The new Burr distribution and its application

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

This paper derives a new family of Burr-type distributions as new Burr distribution. This particular skewed distribution that can be used quite effectively in analyzing lifetime data. It is observed that the new distribution has modified unimodal hazard function. Various properties of the new Burr distribution, such that moments, quantile functions, hazard function, and Shannon's entropy are obtained. The exact form of the probability density function and moments of ......-order statistics in a sample of size n from new Burr distribution are derived. Estimation of parameters and change-point of hazard function by the maximum likelihood method are discussed. Change-point of hazard function is usually of great interest in medical or industrial applications. The flexibility of the new model is illustrated with an application to a real data set. In addition, a goodness-of-fit test statistic based on the Rényi Kullback-Leibler information is used.

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ORIGINAL RESEARCH

The new Burr distribution and its application

Gholamhossein Yari1 Zahra Tondpour1

Received: 11 August 2016 / Accepted: 14 November 2016 / Published online: 17 January 2017 The Author(s) 2017. This article is published with open access at Springerlink.com

Abstract This paper derives a new family of Burr-type distributions as new Burr distribution. This particular skewed distribution that can be used quite effectively in analyzing lifetime data. It is observed that the new distribution has modied unimodal hazard function. Various properties of the new Burr distribution, such that moments, quantile functions, hazard function, and Shannons entropy are obtained. The exact form of the probability density function and moments of ith-order statistics in a sample of size n from new Burr distribution are derived. Estimation of parameters and change-point of hazard function by the maximum likelihood method are discussed. Change-point of hazard function is usually of great interest in medical or industrial applications. The exibility of the new model is illustrated with an application to a real data set. In addition, a goodness-of-t test statistic based on the Rnyi Kull-backLeibler information is used.

Keywords Burr distributions Change-point Goodness-

of-t Modied unimodal hazard function Lifetime data

analysis Rnyi Entropy

Mathematics Subject Classication 62E15, 62N05, 62F10, 60K10

Introduction

Burr [2] developed the system of Burr distributions. The Burr system of distributions includes 12 types of cumulative distribution functions which yield a variety of density shapes. The attractiveness of this relatively unknown family of distributions for model tting is that it combines a simple mathematical expression for cumulative frequency function with coverage in the skewnesskurtosis plane. Many standard theoretical distributions, including the Weibull, exponential, logistic, generalized logistic, Gompertz, normal, extreme value, and uniform distributions, are special cases or limiting cases of the Burr system of distributions (see [11]). Family of Burr-type distributions is a very popular distribution family for modelling lifetime data and for modelling phenomenon with monotone and unimodal failure rates (see, for example, [13, 18]).

Analogous to the Pearson system of distributions, the Burr distributions are solutions to a differential equation, which has the form:dydx y1 ygx; y; 1:1 where y equal to F(x) and g(x, y) must be positive for y in the unit interval and x in the support of F(x). Different functional forms of g(x, y) result in different solutions F(x), which dene the families of the Burr system. For example, Burr II distribution is obtained when gx; y gx

ke

x

1e xk 1

1e

x

k 1

.

& Gholamhossein Yari [email protected]

1 School of Mathematics, Iran University of Science and

Technology, Tehran, Iran

In this paper, we derive a new distribution of Burr-type distributions which is more exible by replacing g(x, y)

with gx 3px

2e x3 1e x3 p 1 1e x3 p 1

, (p [ 0). We refer to this new distribution as the new Burr distribution. If g(x, y) is taken

123

48 Math Sci (2017) 11:4754

to be g(x), then the solution of the differential Eq. 1.1 is given by

Fx e Gx 1 1; 1:2

where Gx R gxdx.

