The New Burr Distribution and Its Application

Math Sci (2017) 11:47–54 DOI 10.1007/s40096-016-0203-z

ORIGINAL RESEARCH

The new Burr distribution and its application

1 1 Gholamhossein Yari • Zahra Tondpour

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

Abstract This paper derives a new family of Burr-type Introduction distributions as new Burr distribution. This particular skewed distribution that can be used quite effectively in Burr [2] developed the system of Burr distributions. The analyzing lifetime data. It is observed that the new distri- Burr system of distributions includes 12 types of cumula- bution has modified unimodal hazard function. Various tive distribution functions which yield a variety of density properties of the new Burr distribution, such that moments, shapes. The attractiveness of this relatively unknown quantile functions, hazard function, and Shannon’s entropy family of distributions for model fitting is that it combines are obtained. The exact form of the probability density a simple mathematical expression for cumulative fre- function and moments of ith-order statistics in a sample of quency function with coverage in the skewness–kurtosis size n from new Burr distribution are derived. Estimation plane. Many standard theoretical distributions, including of parameters and change-point of hazard function by the the Weibull, exponential, logistic, generalized logistic, maximum likelihood method are discussed. Change-point Gompertz, normal, extreme value, and uniform distribu- of hazard function is usually of great interest in medical or tions, are special cases or limiting cases of the Burr system industrial applications. The flexibility of the new model is of distributions (see [11]). Family of Burr-type distribu- illustrated with an application to a real data set. In addition, tions is a very popular distribution family for modelling a goodness-of-fit test statistic based on the Reńyi Kull- lifetime data and for modelling phenomenon with mono- back–Leibler information is used. tone and unimodal failure rates (see, for example, [13, 18]). Analogous to the Pearson system of distributions, the Keywords Burr distributions Á Change-point Á Goodness- Burr distributions are solutions to a differential equation, of-fit Á Modified unimodal hazard function Á Lifetime data which has the form: analysis Á Reńyi Entropy dy ¼ yð1 À yÞgðx; yÞ; ð1:1Þ dx Mathematics Subject Classification 62E15, 62N05, 62F10, 60K10 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 define the families of the Burr system. For example, Burr II distribution is obtained when gðx; yÞ¼gðxÞ Àx Àx kÀ1 ¼ ke ð1þe Þ . ð1þeÀxÞkÀ1 In this paper, we derive a new distribution of Burr-type distributions which is more flexible by replacing g(x, y) & Gholamhossein Yari 2 Àx3 Àx3 pÀ1 [email protected] with gðxÞ¼3px e ð1þe Þ ,(p [ 0). We refer to this new ð1þeÀx3 ÞpÀ1 1 School of Mathematics, Iran University of Science and distribution as the new Burr distribution. If g(x, y) is taken Technology, Tehran, Iran 123 48 Math Sci (2017) 11:47–54 to be g(x), then the solution of the differential Eq. 1.1 is peaks, see, for example, [5]. In material characterization, a given by study conducted by [4], grain size distribution data reveals a bimodal structure. In meteorology, [19] indicated that, FðxÞ¼ðeÀGðxÞ þ 1ÞÀ1; ð1:2Þ R water vapor in tropics, commonly have bimodal distribu- where GðxÞ¼ gðxÞdx. tions. To see more applications of bimodal distributions, Hence, cdf and pdf of new Burr distribution are, see [7–9, 16]. respectively, given by The reminder of the paper is organized as follows:

