The Beta Generalized Weibull distribution Properties ...

Reliability Engineering and System Safety 102 (2012) 5–15

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The Beta Generalized Weibull distribution: Properties and applications Neetu Singla n, Kanchan Jain, Suresh Kumar Sharma Department of Statistics, Panjab University, Chandigarh 160014, India

a r t i c l e i n f o

abstract

Article history: Received 28 June 2011 Received in revised form 9 February 2012 Accepted 12 February 2012 Available online 25 February 2012

A five-parameter distribution called Beta Generalized Weibull (BGW) distribution is introduced. Beta Generalized Exponential (BGE), Beta Weibull (BW), Generalized or Exponentiated Weibull (GW or EW), Generalized Rayleigh (GR), Beta Exponential (BE), Generalized Exponential (GE), Weibull, Rayleigh and Exponential are its sub models. The cumulative distribution function (cdf) and the probability density function (pdf) have been expressed as mixtures of the Generalized Weibull cdfs and pdfs. The kth order moment has been derived. The non-linear equations for deriving the maximum likelihood estimators and the elements of the observed information matrix are presented. The distribution is found to be superior to the existing sub models on being fitted to two real data sets. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Weibull Beta Weibull distribution Generalized Weibull distribution Maximum likelihood estimation (MLE) Information matrix

1. Introduction For complex electronic and mechanical systems, the failure rate often exhibits non-monotonic (bathtub or upside-down bathtub (unimodal) shaped) failure rates (Lai and Xie [1]). Distributions with such failure rates have attracted a considerable attention of researchers in reliability engineering. In software reliability, bathtub shaped failure rate is encountered in firmware, an embedded software in hardware devices. Firmware plays an important role in functioning of hard drives of large computers, spacecraft and high performance aircraft control systems, advanced weapon systems, safety critical control systems used for monitoring the industrial process in chemical and nuclear plants (Zhang et al. [2]). The upside down bathtub shaped failure rate is used in data of motor bus failures (Mudholkar et al. [3]), for optimal burn-in decisions (Block and Savits [4], Chang [5]) and for ageing properties in reliability (Gupta and Gupta [6], Jiang et al. [7]). Weibull distribution introduced by Weibull [8] is a popular distribution for modeling phenomenon with monotonic failure rates. But this distribution does not provide a good fit to data sets with bathtub shaped or upside-down bathtub shaped (unimodal) failure rates, often encountered in reliability, engineering and biological studies. Hence a number of new distributions modeling the data in a better way have been constructed in literature as ramifications of Weibull distribution. Pham and Lai [9] present a

n

Corresponding author. Tel.: þ91 9888931886. E-mail addresses: [email protected] (N. Singla), [email protected] (K. Jain), [email protected] (S. Kumar Sharma). 0951-8320/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2012.02.003

review of some of the generalizations or modifications of Weibull distribution. Xie and Lai [10] introduced the additive Weibull model obtained by adding two Weibull survival functions. They showed that the failure rate function of the proposed model is expressible as the sum of two Weibull failure rate functions. They demonstrated the graphical estimation technique based on conventional Weibull plot technique. This model is applicable in case of bathtub shaped failure rate function. Xie et al. [11] studied a generalization of the Weibull distribution, the modified Weibull extension, for modeling the systems with a bathtub shaped failure rate function. They explored the distributional properties and found the maximum likelihood estimates of the parameters. They used two lifetime data sets for illustrating the applications of new model. Beta Exponential (BE) distribution, generated from the logit of a Beta random variable, was introduced by Nadarajah and Kotz [12]. They provided mathematical properties of BE model and also derived expressions for the mgf, first four moments, variance, skewness and kurtosis. They provided estimates of the parameters using the methods of moments and maximum likelihood estimate and gave expressions for the Fisher Information matrix. Bebbington et al. [13] proposed a new distribution called the new flexible Weibull distribution. They characterized its failure rate function and considered parametric estimation. They found the new distribution to be quite flexible for modeling IFR, IFRA and modified bathtub (MBT). Jiang et al. [7] studied the ageing properties of unimodal failure rate models. Jiang and Murthy [14] studied three different models, each involving two parameter Weibull distributions, to analyze the failure data and discussed parameter estimation. Bucar et al. [15] proved that the reliability of an arbitrary system can be approximated well by a finite Weibull mixture with