Hence, cdf and pdf of new Burr distribution are, respectively, given by

Fx; p 1 e x3 p; 1\x\1; p [ 0; 1:3 and

f x; p 3px2e x31 e x3 p 1; 1\x\1: 1:4

If the location parameter l and the scale parameter r are introduced in the equation 1.3, we have

Fx; l; r; p 1 e

x l

r

3 p; 1\x\1; p; r [ 0; l 2 R

1:5

and

f x; l; r; p

3p r

x l

r

2e

x l

r

3 1 e

x l

r

3

p 1:

1:6

Hence, Eq. 1.5 is three parameter new Burr distribution. Hazard function associated with the new Burr distribution is

hx; l; r; p

3p r

3 1 pe

x l

r

3

2 1e

x l

r

3

:

2:1

The derivative f 0x;l;r;p exists every where, hence crit

ical point(s) satisfy equation f 0x;l;r;p 0. In 2.1, set

l 0 and r 1, because location and scale parameters will

not affect the distribution shape. Thus, equation f 0x;l;r;p 0 simplies to3x31 pe x3 21 e x3 0: 2:2

Analytical solution of 2.2 is not possible. Numerical approximation of modes using the midpoint method is applied to study the modes. The distance between the two

2e

x l

r

x l

r

3 1 e

x l

r

3

p 1

1 1 e

p

:

x l

r

3

1:7

The shapes of density and hazard functions of the new Burr distribution for different values of shape parameter p are illustrated in Fig. 1.

New Burr distribution has unimodal and bimodal pdfs. None of the 12 types of Burr distributions has this feature. Data that exhibit bimodal behavior arises in many different disciplines. In medicine, urine mercury excretion has two

peaks, see, for example, [5]. In material characterization, a study conducted by [4], grain size distribution data reveals a bimodal structure. In meteorology, [19] indicated that, water vapor in tropics, commonly have bimodal distributions. To see more applications of bimodal distributions, see [79, 16].

The reminder of the paper is organized as follows: properties of the new Burr distribution, such that moments, quantile functions, hazard function, Shannons entropy, and distribution of its order statistics are discussed in Sects. 2, 3, and 4. In Sect. 5, estimation of parameters and change-point of hazard function by the maximum likelihood method are discussed, and in Sect. 6, we establish a goodness-of-t test statistic based on the Rnyi Kullback Leibler information for testing new Burr model. Finally, in Sect. 7, we present an illustrative example. Section 8 provides conclusions.

Properties of the new Burr distribution

New Burr distribution has unimodal and bimodal pdfs. The modes of distribution are provided by differentiating the density of new Burr distribution in 1.6 with respect to x:

f 0x;l;r;p 3

x l

r

Fig. 1 Graphs of density and hazard functions of the new Burr distribution for different values of shape parameter p

p=1/2, p=1, p=2, p=10,

6

20

18

5

16

14

4

12

f(x)

3

h(x)

10

8

2

6

4

1

2

0 0 1 2 3 4 5

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x

x

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Math Sci (2017) 11:4754 49

Table 1 Distance between twomodes of new Burr distribution p Distance

0.3 2.4294

0.5 2.2930

1 2.1388

2 2.0191

3 1.9682

4 1.9409

5 1.9243

The rth moment about origin of the new Burr distribution is given by

lr EXr

Z 1

1

xr 3p r

x l

r

2e

x l

r

p 1

dx;

using the change of variable, t

1 1e

x l

r

3 1 e

x l

r

3

3 , 0\t\1, we

obtain

1

0 ptp 1 r ln

!rdt

pX

r

i0

13

l

EXr

Z

1t 1

modes is demonstrated in Table 1. From Table 1, it is observed that when p increases, the distance between two modes decreases, and for 0\p\1, when p decreases, value of pdf in the second mode decreases to zero and pdf will be almost unimodal, and for p 1, values of pdf in

two modes are the same but for p [ 1, and when p increases, value of pdf in the rst mode decreases to zero and pdf will be almost unimodal. Hence, the new Burr distribution can be used to analyse different kinds of lifetime data sets with unimodal and bimodal shapes of pdf.