3 properties of the new Burr distribution, such that moments, Fðx; pÞ¼ð1 þ eÀx ÞÀp; À1\x\1; ðp [ 0Þ; ð1:3Þ quantile functions, hazard function, Shannon’s entropy, and and distribution of its order statistics are discussed in Sects. 2, 3, and 4. In Sect. 5, estimation of parameters and 2 Àx3 Àx3 ÀpÀ1 f ðx; pÞ¼3px e ð1 þ e Þ ; À1\x\1: ð1:4Þ change-point of hazard function by the maximum likelihood method are discussed, and in Sect. 6, we establish a If the location parameter l and the scale parameter r are goodness-of-fit test statistic based on the Reńyi Kullback– introduced in the equation 1.3, we have Leibler information for testing new Burr model. Finally, in ÀðxÀlÞ3 Àp Fðx; l; r; pÞ¼ð1 þ e r Þ ; À1\x\1; ðp; r [ 0; l 2 RÞ Sect. 7, we present an illustrative example. Section 8 ð1:5Þ provides conclusions. and 2 3 3 ÀpÀ1 Properties of the new Burr distribution 3p x À l À xÀl À xÀl f ðx; l; r; pÞ¼ e ðÞr 1 þ e ðÞr : r r New Burr distribution has unimodal and bimodal pdfs. The ð1:6Þ modes of distribution are provided by differentiating the Hence, Eq. 1.5 is three parameter new Burr distribution. density of new Burr distribution in 1.6 with respect to x: Hazard function associated with the new Burr distribution 3 3 3 0 xÀl À xÀl À xÀl f ðx;l;r;pÞ¼3 1Àpe ðÞr À21þe ðÞr : is r ÀÁ 3 3 ÀpÀ1 2 xÀl xÀl ð2:1Þ 3p xÀl ÀðÞr ÀðÞr r r e 1 þ e hðx; l; r; pÞ¼ : 0 xÀl 3 Àp The derivative f ðx;l;r;pÞ exists every where, hence crit- À 1 À 1 þ e ðÞr 0 ical point(s) satisfy equation f ðx;l;r;pÞ¼0. In 2.1, set ð1:7Þ l ¼ 0 and r ¼ 1, because location and scale parameters will not affect the distribution shape. Thus, equation The shapes of density and hazard functions of the new Burr 0 f ðx;l;r;pÞ¼0 simplifies to distribution for different values of shape parameter p are 3 3 illustrated in Fig. 1. 3x3ð1 À peÀx ÞÀ2ð1 þ eÀx Þ¼0: ð2:2Þ New Burr distribution has unimodal and bimodal pdfs. None of the 12 types of Burr distributions has this feature. Analytical solution of 2.2 is not possible. Numerical Data that exhibit bimodal behavior arises in many different approximation of modes using the midpoint method is disciplines. In medicine, urine mercury excretion has two applied to study the modes. The distance between the two

Fig. 1 Graphs of density and 6 20 hazard functions of the new 18 p=0.01,μ=4,σ=1/3 p=0.01,μ=4,σ=1/3 Burr distribution for different 5 p=1/2,μ=4,σ=1/3 16 p=1/2,μ=4,σ=1/3 values of shape parameter p p=1,μ=4,σ=1/3 p=1, μ=4,σ=1/3 14 4 p=2,μ=4,σ=1/3 p=2, μ=4,σ=1/3 p=10,μ=4,σ=1/3 12 p=10,μ=4,σ=1/3

