rate, the mean residual life, and likelihood ratio order for KM-Lomax random variables. If X and Y are two random variables with cumulative distribution functions Fx and Fy, respectively, then X is said to be smaller than Y in the
• stochastic order (X <st Y) if FX (x) > FY(x) for all x
• hazard rate order (X <hr Y) if hX(x) > hY(x) for all x
• mean residual life order (X <mri Y) if mX(x) > mY(x) for all x
• likelihood ratio order (X <ir Y) if fX^ decreases in x
The implication between the ordering is X <ir Y ^ X <hr Y ^ X <mri Y ^ X <st Y. The KM-Lomax distribution is ordered with respect to the strongest "likelihood ratio" ordering as shown in the following theorem. It shows the flexibility of the proposed distribution.
Theorem 1. Let X ~ KML(ai, and Y ~ KML(«2, £2) if «1 = «2 = « and > and if
= £2 = £ and a1 > «2, then X <lr Y, X <hr Y, X <mrl Y and X <st Y.
Proof. The likelihood ratio is
fx(x) _ «1 £1 (1 + £1 x)-(a1+1)e(1+£1x)-a1
x)-a2
and
fY (x) a2 £2 (1 + £2 x) -(a2+1) e(1+£2x)
log fXx = log«1 + log£1 - («1 + 1) log(1 + £1 x) + (1 + £1 x)-«1 - log«2 - log£2 + (a2 + 1) log(1 + £2x) - (1 + £2x)-«2
(14)
thus,
dL 'og = - («1 + ^r+kx - "1ft(1+hx)-("1+1) ™
+ («2 + 1) 1+h + "2fc(1 + x)-('2+1)
1. Case I: «1 = «2 = «, £1 > £2
dx log f^ < 0 ^ X <irY hence X <hr Y, X <mri Y and X <st Y.
2. Case II: = £2 = «1 > «2
Jx (x) fY (x)
■
d log f№ < 0 ^ X <irY hence X <hr Y, X <mri Y and X <st Y.
6. Estimation of the parameters of the model
In this section we estimate the parameters involved in the distribution using maximum likelihood estimation method. This is one of the most popular methods used for estimation. The likelihood function is defined as,
L(x; A) = n f fa, A) i=1
In our distribution,
L(x; a,fi) = (-Ofi-) " n(1 + fixt)-(a+1)e^+te)-a. \e V i=0
The log-likelihood function of the distribution is given by,
n n
log L(x; a, fi) = -n log(e - 1) + n log a + n log fi - (a + 1) £ log(1 + fix,) + £ (1 + fix{ )-a.
i=1 i=1
We proceed as follows. First we find partial derivatives of the log-likelihood function with respect to the parameters a and fi. The partial derivatives are
d log L n n n
= n - £ log(1 + fixi) + £ log(1 + fix,)-a log(1 + fiXi),
and
da a ■ , ■ -,
i=i i=i
= I - (a + 1) XlT+fc - aSXi(1 + ^)-(a+1).
Two non-linear equations can be obtained by equating these partial derivatives to zero, the solutions for which provide the maximum likelihood estimates of the parameters. The Newton-Raphson method can be used to solve this equation with the help of the available statistical packages. We use R [17] language for finding the numerical solution of the non-linear system of equations.
n
7. Application
In this section we are showing the flexibility of the proposed distribution using two real-life data sets. The first data set is the uncensored data set corresponding to intervals in days between 109 successive coal-mining disasters in Great Britain, for the period 1875-1951, published by Maguire et al. [15] and the data set is given in Table 1. The second data set is of Wheaton River obtained from Choulakian and Stephens [4] and presented in Table 2.
We use AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), HQC (Hannan-Quinn Information Criterion) and K-S (Kolmogorov-Smirnov) test value for the comparison. The distribution which shows minimum AIC, BIC, HQC and K-S test value is the sign of a better fit for the data set. The AIC, BIC and HQC are defined as
AIC = -2 log(L) + 2m,
BIC = -2 log(L) + m log(n),
and
HQC = -2 log(L) + 2m log(log(n)),
where n is the sample size, m is the number of parameters, and L is the maximum value of the likelihood function for the considered distribution. Here R [17] language is used for all the computation. We compare the proposed distribution with the following distributions,
Table 1: Flood Level Data.
1 4 4 7 11 13 15 15 17 18 19 19
20 20 22 23 28 29 31 32 36 37 47 48
49 50 54 54 55 59 59 61 61 66 72 72
75 78 78 81 93 96 99 108 113 114 120 120
120 123 124 129 131 137 145 151 156 171 176 182
188 189 195 203 208 215 217 217 217 224 228 233
255 271 275 275 275 286 291 312 312 312 315 326
326 329 330 336 338 345 348 354 361 364 369 378
390 457 467 498 517 566 644 745 871 1312 1357 1613
1630
Table 2: Wheaton River Data.