The new Burr distribution has modied unimodal (uni-modal followed by increasing) hazard function, and when p increases, hazard function will be almost increasing.

The main purpose in this paper is to describe and t the data sets with non-monotonic hazard function, such as the bathtub, unimodal and modied unimodal hazard function. Many modications of important lifetime distributions have achieved the above purpose, but unfortunately, the number of parameters has increased, the forms of survival and hazard functions have been complicated, and estimation problems have risen. More over some of the modications do not have a closed form for their cdfs. However, this new distribution with one parameter and simple form of cdf achieves this purpose.

Now, we discuss the reverse hazard function of the new Burr distribution. The reverse hazard function of any distribution function F(x) can be dened as rx fxFx. Consequently, the reversed hazard function of new Burr distribution with zero location parameter and unit scale parameter is given by

rx; p

3x2e x3

1 e x3

1

0 ptp 1 ln

r i

!

i 3

rilr i

Z

1t 1

dt:

Now, using 1t 1 eu, 0\u\1, we obtain

EXr p X

r

i0

r i

rilr i

Z 1 0

eu 1 p 1ui 3eudu

p X

r

i0

r i

rilr iEqgX;

where Eq: denotes expectation for X q and q is the

standard exponential distribution and gx ex 1 p 1xi

3 e2x. Using the importance sampling method, the importance sampling estimate of lr is given by

^

lrq p X

r

i0

r i

rilr i

1n X

!; Xk q:

n

k1

gXk

2:3

Using n 1000, the importance sampling estimate of

mean and variance of the new Burr distribution as l 0

and r 1 for different values of p is demonstrated in

Table 2. From Table 2, it is observed that when p increases, mean increases and variance decreases. Mean and variance ^

lrq are given by

E ^

lrq lr; var ^

lrq p2 X

r

i0

rilr i

2var

^

EqgX;

r i

where var

^

EqgX EqgX EqgX2.

To form a condence interval for lr, we need to estimate var

^

EqgX. Because Xk are sampled from q, the

natural variance estimate is

p:

The reversed hazard function has recently attracted considerable interest of researchers (see, for example, [1, 3]). In a reliability setting, the reversed hazard function (multiplied by dx) denes the conditional probability of a failure of an object in x dx; x given that the failure had

occurred in [0, x]. The reversed hazard function of new Burr distribution with zero location parameter and unit scale parameter is a linear function of p.

Table 2 Importance sampling estimate of mean and variance of the new Burr distribution

p Mean Variance

1 -0.4984 0.3165

2 -0.2091 0.1475

3 -0.0921 0.0683

4 -0.0428 0.0296

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50 Math Sci (2017) 11:4754

^

^ EqgX

1n X

n

k1

gXk

EqgX2:

Then, an approximate 99% condence interval for lr is

^

lrq 2:58

^

^ p .

lrq

n

The quantile function, Q(u), 0\u\1, for the new Burr distribution can be computed using the formula:

Qu r ln u

1p

13

1

l:

The median of a new Burr distribution occurs at

r ln12

1p

l, and clearly, it is a decreasing

function of p as p 1 but an increasing function of p as

p 1.

Skewness and kurtosis of a parametric distribution are often measured by a3 l3r3 and a4 l4r4, respectively. When

the third or fourth moment does not exist, for example, Cauchy, Lvy, and Pareto distributions, a3 and a4, cannot be computed. For the new Burr distribution, skewness and kurtosis can be approximated by approximations of l3 and l4 or alternative measures for skewness and kurtosis, based on quantile functions. The measure of skewness S dened by [6] and the measure of kurtosis K dened by [12] are based on quantile functions and they are dened as

S

Q 68 2Q 48 Q 28

Q 68 Q 28

1

13

3 X

C01 lnpi

pi

2p

!

pX

1 p 1

i

1

pi2

1 p 1

i

1 p1:

Proof

HshX

Z 1

f x ln f xdx ln

; 2:4

K

3pr 2E ln

X l

r

!

p 1E ln 1 e

3

E

1 X l

r

:

Q 78 Q 58 Q 38

Q 68 Q 28

X l

r

3

: 2:5

To investigate the effect of the shape parameter p on the new Burr density function, Eqs. 2.4 and 2.5 are used to obtain Galtons skewness and Moors kurtosis. Figure 2 displays the Galtons skewness and Moors kurtosis for the

new Burr distribution in terms of the parameter p when l 0 and r 1.