3 10 f(x) h(x) 8 2 6

4 1 2

0 0 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x x

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Table 1 Distance between two p Distance The rth moment about origin of the new Burr distribu- modes of new Burr distribution tion is given by Z 0.3 2.4294 1 ÀpÀ1 3p x À l 2 xÀl 3 xÀl 3 r r ÀðÞr ÀðÞr 0.5 2.2930 lr ¼ EðX Þ¼ x e 1 þ e dx; À1 r r 1 2.1388 2 2.0191 1 xÀl using the change of variable, t ¼ Àð Þ3 ,0\t\1, we 3 1.9682 1þe r obtain 4 1.9409 ! Z 1 r 5 1.9243 1 1 3 EðXrÞ¼ ptpÀ1 r À ln À 1 þl dt 0 t !i modes is demonstrated in Table 1. From Table 1,itis Z 3 Xr r 1 1 observed that when p increases, the distance between two ¼ p ðÀrÞilrÀi ptpÀ1 ln À 1 dt: i t modes decreases, and for 0\p\1, when p decreases, i¼0 0 value of pdf in the second mode decreases to zero and pdf 1 u will be almost unimodal, and for p ¼ 1, values of pdf in Now, using t À 1 ¼ e ,0\u\1, we obtain Z two modes are the same but for p [ 1, and when p in- Xr 1 r r i rÀi u ÀpÀ1 i u creases, value of pdf in the first mode decreases to zero and EðX Þ¼p ðÀrÞ l ðe þ 1Þ u3e du i pdf will be almost unimodal. Hence, the new Burr distri- i¼0 0 Xr bution can be used to analyse different kinds of lifetime r ¼ p ðÀrÞilrÀiE ðgðXÞÞ; data sets with unimodal and bimodal shapes of pdf. i q i¼0 The new Burr distribution has modified unimodal (unimodal followed by increasing) hazard function, and when where Eqð:Þ denotes expectation for X q and q is the p increases, hazard function will be almost increasing. standard exponential distribution and x ÀpÀ1 i 2x The main purpose in this paper is to describe and fit the gðxÞ¼ðe þ 1Þ x3e . Using the importance sampling data sets with non-monotonic hazard function, such as the method, the importance sampling estimate of lr is given by ! bathtub, unimodal and modified unimodal hazard function. Xr r 1 Xn Many modifications of important lifetime distributions l^ ¼ p ðÀrÞilrÀi gðX Þ ; X q: rq i n k k have achieved the above purpose, but unfortunately, the i¼0 k¼1 number of parameters has increased, the forms of survival ð2:3Þ and hazard functions have been complicated, and estimation problems have risen. More over some of the modifi- Using n ¼ 1000, the importance sampling estimate of cations do not have a closed form for their cdfs. However, mean and variance of the new Burr distribution as l ¼ 0 this new distribution with one parameter and simple form and r ¼ 1 for different values of p is demonstrated in of cdf achieves this purpose. Table 2. From Table 2, it is observed that when p increases, Now, we discuss the reverse hazard function of the new mean increases and variance decreases. Mean and variance Burr distribution. The reverse hazard function of any dis- l^rq are given by f ðxÞ Xr tribution function F(x) can be defined as rðxÞ¼ . Con- r 2 FðxÞ Eðl^ Þ¼l ; varðl^ Þ¼p2 ðÀrÞilrÀi varðE^ ðgðXÞÞÞ; rq r rq i q sequently, the reversed hazard function of new Burr i¼0 distribution with zero location parameter and unit scale 2 parameter is given by where varðE^qðgðXÞÞÞ ¼ EqððgðXÞÀEqðgðXÞÞÞ Þ. 3 To form a confidence interval for l , we need to esti- 3x2eÀx r rðx; pÞ¼ 3 p: mate varðE^ ðgðXÞÞÞ. Because X are sampled from q, the 1 þ eÀx q k natural variance estimate is The reversed hazard function has recently attracted con- siderable interest of researchers (see, for example, [1, 3]). Table 2 Importance sampling p Mean Variance In a reliability setting, the reversed hazard function (mul- estimate of mean and variance tiplied by dx) defines the conditional probability of a failure of the new Burr distribution 1 -0.4984 0.3165 of an object in ðx À dx; x given that the failure had 2 -0.2091 0.1475 x occurred in [0, ]. The reversed hazard function of new 3 -0.0921 0.0683 Burr distribution with zero location parameter and unit 4 -0.0428 0.0296 scale parameter is a linear function of p.