1.7 2.2 14.4 1.1 0.4 20.6 5.3 0.7 1.9 13.0
12.0 9.3 1.4 18.7 8.5 25.5 11.6 14.1 22.1 1.1
2.5 14.4 1.7 37.6 0.6 2.2 39.0 0.3 15.0 11.0
7.3 22.9 1.7 0.1 1.1 0.6 9.0 1.7 7.0 20.1
0.4 2.8 14.1 9.9 10.4 10.7 30.0 3.6 5.6 30.8
13.3 4.2 25.5 3.4 11.9 21.5 27.6 36.4 2.7 64.0
1.5 2.5 27.4 1.0 27.1 20.2 16.8 5.3 9.7 27.5
2.5 27.0
1. DUS-Lomax distribution Deepthi and Chacko [6] with cdf,
F(x)
(e -1)
e(1-(1+9x)-a) _ 1
x > 0, a, 9 > 0
2. Lomax distribution Lomax [13] with cdf,
F(x) = 1 - (1 + 9x)-a, x > 0, a, 9 > 0
1
3. KM-Exponential (KME) distribution Kavya and Manoharan [10] with cdf,
F(x) = [1 - e-(1-e-Ax)], x > 0, A > 0
4. KM-Weibul (KMW) distribution Kavya and Manoharan [10] with cdf,
F(x) = e^j [1 - e-(1-e-(xßr )], x > 0, a,ß > 0
5. Weibull distribution Weibull [22] with cdf,
F(x) = 1 - e-(ßx)a, x > 0, a, ß > 0
The values of AIC, BIC, HQC and K-S test for distributions based on the first data set are
given in Table 3.
From Table 2, we can see that the new model shows the lowest AIC, BIC and HQC values among all the distributions considered here. The K-S test value of Lomax distribution is smaller than KM-Lomax distribution. In general we can say that our proposed model shows better fit to the data compared to other distributions given in this study. The plot of empirical cdf along with other cdf of the distributions for the first data set is given in Fig. 3. for a better understanding of
(a) (b)
(c) (d)
(e) (f)
Figure 3: Comparison piotfor the first data set.
the result. Compared to the distributions mentioned in Mahdavi [14] for this particular data set, the proposed model gives better result.
The comparison table of the considered models for the second data set is given in Table 4. Based on the AIC, BIC and HQC values, we can conclude that the proposed model gives the best fit to the data set compared to other Lomax, KM family and Weibull distributions considered here. The K-S test value of the KMW distribution is slightly lower than the KM-Lomax distribution. The empirical and fitted cdf plot of distributions considered for the comparison for the second data set is given in Fig. 4.
10 20 30 40 50 60
x
(a)
70
10 20 30 40 50 60 70
x
(b)
10 20
30 40
x
(c)
50 60
70
10 20
30 40
x
(d)
50 60
70
0 10 20 30 40 50 60 70
x
(e)
0 10 20 30 40 50 60 70
x
(f)
0
0
0
0
Figure 4: Comparison plot for the second data set.
8. Conclusions
In the present work, we introduce a new modified Lomax distribution using KM transformation. The main advantage of the new model is that which is parsimonious in parameter. So we can use the model more conveniently. The behavior of the hazard rate function is studied and the shape of the hazard function is shown both graphically and theoretically. Some of properties of the new model like moments, mgf, characteristic function, quantile function and order statistics are derived. stochastic ordering of the KM Lomax distribution is discussed and established the condition for the stronger mode of ordering viz likelihood ratio. We consider two real data sets to
Table 3: Maximum likelihood (ML) estimates, K-S test value, AIC, BIC, and HOC of the fitted models.
Model ML estimates K-S value AIC BIC HQC
KM-Lomax a = = 9.4097, j8 = 0.0004 0.0720 1040.647 1046.03 1042.83
DUS-Lomax a = = 9.4008, e = 0.0004 0.2703 1187.936 1193.319 1190.119
Lomax a = = 4.9251, e = 0.0011 0.0642 1405.426 1410.809 1407.609
KME A = 0.0033 0.0761 1404.864 1407.555 1405.955
KMW a = = 0.9685, fi = 0.0033 0.0772 1406.654 1412.037 1408.837
Weibull a = = 0.8848,e = 0.0046 0.0784 1407.545 1412.927 1409.728
Table 4: Maximum likelihood (ML) estimates, K-S test value, AIC, BIC, and HOC of the fitted models.
Model ML estimates K-S value AIC BIC HQC
KM-Lomax a = 243.7254,e = 0.000258 0.11 286.6547 291.2081 288.4674
DUS-Lomax a = 251.4388, e = 0.00025 0.24 364.0633 368.6166 365.876
Lomax e = 80.7719, e = 0.00102 0.14 508.2621 512.8155 510.0748
KME A = 0.0632 0.11 506.025 508.3017 506.9313
KMW e = 0.9722,e = 0.0633 0.10 507.9317 512.4851 509.7444
Weibull e = 0.9012,e = 0.08597 0.11 506.9973 511.5506 508.81
show the suitability of the model. The proposed model shows better fit to the data sets compared to other models in the literature. We can conclude that the proposed model can be used as a useful lifetime model for decreasing failure rate.
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