6

N. Singla et al. / Reliability Engineering and System Safety 102 (2012) 5–15

positive component weights. They estimated the unknown parameters and the weights of the Weibull mixture. Calculation of AICs was done for comparing the fitted models to the empirical ones. Some other variants of Weibull distribution are Generalized or Exponentiated Weibull (GW or EW) (Mudholkar et al. [16,17]), the modified Weibull (MW) distribution (Lai et al. [18], Nadrajah and Kotz [19]), the Beta Weibull (BW) distribution (Lee et al. [20]) and the Generalized Modified Weibull (GMW) distribution (Carrasco et al. [21]). Beta Generalized (Beta G), introduced by Singh et al. [22] is a rich class of generalized distributions. This class has captured a considerable attention over the last few years. Sepanshi and Kang [23] applied the Beta G distribution to model the size distribution of income. This distribution has been studied in literature for various forms of G. The distributions that have been explored are the Beta Normal (BN) (Eugene et al. [24]), the Beta Gumbel (BGu) distribution (Nadarajah and Kotz [25]), the Beta Frećhet (BFr) distribution (Nadarajah and Gupta [26]), the Beta Exponential (BE) distribution (Nadarajah and Kotz [12]), the Beta Weibull (BW) distribution (Famoye et al. [27], Lee et al. [20] and Cordeiro et al. [28]). Souza et al. [29] introduced the Beta Generalized Exponential (BGE) distribution and the Beta Modified Weibull (BMW) distribution was discussed by Silva et al. [30]. Beta Inverse Weibull (BInw) (Khan [31]), Beta Generalized Pareto (BGP) (Mahmoudi [32]) and Beta Dagum (BDa) (Domma and Contino [33]) are some new extensions of the Beta G class of distributions. The cumulative distribution function (cdf) of the Beta G distribution has the form Z GðxÞ BGðxÞ ða,bÞ 1 FðxÞ ¼ ¼ IGðxÞ ða,bÞ,a,b4 0, wa1 ð1wÞb1 dw ¼ Bða,bÞ 0 Bða,bÞ ð1Þ where G(x) is an arbitrary parent/baseline cdf of a random variable; Ry By ða,bÞ ¼ 0 wa1 ð1wÞb1 dw is the incomplete beta function with B(a,b) ¼B1(a,b) and Iy(a,b)¼(By(a,b)/B(a,b)) is the incomplete beta function ratio. The density corresponding to (1) is written as 1 GðxÞa1 ð1GðxÞÞb1 gðxÞ, f ðxÞ ¼ Bða,bÞ

proposed distribution provides a better fit than all its sub models. Hence, the new distribution is expected to provide a more general and flexible framework having wider applications in reliability engineering and survival studies. The paper is organized as follows. In Section 2, the proposed BGW distribution is introduced. In Section 3, we derive some alternative forms of the cdf and the pdf of the BGW distribution. It is shown that the BGW pdf can be expressed as an infinite (or finite) weighted linear combination of the pdfs of the GW random variables. The pdf of order statistics is also derived. Section 4 derives the moments through the definition as well as Laplace transformation. In Section 5, the nonlinear equations for deriving the maximum likelihood estimators and the elements of the observed information matrix are presented. The importance of the proposed distribution is further emphasized in Section 6 by fitting it to two real data sets. It is observed that the BGW provides a better fit than BGE, BW, GW, BE, GE and Weibull distributions. Concluding remarks are presented in Section 7. Some proofs and mathematical expressions are included in the appendix.

2. Beta Generalized Weibull distribution In this section, we introduce the five-parameter Beta Generalized Weibull (BGW) distribution by assuming G(x) to be the cdf of the Generalized Weibull (GW) distribution. The Generalized Weibull distribution with two shape parameters a, b 40 and a scale parameter l 40 was introduced by Mudholkar et al. [16]. The cdf of the GW is given by b GGW ðxÞ ¼ ð1eðlxÞ Þa , x 4 0, a 4 0, b 4 0, l 4 0:

The pdf corresponding to (3) is b b b g GW ðxÞ ¼ abl xb1 ð1eðlxÞ Þa1 eðlxÞ , x 4 0, a 40, b 4 0, l 40:

ð4Þ Using (3) in (1), the cdf of the BGW distribution can be written as FðxÞ ¼

1 Bða,bÞ

Z

b

ð1eðlxÞ Þa

wa1 ð1wÞb1 dw, x 40,a,b, a, b and l 4 0:

0

ð2Þ

where g(x)¼(dG(x)/dx) is the probability density function (pdf) of the parent/baseline distribution. In this paper, we introduce a new five-parameter distribution called Beta Generalized Weibull (BGW) distribution by taking G(x) to be the cumulative distribution function of Generalized Weibull (GW). The GW family was applied for analyzing bathtub failure data for bus motor failure data, head and neck cancer clinical trial data (Mudholkar and Srivastava [3]) and flood data (Mudholkar and Hutson [34]). The BGW distribution unifies many existing distributions as its sub models with applications in reliability engineering. These distributions are Beta Generalized Exponential (BGE), Beta Weibull (BW), Generalized or Exponentiated Weibull (GW or EW), Generalized Rayleigh (GR) (Kundu and Rakab [35]), Beta Exponential (BE), Generalized Exponential (GE) (Gupta and Kundu [36]), Weibull, Rayleigh and Exponential distributions. The addition of new parameters helps in controlling the skewness and the tail weight. This new distribution exhibits increasing, decreasing, bathtub and upside down bathtub shaped failure rate functions for different parametric combinations. The motive behind introducing this new distribution is its flexibility in modeling a wide spectrum of real data sets in reliability and biological sciences. Although other variants of Weibull or Beta G distributions also possess various shapes of the failure rate function, it has been observed through two data sets and simulations that the

ð3Þ

ð5Þ The pdf and the failure rate function of the new distribution take the form f ðxÞ ¼

ablb xb1 Bða,bÞ

b

1eðlxÞ

aa1 n

o b a b1 b 1 1eðlxÞ eðlxÞ ,

x 40 ð6Þ

and hðxÞ ¼

n o b aa1 b a b1 b 1eðlxÞ 1 1eðlxÞ eðlxÞ : a ða,bÞ b

ablb xb1 Bða,bÞI

1 1eðlxÞ

ð7Þ If X is a random variable with pdf (6), we use the notation X BGW (a, b, l, a, b). Special cases: 1. For b ¼1, we get the Beta Generalized Exponential (BGE (a, b, l, a)) distribution. 2. If a¼b ¼1, then (6) reduces to the Generalized Weibull (GW) distribution. If in addition b ¼2, we obtain Generalized Rayleigh (GR) distribution. 3. Beta Weibull (BW) distribution arises as a special case of BGW by taking a ¼1. 4. Applying a ¼b¼ b ¼1, we can obtain Generalized Exponential (GE) distribution.