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1 Xn new Burr distribution in terms of the parameter p when var^ ðE^ ðgðXÞÞÞ ¼ ðgðX ÞÀE^ ðgðXÞÞÞ2: q n k q l ¼ 0 and r ¼ 1. k¼1

Then, an approximate 99% confidence interval for lr is var^ ðl^ Þ pffiffirq Shannon’s entropy l^rq Æ 2:58 n . The quantile function, Q(u), 0\u\1, for the new Burr The entropy of a random variable X is a measure of vari- distribution can be computed using the formula: ation of uncertainty. Shannon’s entropy [17] for a random 1 À1 3 variable X with pdf f(x) is defined as EðÀ logðf ðxÞÞÞ.In QðuÞ¼r À ln u p À 1 þl: recent years, Shannon’s entropy has been used in many The median of a new Burr distribution occurs at applications in fields of engineering, physics, and 1 1 economics. 1 Àp 3 rðÀ lnðð2Þ À 1ÞÞ þ l, and clearly, it is a decreasing Denote by HshðXÞ the well-known Shannon’s entropy. function of p as p 1 but an increasing function of p as The following theorem gives the Shannon’s entropy of the p 1. new Burr distribution. Skewness and kurtosis of a parametric distribution are l3 l4 Theorem 3.1 The Shannon’s entropy of the new Burr often measured by a3 ¼ r3 and a4 ¼ r4, respectively. When the third or fourth moment does not exist, for example, distribution is given by ! X1 0 Cauchy, Le´vy, and Pareto distributions, a3 and a4, cannot 3p 2p ÀpÀ1 C ð1ÞÀlnðpþiÞ be computed. For the new Burr distribution, skewness and HshðXÞ¼Àln À r i p i 3 i¼0 þ kurtosis can be approximated by approximations of l3 and X1 l or alternative measures for skewness and kurtosis, based ÀpÀ1 1 1 4 Àp þ þ1: i 2 p on quantile functions. The measure of skewness S defined i¼0 ðpþiÞ by [6] and the measure of kurtosis K defined by [12] are based on quantile functions and they are defined as Proof ÀÁ ÀÁ ÀÁ Z 1 3p X À l Q 6 À 2Q 4 þ Q 2 H ðXÞ¼À f ðxÞ ln f ðxÞdx ¼Àln À 2E ln 8 ÀÁ 8 ÀÁ 8 sh r r S ¼ 6 2 ; ð2:4Þ À1 ! Q À Q 8 8 3 3 ÀÁ ÀÁ ÀÁ X À l À XÀl þ E þðp þ 1ÞE ln 1 þ e ðÞr : Q 7 Q 5 Q 3 r 8 ÀÀÁ 8 þÀÁ 8 K ¼ 6 2 : ð2:5Þ Q 8 À Q 8 ð3:1Þ

To investigate the effect of the shape parameter p on the We need to ﬁnd the expressions EðlnðXÀlÞÞ, EððXÀlÞ3Þ and new Burr density function, Eqs. 2.4 and 2.5 are used to r r ÀðXÀlÞ3 obtain Galton’s skewness and Moors’ kurtosis. Figure 2 Eðlnð1 þ e r ÞÞ. First, we calculate the expectation of XÀl r displays the Galton’s skewness and Moors’ kurtosis for the ð r Þ . r Z 1 rþ2 3 3 ÀpÀ1 X À l 3p x À l À xÀl À xÀl E ¼ e ðÞr 1 þ e ðÞr dx; r À1 r r 2.5 1 skewness using the change of variable, t ¼ xÀl 3 ,0\t\1, and 2 1þeÀð r Þ kurtosis 1 u then, the change of variable, t À 1 ¼ e ,0\u\1,we 1.5 obtain, ÀÁ 1 r X1 r X À l r Àp À 1 C 3 þ 1 E ¼ðÀ1Þ p rþ1 : r i 3 0.5 i¼0 ðp þ iÞ ð3:2Þ 0 For r ¼ 2k −0.5 !ÀÁ 2k X1 2k X À l Àp À 1 C 3 þ 1 −1 E ¼ p 2k : ð3:3Þ r i 3 þ1 0 5 10 15 20 i¼0 ðp þ iÞ p Differentiating both sides of 3.3 with respect to k at k ¼ 0 Fig. 2 Galton’s skewness and Moors’ kurtosis for the new Burr leads to distribution 123 Math Sci (2017) 11:47–54 51

