The Beta Modified Weibull Distribution Giovana O. Silva∗ Universidade de S˜ao Paulo
Edwin M. M. Ortega† Universidade de S˜ao Paulo
Gauss M. Cordeiro. ‡ Universidade Federal Rural de Pernambuco
Abstract A five-parameter distribution so called the beta modified Weibull distribution is defined and studied. The new distribution includes, as special cases, some important distributions discussed in the literature, such as the generalized modified Weibull, beta Weibull, exponentiated Weibull, beta exponential, modified Weibull and Weibull distributions, among several others. We derive the moments of the new distribution. We study the order statistics and obtain their moments. The method of maximum likelihood is proposed for estimating the model parameters. We obtain the observed information matrix. A data set is used to illustrate the application of the distribution. Keywords:Beta distribution; Exponentiated exponential; Exponentiated Weibull; Generalized Modified Weibull; Maximum likelihood; Modified Weibull; Observed information matrix; Weibull distribution.
1
Introduction
The Weibull distribution, having exponential and Rayleigh as special cases, is a very popular distribution for modeling lifetime data and for modeling phenomenon with monotone failure rates. When modeling monotone hazard rates, the Weibull distribution may be an initial choice because of its negatively and positively skewed density shapes. However, it does not provide a reasonable parametric fit for modeling phenomenon with non-monotone failure rates such as the bathtub shaped and the unimodal failure rates which are common in reliability and biological studies. Such ∗
Address: ESALQ, Universidade de S˜ ao Paulo, Piracicaba, Brazil. E-mail:
[email protected] Address: ESALQ, Universidade de S˜ ao Paulo, Piracicaba, Brazil. E-mail:
[email protected] ‡ Address: DEINFO, Universidade Federal Rural de Pernambuco, Brazil. E-mail:
[email protected] 1 Address for correspondence: Departamento de Ciˆencias Exatas, ESALQ/USP, Av. P´ adua Dias 11 - Caixa Postal 9, 13418-900, Piracicaba - S˜ ao Paulo - Brazil. e-mail:
[email protected] †
1
bathtub hazard curves have nearly flat middle portions and the corresponding densities have a positive anti-mode. An example of bathtub shaped failure rate is the human mortality experience with a high infant mortality rate which reduces rapidly to reach a low. It then remains at that level for quite a few years before picking up again. Unimodal failure rates can be observed in course of a disease whose mortality reaches a peak after some finite period and then declines gradually. The models that present bathtub shaped failure rate are very useful in survival analysis. But, according to Nelson (1982), the distributions presented in shape literature with this type of data, such as the distributions proposed by Hjorth (1980), are sufficiently complex and, therefore, difficult to be modeled. Later, other works had introduced new distributions for modeling bathtub shaped failure rate. For example, Rajarshi and Rajarshi (1988) presented a revision of these distributions and Haupt and Schabe (1992) considered a lifetime model with bathtub failure rates. But, these models do not present much practicability to be used. However, in the last few years, new classes of distributions were proposed based on modifications of the Weibull distribution to cope with bathtub shaped failure rate. A good review of some of these models is presented in Pham and Lai (2007). Between these, the exponentiated Weibull (EW) distribution introduced by Mudholkar et al. (1995, 1996), the additive Weibull distribution (Xie and Lai, 1995), the extended Weibull distribution presented by Xie et al. (2002), the modified Weibull (MW) distribution proposed by Lai et al. (2003), the beta exponential (BE) distribution by Nadarajah and Kotz (2005), the beta Weibull (BW) distribution studied by Cordeiro et al.(2008), the extended flexible Weibull distribution by Bebbington et al. (2007) and the generalized modified Weibull (GMW) by Carrasco et al. (2008). In this work we introduce a new distribution with five parameters, so called the beta modified Weibull (BMW) distribution, with the hope it will attract wider application in reliability and biology, as well as in other areas of research. This generalization contains as special cases several distributions such as the EW (Mudholkar et al., 1995, 1996), exponentiated exponential (EE) (Gupta and Kundu, 1999, 2001), MW (Lai et al., 2003), generalized Rayleigh (GR) (Kundu and Rakab, 2005) and the GMW distribution, among several others. The new distribution due to its flexibility in accommodating all the forms of the risk function seems to be an important distribution that can be used in a variety of problems in modeling survival data. The BMW distribution is not only convenient for modeling comfortable bathtub-shaped failure rates but it is also suitable for testing goodness-of-fit of some special sub-models such as the EW, MW and GMW distributions. The rest of the paper is organized as follows. In Section 2, we define the BMW distribution and give some special cases. Section 3 provides expansions for its cumulative distribution function (cdf) and probability density function (pdf). We show that the density function of the BMW distribution can be expressed as a mixture of MW density functions. General expansions for the moments of the new distribution are given in Section 4. Section 5 is devoted to order statistics and their moments. We also present in this section expansions for the L-moments (Hosking, 1986) which are expectations of certain linear combinations of order statistics. They form the basis of a general theory which covers the summarization and description of theoretical probability distributions. We discuss in Section 6 maximum likelihood estimation and calculate the elements of the observed information 2
matrix. Section 7 provides two applications to real data sets. Section 8 ends with some conclusions.