Shannons entropy

The entropy of a random variable X is a measure of variation of uncertainty. Shannons entropy [17] for a random variable X with pdf f(x) is dened as E logf x. In

recent years, Shannons entropy has been used in many applications in elds of engineering, physics, and economics.

Denote by HshX the well-known Shannons entropy.

The following theorem gives the Shannons entropy of the new Burr distribution.

Theorem 3.1 The Shannons entropy of the new Burr distribution is given by

HshX ln

3pr

3:1

We need to nd the expressions ElnX lr, EX lr3 and

Eln1 e

X l

r

3 . First, we calculate the expectation of X lrr.

E X l

r

r

Z 1

1

3p r

r2e

x l

r

x l

r

3 1 e

x l

r

3

p 1dx;

2.5

using the change of variable, t

1 1e

2

x l

r

1.5

3 , 0\t\1, and then, the change of variable, 1

t

1 eu, 0\u\1, we

obtain,

E X l

r

r

1rp X

1

C r3 1

1 p 1

i

1 :

3:2

0.5

p i

r 3

0

0.5

For r 2k

E X l

r

2k

! p X

C 2k3 1

1 p 1

i

3 1 : 3:3

Differentiating both sides of 3.3 with respect to k at k 0

leads to

p i

1

2k

0 5 10 15 20

p

Fig. 2 Galtons skewness and Moors kurtosis for the new Burr distribution

123

Math Sci (2017) 11:4754 51

E ln X l

r

p3 X

C01 lnp i

p i

:

3:4

1 p 1

i

3 r and then differentiating with respect to r at r 0, we obtain

E ln 1 e

X l

r

3

In the same way, by calculating E1 e

Fi:nx IFxi; n i 1;where Ixa; b is lower incomplete gamma function.

There, the 100uth percentile of Xi:n can be obtained by solving

Fi:nx u: 4:3 The percentage points of Xi:n can be evaluated from 4.3 using tables of incomplete beta function (see [15]). However, for i 1, Eq. 4.3 reduces to 1 1 e

x l

r

X l

r

1p : 3:5

By replacing 3.2, 3.4, and 3.5 in relation 3.1, the proof is completed. h

Distribution of order statistics

The pdf of Xi:n i 1; . . .; n is given by fi:nx; l; r; p

3 pn 1 u. Thus, the 100u-percentage

point of the smallest order statistic X1:n is given by

F 11:nu; p; l; r l r ln 1 1 u

1n

13

1 p

1

:

n!

i 1!n i!

Similarly, for i n, the 100u-percentage point of the lar

gest order statistic is

F 1n:nu; p; l; r l r ln u

1 np

f x; l; r; pFi 1

x; l; r; p1 Fx; l; r; pn i;

where f x; l; r; p and Fx; l; r; p are pdf and cdf given in

1.5 and 1.6, respectively:

fi:nx; l; r; p X

n i

j0

13:

1

djn; if x; l; r; pi j; 4:1

where

djn; i

n! 1j i 1!j!n i j!i j

:

Note that djn; ij 0; 1; . . .; n i are coefcients not

dependent on p, l, and r. This means that fi:nx; l; r; p is a

weighted average of other new Burr.From 1.6 and 4.1, we get the rth moment of Xi:n to be

EXri:n X

n i

j0

djn; iEXr

p X

n i

j0

i jdjn; i X

r

k0

Hazard change-point estimation-classical approach

Hazard function plays an important role in reliability and survival analysis. New Burr distribution has modied unimodal (unimodal followed by increasing) hazard function. In some medical situations, for example, breast cancer, the hazard rate of death of breast cancer patients represents a modied unimodal shape.