X1 0 X À l p Àp À 1 C ð1ÞÀlnðp þ iÞ Fi:nðxÞ¼IFðxÞði; n À i þ 1Þ; E ln ¼ : r 3 i p þ i i¼0 where Ixða; bÞ is lower incomplete gamma function. ð3:4Þ There, the 100uth percentile of Xi:n can be obtained by solving ÀðXÀlÞ3 r In the same way, by calculating Eðð1 þ e r Þ Þ and then differentiating with respect to r at r ¼ 0, we obtain Fi:nðxÞ¼u: ð4:3Þ 3 À XÀl 1 The percentage points of Xi:n can be evaluated from 4.3 E ln 1 þ e ðÞr ¼ : ð3:5Þ p using tables of incomplete beta function (see [15]). How- ever, for i ¼ 1, Eq. 4.3 reduces to By replacing 3.2, 3.4, and 3.5 in relation 3.1, the proof is ÀðxÀlÞ3 Àp n ð1 Àð1 þ e r Þ Þ ¼ 1 À u. Thus, the 100u-percentage completed. h point of the smallest order statistic X1:n is given by 1 À1 3 1 p À1 n Distribution of order statistics F1:n ðu; p; l; rÞ¼l þ r À ln 1 Àð1 À uÞ À1 :

The pdf of Xi:n ði ¼ 1; ...; nÞ is given by Similarly, for i ¼ n, the 100u-percentage point of the lar- n! iÀ1 gest order statistic is fi:nðx; l; r; pÞ¼ f ðx; l; r; pÞF ði À 1Þ!ðn À iÞ! 1 1 3 À1 Ànp Âðx; l; r; pÞð1 À Fðx; l; r; pÞÞnÀi; Fn:n ðu; p; l; rÞ¼l þ r À ln u À 1 : where f ðx; l; r; pÞ and Fðx; l; r; pÞ are pdf and cdf given in 1.5 and 1.6, respectively:

XnÀi Hazard change-point estimation-classical approach fi:nðx; l; r; pÞ¼ djðn; iÞf ðx; l; r; pði þ jÞÞ; ð4:1Þ j¼0 Hazard function plays an important role in reliability and where survival analysis. New Burr distribution has modified n!ðÀ1Þj unimodal (unimodal followed by increasing) hazard func- d ðn; iÞ¼ : j ði À 1Þ!j!ðn À i À jÞ!ði þ jÞ tion. In some medical situations, for example, breast cancer, the hazard rate of death of breast cancer patients Note that djðn; iÞðj ¼ 0; 1; ...; n À iÞ are coefficients not represents a modified unimodal shape. dependent on p, l, and r. This means that fi:nðx; l; r; pÞ is a A modified unimodal shape has three phases: first weighted average of other new Burr. increasing, then decreasing, and then again increasing. It th From 1.6 and 4.1, we get the r moment of Xi:n to be can be interpreted as a description of three groups of XnÀi patients, first group is represented by the first phase that r r EðXi:nÞ¼ djðn; iÞEðX Þ contains the weak patients, so the hazard rate of this group j¼0 is increasing, while the second phase represents the group XnÀi Xr r with strong patients, their bodies have became familiar ¼ p ði þ jÞd ðn; iÞ ðÀrÞklrÀkE ðgðYÞÞ; j k q with the disease and they are getting better. The hazard rate j¼0 k¼0 of death of these patients is decreasing. In the third phase, they become weaker and their ability to cope with the where X has new Burr distribution with parameters l, r and disease declines, then the hazard rate of death increases. pði þ jÞ and Y has q distribution, standard exponential For situations, where the hazard function is modified y ÀpðiþjÞÀ1 k 2y distribution, and gðyÞ¼ðe þ 1Þ y3e . Then, the unimodal shaped, usually, we have interest in the estima- importance sampling estimate of the rth moment about tion of lifetime change-point, that is , the point at which the origin of Xi:n is given by !hazard function reaches to a maximum (minimum) and XnÀi Xr Xm then decreases (increase). In reliability, the change-point of r k rÀk 1 p ði þ jÞdjðn; iÞ ðÀrÞ l gðYlÞ ; Yl q: a hazard function is useful in assessing the hazard in the k m j¼0 k¼0 l¼1 useful life phase. One of change-points of hazard function ð4:2Þ of the new Burr distribution is location parameter. In this section, we consider maximum likelihood estimation pro- The cdf of X ð1 i nÞ is given by i:n cedure for change-points of the hazard function.