2
The model definition
The idea of the BMW distribution stems from the following general class: if G denotes the cdf of a random variable then a generalized class of distributions can be defined by Z G(x) 1 wa−1 (1 − w)b−1 dw (1) F (x) = IG(x) (a, b) = B(a, b) 0 for a > 0 and b > 0,R where Iy (a, b) = By (a, b)/B(a, b) denotes the incomplete beta function y ratio, and By (a, b) = 0 wa−1 (1 − w)b−1 dw denotes the incomplete beta function. This class of generalized distributions has been receiving considerable attention over the last years, in particular after the recent works of Eugene et al. (2002) and Jones (2004). Eugene et al. (2002) introduced what is known as the beta normal (BN) distribution by taking G(x) in (1) to be the cdf of the normal distribution and derived some of its first moments. More general expressions for these moments were derived by Gupta and Nadarajah (2004). Nadarajah and Kotz (2004) introduced the beta Gumbel (BG) distribution by taking G(x) to be the cdf of the Gumbel distribution and provided closed form expressions for the moments, the asymptotic distribution of the extreme order statistics and discussed the maximum likelihood estimation procedure. Nadarajah and Gupta (2004) introduced the beta Fr´echet (BF) distribution by taking G(x) to be the Fr´echet distribution, derived the analytical shapes of the probability density function (pdf) and the hazard rate function and calculated the asymptotic distribution of the extreme order statistics. Also, Nadarajah and Kotz (2005) worked with the BE distribution and obtained the moment generating function, the first four cumulants, the asymptotic distribution of the extreme order statistics and discussed the maximum likelihood estimation. Another distribution that happens to belong to equation (1) is the beta logistic distribution, which has been around for over 20 years (Brown et al., 2002), even if it did not originate directly from this equation. We are motivated to introduce the BMW distribution because of the above generalizations, the wide usage of the Weibull distribution and the fact that the current generalization provides means of its continuous extension to still more complex situations. Lai et al. (2003) introduced the MW distribution having three parameters α > 0, λ > 0 and γ ≥ 0 with cdf and pdf given by Gα,γ,λ (x) = 1 − exp{−αxγ exp(λx)}
(2)
gα,γ,λ (x) = αxγ−1 (γ + λx) exp{λx − αxγ exp(λx)}, x > 0,
(3)
and
respectively. The parameters α and γ control the scale and shape of the distribution, respectively. The parameter λ is a kind of accelerating factor in the imperfection time and it works as a factor 3
of fragility in the survival of the individual when the time increases. The Weibull (W) distribution is a special case of (3) when λ = 0. If, in addition to λ = 0, γ = 1 and γ = 2, we obtain the exponential (E) and Rayleigh (R) distributions, respectively. The pdf corresponding to (1) can be written in the form f (x) =
1 G(x)a−1 {1 − G(x)}b−1 g(x), B(a, b)
(4)
where g(x) = dG(x)/dx is the pdf of the parent distribution. The pdf f (x) will be most tractable when the functions G(x) and g(x) have simple analytic expressions as is the case of the MW distribution. Except for some special choices for G(x) in (1), it would appear that the formula (4) will be difficult to deal with in generality. We now introduce the BMW distribution by taking G(x) in (1) to be the cdf (2) of the MW distribution. Then, the general form for the BMW cdf is 1 F (x) = B(a, b)
Z
1−exp{−αxγ exp(λx)}
ω a−1 (1 − ω)b−1 dω.