A modied unimodal shape has three phases: rst increasing, then decreasing, and then again increasing. It can be interpreted as a description of three groups of patients, rst group is represented by the rst phase that contains the weak patients, so the hazard rate of this group is increasing, while the second phase represents the group with strong patients, their bodies have became familiar with the disease and they are getting better. The hazard rate of death of these patients is decreasing. In the third phase, they become weaker and their ability to cope with the disease declines, then the hazard rate of death increases.

For situations, where the hazard function is modied unimodal shaped, usually, we have interest in the estimation of lifetime change-point, that is , the point at which the hazard function reaches to a maximum (minimum) and then decreases (increase). In reliability, the change-point of a hazard function is useful in assessing the hazard in the useful life phase. One of change-points of hazard function of the new Burr distribution is location parameter. In this section, we consider maximum likelihood estimation procedure for change-points of the hazard function.

r k

rklr kEqgY;

where X has new Burr distribution with parameters l, r and pi j and Y has q distribution, standard exponential

distribution, and gy ey 1 pij 1yk

3e2y. Then, the importance sampling estimate of the rth moment about origin of Xi:n is given by

p X

n i

j0

i jdjn; i X

r

k0

r k

rklr k

1m X

gYl

!; Yl q:

4:2

m

l1

The cdf of Xi:n 1 i n is given by

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52 Math Sci (2017) 11:4754

Let us assume that x1; . . .; xn is a random sample of size n of lifetimes generated by a new Burr distribution with parameters l, r, and p. The log-likelihood function is given by

ll;r;p nlog

3p r

information can be consider as a goodness-of-t test statistic. For that purpose, the Rnyi KullbackLeibler information can be estimated by

Drf ; f 0 HrX1:n; . . .; Xn:n X

n

j1

2X

n

i1

log xi l r

X

n

i1

xi l r

3

f 0xj:

p1X

n

i1

:

The maximum likelihood estimates for l, r, and p denoted by ^

l, ^

r, and ^

p, respectively, are obtained solving the likelihood equations, (olol 0,

ol or

Thus, the test statistics based on Drf;f

0

n is given by

log 1e

xi l

r

3

!;

where ^

l, ^

r, and ^

p are MLEs of l, r, and p, respectively,

and ^

HrX1:n; . . .; Xn:n is an estimate of Rnyi entropy for

sample X1:n\X2:n\ \Xn:n. Under the null hypothesis,

Tr for r close to 1 will be close to 0, and therefore, large values of Tr will lead to the rejection of H0.

In this paper, we use estimation of Rnyi entropy based on generalized nearest-neighbor graphs that is introduced by [14]. The basic tool to dene their estimator was the generalized nearest-neighbor graph. This graph on vertex set V is a directed graph on V. The edge set of it contains for each i 2 S (S is a nite non-empty set of positive

integers), an edge from each x 2 V to its ith nearest

neighbor according to the Euclidean distance to x.

For p 0 denote by LpV, the sum of the pth powers of

Euclidean lengths of its edges. According to proven theorem in [14]

lim

n!1

LpX1:n; . . .; Xn:n

n1 pd c [

Tr

Drf ; f 0n

1n

^

HrX1:n; . . .; Xn:n X

n

j1

f 0xj; ^

l; ^

r; ^

p

0, and

ol op

0.

According to the above, maximum likelihood estimator of one of change-points is ^

l.

From the invariance property of maximum likelihood estimators, we can obtain maximum likelihood estimators for functions of l, r and p. For / gl; r; p, a one-to-one

function of l, r, and p, and we have ^

/ g^

l; ^

r; ^

p. Taking

/ hx, dened in 1.7, the change-point of new Burr

hazard function is obtained as solution of equation

ddx log/ 0. The maximum likelihood estimator of the

change-point is the solution of ddx log/ 0 with l, r, and

p replaced by maximum likelihood estimates. We observe that ddx log/ 0 is non-linear in x, so the change-point of

the hazard function estimate should be obtained using some one-dimensional root nding techniques, such as Newton Raphson.