5. With a ¼ b ¼1 in (6), Beta Exponential distribution can be obtained. 6. For a¼ b¼ a ¼1, we obtain the Weibull distribution with parameters l and b. If in addition b ¼2, the BGW distribution becomes Rayleigh distribution.

If V follows Beta distribution with parameters a and b, then 1=b

X ¼ G1 ðVÞ ¼

flogð1V 1=a Þg

l

follows BGW distribution with parameters a, b, l, a and b. This follows easily from the result of Jones [37] which states that if Y follows Beta distribution with parameters a and b, then X¼G 1(Y) follows Beta Generalized distribution given by (1).

The above special cases are also depicted in pictorial form in Fig. 1. The following result helps in simulating observations from the BGW distribution.

=1

=1

BGW

a = b =1 BGE =1

=1

BW

GW =1

a = b =1

GE

a = b =1 E

=1

=1

a = b =1

= 2

=1 BE

7

W

GR =1

=2

BE

W

a = b =1

=1

R E Fig. 1. Sub models of BGW.

Fig. 2. Plot for comparison of exact and empirical cdf of BGW to simulated random numbers.

Fig. 3. Plots for BGW densities for simulated data sets: (a) a ¼8, b¼ .5, a ¼ .5, l ¼1, b ¼ 1 and (b) a ¼.5, b¼ .5, a ¼ 1.5, l ¼ .5, b ¼ 4.5.

8


Fig. 4. Plots of BGW density for some values of the parameters.

Fig. 5. Plots of failure rate of BGW for some specific values of the parameters.

For checking the correctness of the procedure for simulating a data set from BGW distribution, we plot, the exact and the empirical cdf of BGW in Fig. 2 using a pseudo random sample of size 1000. The histograms for two generated data sets and the exact BGW density plots are displayed in Fig. 3. These plots show that the simulated values are consistent with the BGW distribution. For some values of the parameters, the plots of the density and the failure rate function are shown in Figs. 4 and 5 respectively. It is depicted by Fig. 5 that the failure rate function of the BGW distribution can take monotonic, bathtub and upside-down bathtub shapes for different parametric combinations.

The mathematical relation given below will be used in Theorem 1 and Section 4. ð1zÞb1 ¼

1 X j¼0

ð1Þj GðbÞ zj GðbjÞGðj þ 1Þ

! ð8Þ

when b is a positive real non-integer and 9z9o1. The following theorem shows that the cdf (pdf) of the BGW can be expressed as an infinite (finite for integer b) weighted sum of the cdfs (pdfs) of the GW distribution. Alternative expressions for the pdf and cdf are given. The density function of the ith order statistic is also presented.

3. Some results for the cumulative distribution and density function For any device/equipment, the interest might be in determining the survival probabilities which can be obtained directly or through the cdf. The probability of failure of a device within an interval of infinitesimal length dx is given by f(x)dx. Hence, we present various representations of the cdf, the survival function and the pdf which can be helpful in numerical applications.

Theorem 1. Let Gl, b, aða þ jÞ ðxÞ and g l, b, aða þ jÞ ðxÞ be the cdf and pdf of the GW distribution with scale parameter l and shape parameters b and aða þjÞ and

oj ¼

ð1Þj Gða þ bÞ be the simple weights: GðaÞGðbjÞGðj þ 1Þða þ jÞ

Then


2

(A) For real and non-integer b40, (i) FðxÞ ¼

1 X

¼ ðoj Gl, b, aða þ jÞ ðxÞÞ,

j¼0

(ii) f ðxÞ ¼

1 X

Bða,bÞ

ðoj g l, b, aða þ jÞ ðxÞÞ,

ai ¼

dðd1Þ. . .ðdi þ 1Þ i!

ablb xb1 Bða,bÞ

b

eðlxÞ Gl, b,aa1 ðxÞ

1 X j¼0

! ð1Þj GðbÞ Gl, b, aj ðxÞ : GðbjÞj!ða þ jÞ

1 ðGðxÞÞ Bða,bÞ a

1 aGðbÞ X

EðX k Þ ¼

Bða,bÞl

j¼0

aGðbÞ k

Bða,bÞl

G

ð9Þ

1

k

ðyÞb 1 þ 0

#

!

ai eyi ey dy

i¼1

Z

j

1 X

1

k=b y

e

ðyÞ

dy þ

0

1 X

ai

Z

1

!# ðyÞk=b eyði þ 1Þ dy

0

i¼1

" !# X 1 1 X ð1Þj þ1 ai ði þ 1Þððk=bÞ þ 1Þ : 1þ b GðbjÞj! j¼0 i¼1 k

ð10Þ

Eq. (10) gives the general expression for the kth moment of BGW distribution. If a is a positive integer, then the last expression in the parentheses is 1þ

a1 X dðd1Þ. . .ðdiþ 1Þ

¼

ð1Þi ði þ1Þððk=bÞ þ 1Þ

i!

a1 d X ð1Þi ði þ1Þððk=bÞ þ 1Þ : i i¼0

b

1 abl xb1 eðlxÞ G ðxÞ Bði,niþ 1Þ ðBða,bÞÞn ai1 bni l, b,aa1

Substituting in (10), this gives

ð1Gl, b, a ðxÞÞbð1 þ niÞ1 :Gl, b, aaði1Þ ðxÞ f2 F 1 ða,1b,a þ1;