(5)
0
The BMW density function (for x > 0) can be written from (2) and (4) as f (x) =
αxγ−1 (γ + λx) exp(λx) [1 − exp{−αxγ exp(λx)}]a−1 exp{−bαxγ exp(λx)}. B(a, b)
(6)
The BMW distribution contains as special cases several well-known distributions. For example, it simplifies to the BW distribution when λ = 0. If γ = 1 in addition to λ = 0, it reduces to the BE distribution. The GMW distribution is also a special case when b = 1. If a = 1 in addition to b = 1, it gives as special case the MW distribution. For b = 1 and λ = 0, the BMW distribution reduces to the EW distribution. If γ = 1 in addition to b = 1 and λ = 0, the BMW distribution becomes the EE distribution. For γ = 2, λ = 0 and b = 1, the BMW distribution becomes the generalized Rayleigh (GR) distribution (Kundu and Rakab, 2005). The Weibull distribution is clearly a special case for a = b = 1 and λ = 0. Several special sub-models of the BMW model are illustrated in Figure 1, where the sub-models widely known not defined before are: beta modified Rayleigh (BMR), beta modified exponential (BME), generalized modified Rayleigh (GMR), generalized modified exponential (GME), beta Rayleigh (BR), modified Rayleigh (MR) and modified exponential (ME). If X is a random variable with density (6), we write X ∼ BMW(a, b, α, λ, γ). The BMW distribution is easily simulated from (5) as follows: if V has a beta B(a, b) distribution, then the solution of the nonlinear equation X γ exp(λX) = −α−1 log(1 − V ) has the BMW(a, b, α, γ, λ) distribution. To simulate data from this nonlinear equation, we can use the matrix programming language Ox through SolveNLE subroutine (see Doornik, 2001). The plots comparing the exact 4
BMW BMW BME
BMR
GMR
GMW
BR
MR
MW
GME
BW
BE
ME
EW GR
R
EE
W
E
Figure 1: Relationships between the BMW distribution and other commonly used distributions.
BMW densities and histograms from two simulated data sets for some parameter values are given in Figure 2. We note from this figure that the simulated values are consistent with the BMW distribution. The hazard function of the BMW distribution depends on the incomplete beta function ratio. It is given by (for x > 0) h(x) =
αxγ−1 (γ + λx) exp(λx) [1 − exp{−αxγ exp(λx)}]a−1 exp{−bαxγ exp(λx)}. B(a, b)Iexp{−αxγ exp(λx)} (a, b)
(7)
Figures 2 and 3 illustrate some of the possible shapes of the pdf (6) and hazard function (7), respectively, for selected parameter values, including some special cases. A characteristic of the BMW distribution is that its hazard function can be bathtub shaped, monotonically increasing or decreasing and upside-down bathtub depending basically on the parameter values.
5
(b)
0.005
0.010
Density
1.0 0.0
0.000
0.5
Density
1.5
0.015
2.0
0.020
(a)
0.5
1.0
0
1.5
200
400
600
x
x
Figure 2: Plots of the BMW densities for simulated data sets: (a) a = 1.5, b = 0.5, α = 0.5, λ = 1.5, γ = 3.0 and (b) a = 8, b = 1.0, α = 0.43, λ = 0, γ = 0.5.
3
Expansions for the distribution, survival and density functions
We provide simple expansions for the BMW cdf depending on whether the parameter b (or a) is real non-integer or integer. The pdf in (6) is straightforward to compute using any statistical software. However, we show that the BMW density can be expressed as an infinite (or finite) weighted linear combination of pdf’s of MW distributions. This is important to obtain some mathematical properties of the BMW distribution directly from the corresponding properties of the MW distribution. We now give an expansion for F (x) in terms of an infinite sum of MW cdf’s. For a > 0 real non-integer, using the power series representation, yields Z
x
w
a−1
b−1
(1 − w)
dw =
0
=
Z ∞ X (−1)j Γ(a) j=0 ∞ X j=0
Γ(a − j)j!
x
(1 − w)b+j−1 dw
0
(−1)j Γ(a) {1 − (1 − x)b+j }, Γ(a − j)(b + j)j!
6
0.05
a = 0.5,b = 1,α = 0.001,λ = 0.2,γ = 0.5 (GMW a = 0.5,b = 1,α = 0.01,λ = 0,γ = 1 (EE) a = 8,b = 1,α = 0.43,λ = 0,γ = 0.5 (WE) a = 1,b = 1,α = 0.43,λ = 0,γ = 0.5 (Weibull ) a = 0.1,b = 1,α = 0.001,λ = 0,γ = 2 (GR)
0.03
f(x) 0.0
0.00
0.01
0.5
0.02
1.0
f(x)
1.5
0.04
2.0
2.5
a = 0.8,b = 0.8,α = 0.8,λ = 0,γ = 1.5 (BW ) a = 0.8,b = 0.8,α = 0.8,λ = 0,γ = 1 (BE) a = 4,b = 4,α = 0.5,λ = 0.5,γ = 0.5 a = 1.5,b = 0.5,α = 0.5,λ = 1.5,γ = 3 a = 0.5,b = 0.5,α = 0.5,λ = 0.5,γ = 0.5
0.0
0.5
1.0
1.5
2.0
2.5
0
10
x
20
30
40
50
60
x
Figure 3: Plots of the BMW density for some parameter values.
where Γ(.) is the gamma function. Hence, ∞
F (x) =
X 1 (−1)j Γ(a) {1 − [1 − Gα,γ,λ (x)]b+j } B(a, b) Γ(a − j)(b + j)j! j=0
and then F (x) =
∞ X
wj Gα(b+j),γ,λ (x),
(8)
j=0
where wj = wj (a, b) =
(−1)j Γ(a) B(a, b)Γ(a − j)(b + j)j!