Testing new Burr model based on the Rnyi KullbackLeibler information

Test statistics

Suppose that we are interested in a goodness-of-t test for

H0 : f x f 0x; l; r; p

3p r

0 a:s:;

where p d1 r and d is dimension of sample

members.

Based on described graph, they estimated Rnyi entropy by

^

HrX1:n; . . .; Xn:n

11 r

log LpX1:n; . . .; Xn:n

cn1 pd :

8 <

:

where l, r, and p are unknown.We will denote the complete samples as

X1:n\X2:n\ \Xn:n. For a null pdf f 0x, the Rnyi

KullbackLeibler information from complete data is dened as

Drf ; f 0

1r 1

2e

x l

r

x l

r

3 1 e

x l

r

3

p 1

H1 : f x 6 f 0x; l; r; p;

log Z 1

1

Z

Application

In this section, we consider an uncensored data set corresponding to remission times (in months) of a random sample of 128 bladder cancer patients. These data were previously reported in [10]. TTT plot for considered data is concave then convex indicating an increasing then decreasing hazard function, and is properly accommodated by new Burr distribution. Because in the system of Burr distributions, only Burr X and Burr XII distributions have unimodal hazard functions, and because of the similarity of cdf of the new Burr distribution with the Burr II distribution compared to the rest of distributions in Burr family,

x fX ;...;X x1:n; . . .; xn:na

1 f 0X ;...;X x1:n; . . .; xn:na 1

dx1 dxn;

where r [ 0 and r 6 1. Because Drf ; f 0 has the property that Drf ; f 0 0, and the equality holds if and only if

f f 0, the estimate of the Rnyi KullbackLeibler

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Math Sci (2017) 11:4754 53

1

0.9

0.8

0.7

0.6

F(x)

0.5

0.4

0.3

0.2

0.1

0 0 10 20 30 40 50 60 70 80

x

Fig. 3 cdfs of the new Burr, Burr X, Burr XII, Burr II, and generalized Burr II models for the remission times of bladder cancer data

Table 3 Values of test statistics for the remission times of bladder cancer data

Distribution Test statistic (T0:99)

New Burr 2.5732

Burr XII 4.4819

Burr II 6.6133

Generalized Burr II 11.0319

Burr X 193.9448

compare the ts of the new Burr distribution and those of Burr X, Burr XII, and Burr II and generalized Burr II. Plot of the estimated cdfs of models tted to the data set is given in Fig. 3. Figure 3 and also values of dened test statistics in the previous section that are shown in Table 3 conrm that the new Burr distribution provides a signicantly better t than Burr X, Burr XII, Burr II, and generalized Burr II distributions. The required numerical evaluations are implemented using Matlab (version 2013) and R software (version 3.3.1).

Conclusions

We introduced a new family of Burr-type distributions as new Burr distribution. Various properties of the distribution are investigated. The distribution is found to be uni-modal and bimodal. This new distribution with one parameter and simple form of cdf has modied unimodal (unimodal followed by increasing) hazard function. Hence, this new distribution can be used quite effectively in analyzing lifetime data with non-monotonic hazard function.

The method of maximum likelihood is suggested for estimating the parameters and change-points of hazard function of the new Burr distribution. In application to remission times (in months) of a random sample of 128 bladder cancer patients, the new Burr distribution provided a signicantly better t than Burr X, Burr XII, Burr II, and generalized Burr II distributions. This fact is conrmed by goodness-of-t test statistic based on the Rnyi Kullback Leibler information.

Acknowledgements The authors acknowledge the Department of Mathematics, Iran University of Science and Technology.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/

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