1 aGðbÞGððk=bÞ þ 1Þ X

EðX k Þ ¼

Bða,bÞl

ni

where ðaÞi ¼ aða þ 1Þ. . .ða þ i1Þ denotes the ascending factorial. The representations given in (A) are important as they help in deriving some mathematical properties of the BGW distribution directly from the corresponding properties of the GW distribution. The theorem also shows that the cdf F(x), the survival function FðxÞ and the density function of ith order statistic can be expressed in terms of the well known hypergeometric function. Remark 1. In Theorem 1, the sum stops at b–1 when b is an integer. 4. Moments of the BGW distribution In this section, we derive the expression for kth order moment of BGW. The moments of different orders will help in determining the expected life time of a device and also the dispersion, skewness and kurtosis in a given set of observations arising in reliability applications. The definition of the kth moment of the BGW distribution helps us in writing Z 1 b1 b b ablb xb1 EðX k Þ ¼ ð1eðlxÞ Þaa1 f1ð1eðlxÞ Þa g xk Bða,bÞ 0

Using Eq. (8) and letting y¼(lx) , 2 0 13 !k Z 1 1 aGðbÞ 4 X ð1Þj ðyÞ1=b k y aa þ aj1 y @ ð1e Þ e dyA5 EðX Þ ¼ Bða,bÞ j ¼ 0 GðbjÞj! 0 l

d i

ð1Þi ði þ 1Þððk=bÞ þ 1Þ

# :

For a¼b ¼1, EðX k Þ ¼

a1 aGððk=bÞ þ 1Þ X

a1

k

l

i

i¼0

ð1Þi ði þ1Þððk=bÞ þ 1Þ ,

which gives the kth order moment of the Generalized Weibull distribution given by Mudholkar et al. [3,16]. k For a¼b ¼ a ¼1, EðX k Þ ¼ ðGððk=bÞ þ1Þ=l Þ is the kth order moment of the Weibull distribution. If b is a positive non-integer and a ¼1, then ! 1 GðbÞGððk=bÞ þ1Þ X ð1Þj EðX k Þ ¼ k GðbjÞj! Bða,bÞl j¼0 gives the kth moment of the Beta Weibull distribution. Cordeiro et al. [28] found the moments of the Beta Weibull when a is a positive non-integer. The general expression for the kth moment of BGW can also be found using the Laplace transformation technique as discussed below. The Laplace transform of the BGW is Z 0

b

j¼0

a1 ð1Þj X GðbjÞj! i ¼ 0

Special cases:

LðsÞ ¼

k Z1:

"

k

Gl, b, a ðxÞÞgi1 f2 F 1 ðb,1a,b þ 1; 1Gl, b, a ðxÞÞg

dx,

dyA5

!# X 1 ð1Þj k Gððk=bÞþ 1Þ G þ1 þ ai GðbjÞj! b ði þ1Þðk=bÞ þ 1 i¼1

k

2 F 1 ða,1b,a þ 1; GðxÞÞ,

(C) For a random sample of size n, the density function of ith order statistic X(i) is given by

e

Þ e

ð1e

"

1 aGðbÞ X

Bða,bÞl

Z

ð1Þ GðbjÞj!

j¼0

i¼1

ðlxÞb

"

k

¼

1 ð1GðxÞÞb 2 F 1 ðb,1a,b þ1,1GðxÞÞ: Bða,bÞ b

b

ð1Þj GðbjÞj!

j¼0 1 X

Bða,bÞl

¼

hypergeometric

(ii) The survival function

f i:n ðxÞ ¼

y d y

l

0

13

!k

ð1Þi ,

"

k

aGðbÞ

¼

a

FðxÞ ¼

GðbjÞj!

j¼0

ðyÞ1=b

1

where d ¼ a(aþj) 1. Using binomial expansion and writing

(B) For general a and b and P1 i the 2 F 1 ða, b, g; xÞ ¼ i ¼ 0 ðððaÞi ðbÞi =ðyÞi i!Þx Þ, function (Gradshteyn and Ryzhik [38]) (i) FðxÞ ¼

Z

1 j aGðbÞ 4 X @ ð1Þ

j¼0

(iii) f ðxÞ ¼

0

9

1

exs

ablb xb1 Bða,bÞ

1eðlxÞ

b

aa1 n o b a b1 b 1 1eðlxÞ eðlxÞ dx:

b

Letting v ¼ eðlxÞ and using Eq. (8), we get LðsÞ ¼

1 aGðbÞ X

Bða,bÞ j ¼ 0

ð1Þj GðbjÞj!