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Let us assume that x1; ...; xn is a random sample of size information can be consider as a goodness-of-fit test n of lifetimes generated by a new Burr distribution with statistic. For that purpose, the Reńyi Kullback–Leibler parameters l, r, and p. The log-likelihood function is given information can be estimated by by Xn 0 0 Xn Xn Drðf ; f Þ¼ÀHrðX1:n; ...; Xn:nÞÀ f ðxjÞ: 3p xi Àl xi Àl 3 lðl;r;pÞ¼nlog þ2 log À j¼1 r r r i¼1 i¼1 0 n Drðf ;f Þ X x l 3 Thus, the test statistics based on is given by À iÀ n Àðpþ1Þ log 1þe ðÞr : ! Xn i¼1 D ðf ; f 0Þ 1 T ¼ r ¼ ÀH^ ðX ; ...; X ÞÀ f 0ðx ; l^; r^; p^Þ ; r n n r 1:n n:n j j¼1 The maximum likelihood estimates for l, r, and p denoted by l^, r^, and p^, respectively, are obtained solving the where l^, r^, and p^ are MLEs of l, r, and p, respectively, ol ol ol likelihood equations, (o ¼ 0, o ¼ 0, and o ¼ 0Þ. l r p and H^rðX1:n; ...; Xn:nÞ is an estimate of Reńyi entropy for According to the above, maximum likelihood estimator of sample X1:n\X2:n\ ÁÁÁ\Xn:n. Under the null hypothesis, one of change-points is l^. Tr for r close to 1 will be close to 0, and therefore, large From the invariance property of maximum likelihood values of Tr will lead to the rejection of H0. estimators, we can obtain maximum likelihood estimators In this paper, we use estimation of Reńyi entropy based for functions of l, r and p. For / ¼ gðl; r; pÞ, a one-to-one on generalized nearest-neighbor graphs that is introduced function of l, r, and p, and we have /^ ¼ gðl^; r^; p^Þ. Taking by [14]. The basic tool to define their estimator was the / ¼ hðxÞ, defined in 1.7, the change-point of new Burr generalized nearest-neighbor graph. This graph on vertex hazard function is obtained as solution of equation set V is a directed graph on V. The edge set of it contains d dx logð/Þ¼0. The maximum likelihood estimator of the for each i 2 S (S is a finite non-empty set of positive d integers), an edge from each x 2 V to its ith nearest change-point is the solution of dx logð/Þ¼0 with l, r, and p replaced by maximum likelihood estimates. We observe neighbor according to the Euclidean distance to x. th d For p 0 denote by LpðVÞ, the sum of the p powers of that dx logð/Þ¼0 is non-linear in x, so the change-point of the hazard function estimate should be obtained using some Euclidean lengths of its edges. According to proven theo- one-dimensional root finding techniques, such as Newton– rem in [14]