P are constants such that ∞ j=0 wj = 1 and Gα(b+j),γ,λ (x) is the cdf of the MW distribution with scale parameter α(b + j), shape parameter γ and accelerated parameter λ. Is a is an integer, the index j in the previous sum stops at a − 1. Expansion (8) is used throughout the paper for any a.
7
0.10 0.08
a = 0.5,b = 1,α = 0.001,λ = 0.2,γ = 0.5 (GMW a = 0.5,b = 1,α = 0.01,λ = 0,γ = 1 (EE) a = 8,b = 1,α = 0.43,λ = 0,γ = 0.5 (WE) a = 1,b = 1,α = 0.43,λ = 0,γ = 0.5 (Weibull ) a = 0.1,b = 1,α = 0.001,λ = 0,γ = 2 (GR)
0.0
0.00
0.5
0.02
1.0
0.04
f(x)
h(x)
1.5
0.06
2.0
2.5
a = 0.8,b = 0.8,α = 0.8,λ = 0,γ = 1.5 (BW ) a = 0.8,b = 0.8,α = 0.8,λ = 0,γ = 1 (BE) a = 4,b = 4,α = 0.5,λ = 0.5,γ = 0.5 a = 1.5,b = 0.5,α = 0.5,λ = 1.5,γ = 3 a = 0.5,b = 0.5,α = 0.5,λ = 0.5,γ = 0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0
50
x
100
150
x
Figure 4: Plots of the BMW hazard function for some parameter values.
From
∞ P j=0
wj = 1, the BMW survival function has the following expansion
S(x) = 1 − F (x) =
∞ X
wj Sα(b+j),γ,λ (x),
(9)
j=0
where Sα(b+j),γ,λ (x) = exp{−α(b + j)xγ exp(λx)} is the survival function of the MW distribution with parameters α(b + j), γ and λ. The density of the BMW distributions follows immediately as f (x) =
∞ X
wj gα(b+j),γ,λ (x).
(10)
j=0
Hence, the ordinary, central, factorial and inverse moments and the moment generating function of the BMW distribution could in principle follow from the same weighted infinite (or finite if a is an integer) linear combination of the corresponding quantities for MW distributions. The MW density thus represents a particular case of (10) when a = b = 1. The GMW density is also a special case of (10) when only b = 1. Besides, while the transformation (1) is not analytically tractable in the general case, the formulae related with the BMW distribution turn out manageable (as it is shown in the rest of this paper), and with the use of modern computer 8
resources with analytic and numerical capabilities, may turn into adequate tools comprising the arsenal of applied statisticians. Equations (8) and (10) represent the main results of this section. The advantage of these expressions is that they can be used to determine moments without any restrictions or conditions on the five parameters.
4
Reliability
In the area of stress-strength models there has been a large amount of work as regards estimation of the reliability R = P r(X2 < X1 ) when X1 and X2 are independent random variables belonging to the same univariate family of distributions. The algebraic form for R has been worked out for the majority of the well-known standard form distributions. However, there are still many other distributions (including generalizations of the well-known distributions) for which the form of R has not been derived. We now derive the corresponding form for the reliability R when X1 and X2 are independent and have the same BMW distribution. The form of R can be expressed as Z ∞ R= f (x)F (x)dx. (11) 0
Substituting the equations (6) and (8) in (11) gives ∞
α X wj (a, b) R= B(a, b) j=0
Z∞ xγ−1 (γ + λx)eλx (1 − ub+j )(1 − u)a−1 ub dx, 0
where u = exp{−αxγ eλx } and wj (a, b) is defined in Section 3. Hence, du = u log(u)[(γ + λx)/x]dx and then Z1 ∞ X 1 wj (a, b) (1 − ub+j )(1 − u)a−1 ub−1 dx. R= B(a, b) j=0
0
The last integral computed using Maple yields " # ∞ X 1 Γ(a)Γ(2b + j) R= wj (a, b) B(a, b) − . B(a, b) Γ(2b + j + a)
(12)
j=0
5
General formulae for the moments
We hardly need to emphasize the necessity and importance of moments in any statistical analysis especially in applied work. Some of the most important features and characteristics of a distribution 9
can be studied through moments (e.g., tendency, dispersion, skewness and kurtosis). We now derive an infinite sum representation for the rth ordinary moment of the BMW distribution, µ0r say. From (10) we can obtain elementary expression µ0r
=
∞ X
wj τr (j),
(13)
j=0
R∞ where τr (j) = 0 xr gα(b+j),γ,λ (x)dx is the rth moment of the MW distribution with parameters α(b + j), γ and λ. Carrasco et al. (2008, Section 4) obtained an infinite representation for the rth moment of the MW distribution which can be written as ∞ X Ai1 ,...,ir Γ(sr /γ + 1) τr (j) = , (14) [α(b + j)]sr /γ i ,...,i =1 1
r
where Ai1 ,...,ir = ai1 . . . air and sr = i1 + . . . + ir , and
(−1)i+1 ii−2 ai = (i − 1)!