Z 0

1

sððlogvÞ1=b =lÞ

e

! ð1vÞ

aa þ aj1

dv ,

10


Interchanging the summation and integration and using binomial theorem, " 1 1 aGðbÞ X ð1Þj X ðsÞr ð1Þr=b LðsÞ ¼ Bða,bÞ j ¼ 0 GðbjÞj! r ¼ 0 lr r! ! !# Z 1 1 X 1þ ai vi ðlog vÞr=b dv , 0

i¼1

where ai is defined in (9). R1 P r=b i Writing I ¼ 0 ð1 þ 1 dv and substituting i ¼ 1 ai v Þðlog vÞ y e ¼ v, we get Z 1 1 Z 1 X ðyÞr=b ey dy þ ai ðyÞr=b eyði þ 1Þ dy I¼ 0

¼ ð1Þr=b

0

i¼1

(

X 1 r G þ1 þ

ai

Gððr=bÞ þ1Þ

!)

b

b b ðb1Það1eðlxÞ Þa1 eðlxÞ ðlxÞb logðlxÞ b 1ð1eðlxÞ Þa

ðlxÞb logðlxÞ: The expected value of the unit score vector vanishes leading to the following equations: h i cðaÞcða þ bÞ b E logð1eðlxÞ Þ ¼

a

h n oi b E log 1ð1eðlxÞ Þa ¼ cðbÞcða þ bÞ; " # b b ð1eðlxÞ Þa logð1eðlxÞ Þ aðcðaÞcða þbÞÞþ 1 : E ¼ b aðb1Þ 1ð1eðlxÞ Þa For a random sample x ¼(x1,y,xn) of size n, the total loglikelihood is written as

ði þ 1Þðr=bÞ þ 1 i¼1 ) ( 1 X r r=b ððr=bÞ þ 1Þ þ1 1þ ai ði þ1Þ ¼ ð1Þ G

b

n X

ln ¼ ln ða,b, l, a, bÞ ¼

i¼1

ðiÞ

l ,

i¼1

Hence, 1 aGðbÞ X

LðsÞ ¼

where l denotes the log-likelihood for the ith observation (i¼ 1,y, n). P The total score function is U n ¼ ni¼ 1 U ðiÞ where U(i) is the score vector for the ith observation. The MLE h^ of y can be obtained numerically from the non-linear equations Un ¼ 0. For interval estimation and testing of hypotheses for the parameters in y, the 5 5 information matrix is obtained as

(

Bða,bÞ j ¼ 0 1þ

(i)

1 X

1

ð1Þj X ðsÞr r G þ 1 GðbjÞj! r ¼ 0 lr r! b !#

ai ði þ 1Þððr=bÞ þ 1Þ

g:

i¼1

This gives k d LðsÞ aGðbÞ k ¼ G þ 1 EðX Þ ¼ ð1Þ k b dsk Bða,bÞl s¼0 " !# 1 1 j X X ð1Þ 1þ ai ði þ 1Þððk=bÞ þ 1Þ , GðbjÞj! j¼0 i¼1 k

2

K a,a 6K 6 a,b 6 K K ¼ KðyÞ ¼ 6 6 a, l 6 K a, a 4

k

K a, b

which is same as (10). Hence, the moments can be determined using any one of the two approaches.

5. Estimation and inference In this section, we derive the non-linear equations for finding the maximum likelihood estimates (MLEs) of the parameters. It is assumed that X follows BGW (a, b, l, a, b) and h ¼(a, b, l, a, b)T denotes the parameter vector. The log likelihood for a single observation x of X can be written as l ¼ lða,b, l, a, bÞ ¼ log a þ log b þ b log l þ ðb1Þlog xlog Bða,bÞ þ ðaa1Þlogð1 ðlxÞb

e

ðlxÞb a

Þ þ ðb1Þlog½1ð1e

b

Þ ðlxÞ ,

b @l ¼ cðaÞ þ cða þ bÞ þ alogð1eðlxÞ Þ, @a n o b @l ¼ cðbÞ þ cða þ bÞ þ log 1ð1eðlxÞ Þa , @b b b b @l b ðaa1ÞeðlxÞ bðlxÞb1 x ðb1Þabð1eðlxÞ Þa1 eðlxÞ ðlxÞb1 x ¼ þ b b a ð l xÞ ð l xÞ l @l Þ Þ ð1e 1ð1e

b

b

@l 1 ðaa1ÞeðlxÞ ðlxÞb logðlxÞ ¼ þ log l þlog x þ b @b b ð1eðlxÞ Þ

K b, l K b, a K b, b

Table 1 Mean estimates and Root mean squared errors based on Monte Carlo simulation study. n

Parameter

Mean

RMSE

50

a^ b^

7.9999 .5020

.000250 .00672

l^

1.0018

.00859

a^ b^

.4996 1.5480

.00384 .08640

a^ b^

7.9998 .5013

.000010 .003490

l^

1.0012

.003681

a^ b^

.4997 1.5342

.001691 .05969

a^ b^

7.9999 .5005

.000045 .001338

l^

1.0005

.001418

a^ b^

.4998 1.5229

.000702 .03927

a^ b^

7.9999 .5003

.000032 .001134

l^

1.0003

.001180

a^

.4999 1.5186

.00050 .03394

100

200

bðlxÞb1 x, b

b @l 1 ðb1Þð1eðlxÞ Þa logð1eðlxÞ Þ ¼ þ a logð1eðlxÞ Þ , b @a a 1ð1eðlxÞ Þa

K b,b

3 K a, l K a, a K a, b K b, l K b, a K b, b 7 7 7 K l, l K l, a K l, b 7: 7 K l, a K a, a K a, b 7 5 K l, b K a, b K b, b

The expressions for the elements of K are given in the appendix. The total information matrix is Kn ¼Kn(y)¼nK(y). Under certain regularity conditions which include that the true parameter values must lie in the interior of the parametric space,

x 4 0:

Let c(.) denote the digamma function, the logarithmic derivative of the gamma function. Then the components of the unit score vector U¼((@l/@a),(@l/@b),(@l/@l),(@l/@a),(@l/@b))T are given by

K a,b

300

b^


11

Fig. 6. TTT plot to time-to-failure of turbocharger of one type of engine data set.

Table 2 Estimates of the parameters for some models fitted to given data set and the values of the AIC, BIC and AICc. Model

MLEs a

BGW BGE BW GW BE GE

.011 .171 .907 1 7.701 1

b

l

a

b

70.46 91.54 .380 1 17.47 1

.115 .224 .177 .481 .059 .449

22.021 32.27 1 11.93 1 9.514

10.28 1 4.282 1.010 1 1

AIC

BIC

AICc

165.561 170.307 172.258 186.859 180.929 184.285

174.005 177.062 179.014 191.926 185.995 187.663

167.325 171.449 173.401 187.526 181.595 184.609

Fig. 7. Estimated densities of the BGW, BGE, BW, GW, BE and GE distributions for a real data set.

pffiffiffi ^ nðhhÞis N5 ð0,KðhÞ1 Þ:

with significance level g for each parameter yi is given by qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi y^ i zg=2 kyi , yi , y^ i þzg=2 kyi , yi ,

This provides a basis for the construction of tests of hypotheses and confidence regions. An asymptotic confidence interval

where k i i is the ith diagonal element of Kn(h) 1 for i¼1, 2, 3, 4, 5 and zg/2 is the upper g/2 point of standard normal distribution.

the asymptotic distribution of

y ,y

12


To select the best model among a range of models, the criteria used are Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) and the second order Akaike information criterion (AICc). For k as the number of free parameters in the model and sample size n, these are defined as AIC ¼ 2k2 lnðlikelihoodÞ; BIC ¼ k ln n2 lnðlikelihoodÞ; AICc ¼ AIC þ

2kðk þ 1Þ : nk1

statistic. This statistic can be written as likelihood under null hypothesis D ¼ 2 ln : likelihood under alternative hypothesis Under H0, D follows approximately Chi-square distribution with d degrees of freedom, where d is the dimension of vector y1 of interest. We apply the likelihood ratio test to a real data set in the next section.

6. Simulations and applications

The likelihood can be increased by adding parameters, but doing so may result in overfitting. The BIC resolves this problem by introducing a penalty term for the number of parameters in the model. AICc, the second order information criterion, takes into account the sample size by increasing the relative penalty for model complexity with small data sets. The best model is the one with least values of AIC, BIC and AICc. In case of small sample size or large no. of parameters, AICc is preferred over AIC. For partitioned h ¼ ðhT1 , hT2 ÞT , testing for ð0Þ H0. h1 ¼ hð0Þ 1 versus H1 h1 a h1

can be carried out using one of the three well known asymptotically equivalent test statistics-namely, the likelihood ratio (LR), Rao score (SR) and Wald (W) statistics. For testing the validity of some sub models of the BGW distribution for fitting to some available data, the maximum values of the restricted and unrestricted likelihoods are used for construction of the LR

For validating the theoretical results reported in Section 5, we carry out simulations by generating n observations from BGW distribution with parametric values a¼8, b¼ .5, l ¼1, a ¼.5 and b ¼1.5. The estimated values of the parameters are found using the BFGS quasi-newton method in R package. The considered sample sizes are n ¼50, 100, 200 and 300 and 10000 is the number of repetitions. The estimates for different sample sizes are reported in Table 1. The values in Table 1 substantiate the theoretical results reported in Section 5. It is observed that for n¼300, the estimated parameters are quite close to the assumed parametric values. We demonstrate the superiority of the new distribution over its sub models using two real data sets arising in reliability engineering. Data set 1: This data set gives the time-to-failure (103 h) of turbocharger of one type of engine given in Xu et al. [39]. The data

Table 4 Estimates of the parameters for some models fitted to Aarset data set and the values of the AIC, BIC and AICc.

Table 3 Values of LR statistic and p-values for comparison of models.

Model Models

Value of LR Statistics

p-value

BGE versus BGW BW versus BGW GW versus BGW BE versus BGW GE versus BGW

6.745 8.697 25.298 19.368 24.724

.0093 .0031 .0000 .0000 .0000

BGW BGE BW GW

MLEs a

b

l

a

b

.587 .221 .124 1

.315 .043 .313 1

.016 .418 .015 .013

.136 3.41 1 .088

5.64 1 5.63 5.63

Fig. 8. TTT plot to time-to-failure of components of Aarset data.