µ ¶i−1 λ . γ
Hence, the moments of the BMW distribution can be obtained directly from equations (13) and (14). Graphical representation of skewness and kurtosis when α = 1.2, λ = 4.2 and γ = 3.2, as a function of parameter a for some choices of parameter b, and as a function of parameter b for some choices of parameter a, are given in Figures 5 and 6, respectively. These figures show that the skewness and kurtosis increase when b decreases for fixed a and when a increases for fixed b.
6
Moments of order statistics
The density of the ith order statistic Xi:n , fi:n (x) say, in a random sample of size n from the BMW distribution, is given by (for i = 1, . . . , n) fi:n (x) =
1 f (x) F (x)i−1 {1 − F (x)}n−i . B(i, n − i + 1)
(15)
The cdf of the ith order statistic is simply Fi:n (x) = IF (x) (i, n − i + 1). Alternatively, we can write Fi:n (x) as binomial sums ¶ n µ ¶ n−i µ X X © ªn−k ªk n i+k−1 © k i Fi:n (x) = F (x) 1 − F (x) = F (x) 1 − F (x) . k k k=1
k=0
10
(a)
(b)
s k w n e s s
k u r t o s i s
Figure 5: Skewness and kurtosis of the BMW distribution as a function of parameter a, for some values of parameter b.
We obtain a closed form expression for the moments of the BMW order statistics by using a general result due to Barakat and Abdelkader (2004) applied to the independent and identically distributed case. For a distribution with pdf f (x) and cdf F (x) we can write r E(Xi:n )
=r
n X
j−n+i−1
(−1)
j=n−i+1
where
Z Ij (r) =
∞
µ ¶µ ¶ j−1 n Ij (r), n−i j
xr−1 {1 − F (x)}j dx.
0
Expansion (9) for the BMW survival function yields 1 − F (x) =
∞ X
ws ub+s ,
s=0
where u was defined in Section 4. Then, Z Ij (r) =
∞
à xr−1
0
∞ X s=0
11
!j ws ub+s
.
(16)
(a)
s k w n e s s
(b)
k u r t o s i s
Figure 6: Skewness and kurtosis of the BMW distribution as a function of parameter b, for some values of parameter a.
We now use an equation of Gradshteyn and Ryzhik (2000, Section 0.314) for power series raised to integer powers. We have for any j positive integer Ã∞ !j ∞ X X s ws u = cj,s us , (17) s=0
s=0
where the coefficients cj,s for s = 1, 2, . . . are easily obtained from the recurrence equation cj,s = (sa0 )−1
s X
(jm − s + m)wm cj,s−m
(18)
m=1
w0j .
with cj,0 = Using (18), the coefficients cj,s are calculated from cj,1 , . . . , cj,s−1 and, therefore, from the constants w0 , . . . , ws . The coefficients cj,s can be given explicitly in terms of these constants, although it is not necessary for programming numerically our expansions in any algebraic or numerical software such as Matlab, Maple or Mathematica. From the last integral and (17) we have Ã∞ !j Z ∞ Z ∞ ∞ X X r−1 b+s Ij (r) = x ws u dx = cs,j xr−1 ubj+s dx. 0
s=0
s=0
12
0
Hence, Ij (r) =
∞ X
Z cj,s
s=0
∞
xr−1 exp{−α(bj + s)xγ exp(λx)}dx.
(19)
0
For obtaining Ij (r) we use the same algebraic development by Carrasco et al. (2008, Section 4). Transforming y = xγ eλx we can invert the transformation to obtain x as a polynomial function in y when both λ and γ are positive. We have x= where F (w) =
γ ³ λy 1/γ ´ F , λ γ
(20)
∞ X (−1)m+1 mm−2 wm . (m − 1)!
m=1
Hence, we can express x in terms of y from (20) as x=
∞ X
am y m/γ ,
m=1
where am
(−1)m+1 mm−2 = (m − 1)!