AIC

BIC

AICc

450.673 481.542 455.886 481.268

460.233 489.190 463.535 487.004

452.037 482.430 456.776 481.789


set is: 1:6, 2:0, 2:6, 3:0, 3:5, 3:9, 4:5, 4:6, 4:8, 5:0, 5:1, 5:3, 5:4, 5:6, 5:8, 6:0, 6:0, 6:1, 6:3, 6:5, 6:5, 6:7, 7:0, 7:1, 7:3, 7:3, 7:3, 7:7, 7:7, 7:8, 7:9, 8:0, 8:1, 8:3, 8:4, 8:4, 8:5, 8:7, 8:8, 9:0: As displayed in Fig. 6, the TTT plot for the above data has concave shape implying that this data set possesses an increasing failure rate. The superiority of BGW is established using AIC, BIC and the AICc, LR statistic and plots of the histogram and the estimated densities. Table 2 displays the values of the MLEs of the parameters and the values of the AIC, BIC and AICc for six fitted models. It is seen that the values of AIC, BIC and AICc for the BGW model are the lowest. Hence our model can be labeled as the best model. The histogram of the data set and the plots of the estimated densities of all six distributions are shown in Fig. 7. This figure also depicts that the BGW distribution produces a better fit than all its sub models. The values of the LR statistic for testing the goodness of fit of BGW in comparison with its sub models BGE, BW, GW, BE and GE are given in Table 3. For example, to test BGE versus BGW, the hypotheses are H0: h1 : b ¼ 1 versus H1: y1:b a1 and for testing GW versus BGW, the hypotheses are H0: y1:(a,b)¼1 versus H1: H0 is not true. The values in Table 3 indicate that the null hypothesis will be rejected in all the cases. This helps in concluding that the proposed distribution fits the data set more closely as compared to some of its sub models. Data set 2: We consider another data set from Aarset [40], on lifetimes of 50 components. The data set is: 0:10,0:20,1,1,1,1,1,2,3,6,7,11,12,18,18,18,18,18,21,32,36,40, 45,46,47,50,55,60,63,63,67,67,67,67,72,75,79,82,82,83, 84,84,84,85,85,85,85,85,86,86: The TTT plot in Fig. 8 has first a convex shape and then a concave shape. It depicts that the data set possesses the bathtub shaped failure rate. Table 4 displays the values of the MLEs of the parameters and the values of the AIC, BIC and AICc for BGW, BGE, BW and GW models. On the basis of tabulated values, we conclude that BGW provides the best fit as compared to its sub models.

13

Acknowledgment The authors are grateful to the anonymous referees for their valuable suggestions that helped in improving the paper to a great extent. The first author is also thankful to University Grants Commission, Government of India, for providing financial support for this work. Appendix A Proof of Theorem 1. (A) (i) Using (8) in (5), the cdf of the BGW distribution is Z ð1eðlxÞb Þa 1 FðxÞ ¼ wa1 ð1wÞb1 dw Bða,bÞ 0 ! Z ð1eðlxÞb Þa 1 GðbÞ X ð1Þj a þ j1 w dw ¼ Bða,bÞ j ¼ 0 GðbjÞGðj þ 1Þ 0 ! b 1 Gða þ bÞ X ð1Þj ð1eðlxÞ Þaða þ jÞ : ¼ GðaÞ j ¼ 0 GðbjÞGðj þ 1Þ ða þ jÞ P Hence FðxÞ ¼ 1 j ¼ 0 ðoj Gl, b, aða þ jÞ Þ: (ii) The result follows from A (i). (iii)

ablb xb1

b

b

b

b1

eðlxÞ ð1eðlxÞ Þaa1 f1ð1eðlxÞ Þa g Bða,bÞ ! 1 X ablb xb1 ðlxÞb ð1Þj GðbÞ e Gl, b, aj ðxÞ : ¼ Gl, b,aa1 ðxÞ Bða,bÞ GðbjÞj!ðaþ jÞ j¼0

f ðxÞ ¼

(B) (i) For general a and b, using in (1), the relation Z GðxÞ ðGðxÞÞa wa1 ð1wÞb1 dw ¼ F ða,1b,a þ 1; GðxÞÞ, a 2 1 0 we get 1 ðGðxÞÞa F ða,1b,a þ1; GðxÞÞ: Bða,bÞ a 2 1 P i where 2 F 1 ða, b, g; xÞ ¼ 1 i ¼ 0 ðððaÞi ðbÞi =ðyÞi i!Þx Þ is the hypergeometric function and (a)i ¼ a(a þ1)y(a þi 1) denotes the ascending factorial. (ii) The survival function FðxÞ ¼

FðxÞ ¼

1 ð1GðxÞÞb 2 F 1 ðb,1a,bþ 1,1GðxÞÞ: Bða,bÞ b

7. Conclusions

(iii)

We propose a new five-parameter distribution, named as the Beta Generalized Weibull (BGW) distribution, which is a generalization of the BGE, BW, GW, GR, BE, GE, Weibull, Rayleigh and Exponential distributions. As the proposed distribution can have monotone, bathtub shaped and unimodal failure rate for different parametric combinations, it is expected that this generalization will be widely applicable in reliability engineering and biological sciences. It is shown that the BGW density can be expressed as a mixture of GW densities. Some expansions for the cdf and pdf are provided and the kth order moment of the BGW distribution has been derived. It is also observed that the method of maximum likelihood estimation can be applied by solving non-linear equations through numerical methods. For testing goodness of fit of the BGW distribution versus some of its special cases, the likelihood ratio test is applied. An application of the BGW distribution to two real data sets is provided to illustrate that this distribution provides a better fit than its sub-models.

FðxÞ ¼ 1FðxÞ ¼ I1GðxÞ ðb,aÞ ¼ Bða,bÞ1

Z

1GðxÞ

ob1 ð1oÞa1 dw

0

¼

1 ð1GðxÞÞb 2 F 1 ðb,1a,b þ 1,1GðxÞÞ: Bða,bÞ b

(C) It is well known that the density function of ith order statistic Xi:n can be written as f i:n ðxÞ ¼

1 f ðxÞFðxÞi1 ð1FðxÞÞni , Bði,ni þ1Þ

for i ¼ 1,. . .,n:

Using (6) and the results of (B) (i) and (ii), b

f i:n ðxÞ ¼

b

1 abl xb1 eðlxÞ G ðxÞð1Gl, b, a ðxÞÞb1 Bði,ni þ 1Þ ðBða,bÞÞn ai1 bni l, b,aa1

:fðGl, b, a ðxÞÞa 2 F 1 ða,1b,a þ 1; Gl, b, a ðxÞÞg

i1

14


"

ni

¼

:fð1Gl, b, a ðxÞÞb 2 F 1 ðb,1a,b þ 1; 1Gl, b, a ðxÞÞg b 1 ablb xb1 eðlxÞ Bði,ni þ 1Þ ðBða,bÞÞn ai1 bni

Gl, b,aa1 ðxÞð1Gl, b, a ðxÞÞbð1 þ niÞ1

:Gl, b, aaði1Þ ðxÞf2 F 1 ða,1b,a þ1;

ni

Gl, b, a ðxÞÞgi1 f2 F 1 ðb,1a,b þ1; 1Gl, b, a ðxÞÞg

:

K l, a ¼ E

# @2 l b ¼ aT 0,1,1,1,0,0 ðb1Þ T 1,1,1,1,0,0 þ T 2,1,1,1,1,0 þ T 1,1,1,1,1,0 , l @l @a

" K l, b ¼ E

# @2 l @l@b

2 3 ðb1ÞafT 1,1,1,1,0,0 þ bðT 1,1,1,1,0,1 þT 1,1,1,2,0,1 ða1ÞT 1,2,2,2,0,1 aT 2,2,2,2,0,1 Þg 16 7 a 1Þf b ðT þT T Þ þT g þ 1ða ¼ 4 0,1,1,1,0,1 0,1,1,2,0,1 0,2,2,2,0,1 0,1,1,1,0,0 5: l þ T 0,0,0,1,0,0 þ bT 0,0,0,1,0,1

A.1. Expressions for elements of information matrix K For i, j, k, l, m, nA{0, 1, 2} and V following Beta distribution with parameters a and b, we define

References

l

T i,j,k,l,m,n ¼ E½ð1VÞi V iðk=aÞ ð1V 1=a Þj flogð1V 1=a Þg 1=b n

ðlog VÞm ðlogflogð1V 1=a Þg

Þ :

0

If c ð:Þ denotes the derivative of the digamma function, then the elements of information matrix K are given by " # @2 l 0 0 K a,a ¼ E ¼ c ðaÞc ða þbÞ, @a2 " # @2 l 0 ¼ c ða þ bÞ, K a,b ¼ E @a@b " # " # b @2 l abeðlxÞ ðlxÞb1 x ab ¼ K a, l ¼ E T , ¼ E b ð l xÞ @a@l l 0,1,1,1,0,0 Þ ð1e " # h i cða þ bÞcðaÞ b @2 l , K a, a ¼ E ¼ E logð1eðlxÞ Þ ¼ @a@a a " # " # b @2 l aeðlxÞ ðlxÞb logðlxÞ ¼ aT 0,1,1,1,0,1 , K a, b ¼ E ¼ E b @a@b ð1eðlxÞ Þ " # @2 l 0 0 K b,b ¼ E ¼ c ðbÞc ða þbÞ, 2 @b "

# " # b b @2 l abeðlxÞ ðlxÞb1 xð1eðlxÞ Þa1 ab ¼ E ¼ K b, l ¼ E T , b l 1,1,1,1,0,0 @b@l 1ð1eðlxÞ Þa " K b, a ¼ E

# " # b b @2 l ð1eðlxÞ Þa logð1eðlxÞ Þ aðcðaÞcða þ bÞÞþ 1 , ¼ E ¼ b a @b@a aðb1Þ 1ð1eðlxÞ Þ

" K b, b ¼ E

# " # b b @2 l aeðlxÞ ð1eðlxÞ Þa1 ðlxÞb logðlxÞ ¼ E ¼ aT 1,1,1,1,0,1 , b @b@b 1ð1eðlxÞ Þa

2 3 # 2 b b @2 l 1 ð1eðlxÞ Þa flogð1eðlxÞ Þg 5 4 K a, a ¼ E ¼ E 2 þ ðb1ÞE b @a2 a ð1ð1eðlxÞ Þa Þ 2 3 2 2 b b fð1eðlxÞ Þa g flogð1eðlxÞ Þg 5 4 þ ðb1ÞE 2 b f1ð1eðlxÞ Þa g "

¼

1 1 þðb1Þ T 1,0,0,0,2,0 þT 2,0,0,0,2,0 ,

a2

"

# @2 l ¼ aT 0,1,1,1,0,1 ðb1Þ T 1,1,1,1,0,1 þ T 2,1,1,1,1,1 þ T 1,1,1,1,1,1 , @a@b " # @2 l 1 ¼ 2 T 0,0,0,1,0,2 þðaa1Þ T 0,1,1,1,0,2 þ T 0,2,2,2,0,2 þ T 0,1,1,2,0,2 K b, b ¼ E 2 b @b

K a, b ¼ E

aðb1Þ T 1,1,1,1,0,2 þ T 1,1,1,2,0,2 ða1ÞT 1,2,2,2,0,2 aT 2,2,2,2,0,2 , " # ( @2 l b K l, l ¼ E ¼ 1 þðaa1Þ ðb1ÞT 0,1,1,1,0,0 þ bT 0,1,1,2,0,0 2 2 @l l h b bT 0,2,2,2,0,0 aðb1Þ ðb1ÞT 1,1,1,1,0,0 bðlÞ T 1,1,1,1,0,0 ) i b þ bða1ÞðlÞ T 1,2,2,1,0,0 abT 2,2,2,2,0,0 ,

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