µ ¶m−1 λ . γ
The last integral in (19) written in terms of y becomes Z I=
∞ ∞n X
0
But
∞ nX
am y m/γ
m=1
am y
m/γ
or−1
∞ nX ap p p/γ−1 o exp{−α(bj + s)y} y dy. γ p=1
or−1
m=1
=
∞ X
am1 . . . amr−1 y (m1 +...+mr−1 )/γ .
m1 ,...,mr−1 =1
Then, I can be rewritten as I = γ −1
∞ X m1 ,...,mr =1
Z mr Am1 ,...,mr
∞
y sr /γ−1 exp{−(j + 1)αy}dy,
0
where the term Am1 ,...,mr = am1 . . . amr 13
(21)
comes easily from the constants in (21) and s r = m1 + . . . + mr . Substituting v = (j + 1)αy in the last integral, we can obtain Z ∞ ∞ X mr Am1 ,...,mr −1 I=γ v sr /γ−1 exp(−v)dv, sr /γ [(j + 1)α] 0 m ,...,m =1 1
r
which in terms of the gamma function reduces to ∞ X
I = γ −1
m1 ,...,mr
mr Am1 ,...,mr Γ(sr /γ). [(j + 1)α]sr /γ =1
Finally, the rth moment of the order statistic Xi:n follows from (16) and (19) as r E(Xi:n )=
r γ
n X
(−1)j−n+i−1
µ ¶µ ¶ ∞ j−1 n X cj,s n−i j s=0
j=n−i+1
∞ X m1 ,...,mr
mr Am1 ,...,mr Γ(sr /γ). sr /γ [(j + 1)α] =1
(22)
Equation 22 gives the moments of the BMW order statistics and can be applied to all sub-models given in Figure 1. An alternative way to compute the moments follows by expressing the density of the ith order statistic as a mixture of MW densities. We have from (15) fi:n (x) =
i−1 X © ªn−i+k 1 (−1)k 1 − F (x) f (x) . B(i, n − i + 1) k=0
The above sum becomes µX ¶n−i+k X i−1 ∞ i−1 ∞ X X (−1)k ws ub+s = (−1)k cn−i+k,s ub(n−i+k)+s . s=0
k=0
k=0
s=0
Plugging (10) we obtain fi:n (x) =
∞ X i−1 X 1 wj (−1)k cn−i+k,s ub(n−i+k)+s gα(b+j),γ,λ (x) B(i, n − i + 1) j,s=0 k=0
and then fi:n (x) =
∞ X i−1 X 1 wj (−1)k cn−i+k,s α(b + j)xγ−1 (γ + λx) B(i, n − i + 1) j,s=0 k=0 n o £ ¤ exp λx − α b(n − i + k + 1) + s + j xγ eλx .
14
Finally, we have ∞ X i−1 X
fi:n (x) =
pn,i,k,j,s gα[b(n−i+k+1)+s+j],γ,λ (x),
j,s=0 k=0
where pn,i,k,j,s = pn,i,k,j,s (a, b) =
wj (−1)k (b + j)cn−i+k,s . b(n − i + k + 1) + s + j
Hence, some mathematical properties (ordinary, central and factorial moments, etc.) of the BMW order statistics follow from the corresponding properties of MW distributions. The L-moments are analogous to the ordinary moments but can be estimated by linear combinations of expected order statistics. They are defined by Hosking (1990) −1
λr+1 = (r + 1)
r X k=0
µ ¶ r (−1) E(Xr+1−k:r+1 ), r = 0, 1, . . . k k
The first four L-moments are: λ1 = E(X1:1 ), λ2 = 21 E(X2:2 − X1:2 ), λ3 = 13 E(X3:3 − 2X2:3 + X1:3 ) and λ4 = 14 E(X4:4 − 3X3:4 + 3X2:4 − X1:4 ). The L-moments have the advantage that they exist whenever the mean of the distribution exists, even though some higher moments may not exist, and are relatively robust to the effects of outliers. From the expansion (22) for the moments of the order statistics we can obtain expansions for the L-moments of the BMW distribution.
7
Maximum Likelihood Estimation
In this section, we consider the maximum likelihood estimates (MLEs) of the BMW(a, b, α, γ, λ) distribution. Let x1 , . . . , xn be a random sample of size n from the BMW(a, b, α, γ, λ) distribution. Then, the log-likelihood function for the vector of parameters θ = (a, b, α, γ, λ)T can be written as n n o £ ¤ £ ¤ X l(θ) = −n log B(a, b) + log(γ + λxi ) + log(vi ) − log(xi ) i=1
+(a − 1)
n X
n n o X log 1 − exp(−vi ) − b vi ,
i=1
i=1
where vi = αxγi exp(λxi ) is a transformed observation.
15
(23)
The components of the score vector U (θ) are given by n £ ¤ X £ ¤ Ua (θ) = −n ψ(a) + ψ(a + b) + log 1 − exp(−vi ) , i=1 n £ ¤ X Ub (θ) = −n ψ(b) + ψ(a + b) − vi , i=1
Uα (θ) =
n X xγ exp(λxi ) i
i=1 n · X
vi
n n X X exp(−vi )xγi exp(λxi ) £ ¤ −b xγi exp(λxi ), + (a − 1) 1 − exp(−vi ) i=1 i=1
n n X X 1 exp(−vi )vi log(xi ) £ ¤ −b Uγ (θ) = + log(xi )] + (a − 1) vi log(xi ), γ + λxi 1 − exp(−vi ) i=1 i=1 i=1 ¶ n n n µ X X X exp(−vi )vi xi xi £ ¤ −b + xi + (a − 1) vi xi , Uλ (θ) = γ + λxi 1 − exp(−vi ) i=1
i=1
i=1
where ψ(.) is the digamma function. For interval estimation and hypothesis tests on the model parameters, we require the observed information matrix. The 5 × 5 unit observed information matrix J = J(θ) is given in appendix A. Under conditions that are fulfilled for parameters in the interior of the parameter space but not on the boundary, the asymptotic distribution of √ b − θ) is N5 (0, I(θ)−1 ), n(θ where I(θ) is the expected information matrix. This asymptotic behavior is valid if I(θ) is replaced b i.e., the observed information matrix evaluated at θ. b The asymptotic multivariate normal by J(θ), −1 b N5 (0, J(θ) ) distribution can be used to construct approximate confidence intervals and confidence regions for the individual parameters and for the hazard and survival functions. The asymptotic normality is also useful for testing goodness of fit of the BMW distribution and for comparing this distribution with some of its special submodels using one of the three well-known asymptotically equivalent test statistics - namely, the likelihood ratio (LR) statistic, Wald and Rao score statistics. We can compute the maximum values of the unrestricted and restricted log-likelihoods to construct the LR statistics for testing some submodels of the BMW distribution. For example, we may use the LR statistic to check if the fit using the BMW distribution is statistically “superior” to a fit using the GMW, MW and EW distributions for a given data set. In any case, hypothesis testing of the type H0 : θ = θ 0 versus H : θ 6= θ 0 , can be performed by using any of the above three asymptotically statistics. For example, the test of H0 : a = b = 1 versus H : H0 is not true is equivalent to compare the BMW distribution with the MW distribution and the LR statistic reduces to b − `(1, 1, α e w = 2{`(b a, bb, α b, γ b, λ) e, γ e, λ)}, b are the MLEs under H and α e are the estimates under H0 . where b a, bb, α b, γ b and λ e, γ e and λ
16
8
Example
We consider an application to a data set on lifetimes of 50 devices given in Aarset (1987) and reported in Mudholkar and Srivastava (1993), Mudholkar et al. (1996) and Wang (2000). The required numerical evaluations were implemented by using an Ox program (sub-rotine MaxBFGS)(see, Doornik, 2001) and SAS (PROC NLMIXED) (see, SAS, 2004). The TTT plot (Barlow and Campo, 1975) for this data set given in Figure 7 indicates a bathtubshaped hazard rate. Hence, an appropriate model for fitting such data is the BMW distribution.
Figure 7: TTT plot on the 50 observations for the Aarset data.
The BMW model includes some sub-models described in Section 2 as especial cases and thus allows their evaluation relative to each other and to a more general model. Further, we can compute the maximum values of the unrestricted and restricted log-likelihoods to obtain the LR statistics for testing some sub-models of the BMW distribution. For example, an analysis under the BMW model provides a check on the appropriateness of the GMW, MW and EW models and indicates the extent to which inferences depend upon the model. The LR statistic was obtained for testing the hypotheses H0 :a=b=1 versus H1 :H0 is not true, i.e. to compare the MW model with the BMW model. The LR statistic w = 2{−220.817 − (−227.982)} = 14.3293 (p-value < 0.01) yields favorable indications for the BMW model. The MLEs of the model parameters with the corresponding standard errors in parentheses as well as the values of the Akaike information criterion (AIC) for the fitted models are given in Table 1. These results indicate that the BMW model has the lowest AIC value among all fitted models, 17
and hence it could be chosen as the best model. Table 1: MLEs for some sub-models for the Aarset data, the corresponding standard errors are given in parentheses and the values of the AIC statistic. Model BMW BW GMW MW EW GR
a 0.1975 (0.0462) 0.18356 (0.0509) 0.2975 (0.0613) 1 0.4668 (0.0889) 0.3643 (0.0624)
Estimated Parameters b α λ 0.1647 0.0002 0.0541 (0.0830) (6.6931e-005) ( 0.0157) 0.0748 0.0007 0 (0.0353) (0.0004) 0 1 0.0002 0.0529 (0.0001) (0.0138) 1 0.0624 0.0233 (0.0267) (0.0048) 1 0.0011 0 (0.0010) 1 0.0002 0 (4.8738e-005)
γ 1.3771 (0.3387) 2.3615 (0.1715) 0.9942 (0.2396) 0.3548 (0.1127) 1.5936 (0.1858) 2
AIC 451.635 463.964 455.765 460.310 480.447 475.904
Additionally, the corresponding survival function for the different models are plotted in Figure 8. It can be seen that the BMW distribution is a very competitive model for describing the bathtubshaped failure rate lifetime data. Besides, the plots of the estimated densities of the different distributions fitted to the Aarset data given in Figure 9 show that the BMW distribution gives a better fit than the other five sub-models.
9
Conclusions
We introduce a five parameter lifetime distribution, named the beta modified Weibull (BMW) distribution, which extends several distributions proposed and widely used in the lifetime literature. The model is much more flexible than the exponentiated Weibull (EW), modified Weibull (MW) and generalized modified Weibull (GMW) sub-models. The new distribution is useful to model lifetime data with a bathtub shaped hazard function. We provide a mathematical treatment of the distribution including the densities of the order statistics. We give explicit closed form expressions for the rth moment which hold in generality for any parameter values. We give infinite weighted sums for the moments of the order statistics. The application of the BMW model are demonstrated in one example.
18
1.0 0.6 0.4 0.0
0.2
Estimated survivor function
0.8
Kaplan−Meier CI for Kaplan−Meier BMW MW BW EW GMW GR
0
20
40
60
80
time
Figure 8: Estimated survival function for some fitted models with empirical survival function for Aarset data.
Appendix The elements of the observed information matrix J(θ) for the parameters (a, b, α, γ, λ) are ¾ ½ 00 h i2 Γ (a) 2 0 2 2 − ψ (a) − Γ(a + b) ψ (a + b) + ψ (a + b) + ψ (a + b) , Jaa (θ) = −n Γ(a) ½ Jbb (θ) = −n
Jαα (θ) = −
¾ h i2 Γ00 (b) − ψ 2 (b) − Γ(a + b) ψ 0 (a + b) + ψ 2 (a + b) + ψ 2 (a + b) , Γ(b)
¸ n · 2γ X x exp(2λxi ) i
i=1
Jγγ (θ) = −
vi2
n n n X X X £ ¤2 £ ¤2 (γ + λxi )−2 + (a − 1) yi log(xi ) − b vi log(xi ) , i=1
Jλλ (θ) = −
¾ n ½ X exp(−vi )x2γ i exp(2λxi ) − (a − 1) , £ ¤2 1 − exp(−vi ) i=1
n µ X i=1
i=1
xi γ + λxi
¶2 + (a − 1)
n X
i=1
yi x2i
−b
i=1
n X i=1
19
vi x2i ,
0.000
0.005
0.010
Density
0.015
0.020
0.025
BMW BW GMW MW WE GR
0
20
40
60
80
x
Figure 9: Estimated densities of the some fitted models for Aarset data.
Jαγ (θ) = (a − 1)
n X
yi vi−1 xγi
exp(λxi ) log(xi ) − b
i=1
Jαλ (θ) = (a − 1)
n X
Jγλ (θ) = (a − 1)
yi vi−1 exp(λxi )xγ+1 −b i
n X
xiγ+1 exp(λxi ),
i=1
yi xi log(xi ) − b
i=1
Jaα (θ) =
xγi exp(λxi ) log(xi ),
i=1
i=1 n X
n X
n X
vi xi log(xi ),
i=1
n X exp(−vi )vi xi £ ¤, Jaλ (θ) = 1 − exp(−v ) i i=1
n n X X exp(−vi )xγi exp(λxi ) exp(−vi )vi log(xi ) £ ¤ , Jaγ (θ) = £ ¤ , Jab (θ) = −nψ 0 (a + b), 1 − exp(−v ) 1 − exp(−v ) i i i=1 i=1
Jbα (θ) = −
n X
xγi
exp(λxi ),
i=1
where yi =
exp(−vi )vi 1−exp(−vi )
Jbγ (θ) = −
n X
vi log(xi ),
i=1
Jbλ (θ) = −
n X
vi xi ,
i=1
h i vi 1− 1−exp(v is another transformed observation Γ00 (.) is the second-derivative i)
of the gamma function and ψ 0 (.) is the derivative of the digamma function. 20
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