Raphson. LpðX1:n; ...; Xn:nÞ lim p ¼ c [ 0 a:s:; 1À n!1 n d Testing new Burr model based on the Reńyi where p ¼ dð1 À rÞ and d is dimension of sample Kullback–Leibler information members. Based on described graph, they estimated Reńyi entropy Test statistics by Suppose that we are interested in a goodness-of-fit test 1 LpðX1:n; ...; Xn:nÞ H^ ðX ; ...; X Þ¼ log p : r 1:n n:n 1À for 1 À r cn d 8 < 2 xÀl 3 xÀl 3 ÀpÀ1 0 3p x À l ÀðÞ ÀðÞ H0 : f ðxÞ¼f ðx; l; r; pÞ¼ e r 1 þ e r : r r H : f ðxÞ 6¼ f 0ðx; l; r; pÞ; 1 Application where l, r, and p are unknown. We will denote the complete samples as In this section, we consider an uncensored data set corre- 0 sponding to remission times (in months) of a random X1:n\X2:n\ ÁÁÁ\Xn:n. For a null pdf f ðxÞ, the Reńyi Kullback–Leibler information from complete data is sample of 128 bladder cancer patients. These data were defined as previously reported in [10]. TTT plot for considered data is Z Z concave then convex indicating an increasing then 1 1 x2:n ðf ðx ; ...; x ÞÞa D ðf ; f 0Þ¼ log ÁÁÁ X1:n;...;Xn:n 1:n n:n dx ÁÁÁdx ; r 0 aÀ1 1 n decreasing hazard function, and is properly accommodated r À 1 À1 À1 ðf ðx1:n; ...; xn:nÞÞ X1:n;...;Xn:n by new Burr distribution. Because in the system of Burr distributions, only Burr X and Burr XII distributions have where r [ 0 and r 1. Because D f ; f 0 has the property 6¼ rð Þ unimodal hazard functions, and because of the similarity of 0 that Drðf ; f Þ0, and the equality holds if and only if cdf of the new Burr distribution with the Burr II distribu- 0 f ¼ f , the estimate of the Reńyi Kullback–Leibler tion compared to the rest of distributions in Burr family,

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1 The method of maximum likelihood is suggested for esti- 0.9 mating the parameters and change-points of hazard func-

0.8 New Burr distribution tion of the new Burr distribution. In application to 0.7 Burr XII distribution remission times (in months) of a random sample of 128 Burr II distribution 0.6 Generalized Burr II distribution bladder cancer patients, the new Burr distribution provided Burr X distribution 0.5 a significantly better fit than Burr X, Burr XII, Burr II, and F(x) generalized Burr II distributions. This fact is confirmed by 0.4 goodness-of-fit test statistic based on the Reńyi Kullback– 0.3 Leibler information. 0.2 0.1 Acknowledgements The authors acknowledge the Department of 0 Mathematics, Iran University of Science and Technology. 0 10 20 30 40 50 60 70 80 x Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://crea Fig. 3 cdfs of the new Burr, Burr X, Burr XII, Burr II, and tivecommons.org/licenses/by/4.0/), which permits unrestricted use, generalized Burr II models for the remission times of bladder cancer distribution, and reproduction in any medium, provided you give data appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Table 3 Values of test statistics for the remission times of bladder cancer data

Distribution Test statistic (T0:99) References New Burr 2.5732 1. Block, H., Savits, T.H., Singh, H.: The reversed hazard rate Burr XII 4.4819 function. Probab. Eng. Inf. Sci. 12, 69–90 (1998) Burr II 6.6133 2. Burr, I.W.: Cumulative frequency distribution. Ann. Math. Stat. Generalized Burr II 11.0319 13, 215–232 (1942) 3. Chandra, N.K., Roy, D.: Some results on reversed hazard rate. Burr X 193.9448 Probab. Eng. Inf. Sci. 15, 95–102 (2001) 4. Dierickx, B., Vleugels, J., Van der Biest, O.: Statistical extreme value modeling of particle size distributions: experimental grain compare the fits of the new Burr distribution and those of size distribution type estimation and parameterization of sintered Burr X, Burr XII, and Burr II and generalized Burr II. Plot zirconia D. Mater. Charect. 45, 61–70 (2000) 5. Ely, J.T.A., Fudenberg, H.H., Muirhead, R.J., LaMarche, M.G., of the estimated cdfs of models fitted to the data set is Krone, C.A., Buscher, D., Stern, E.A.: Urine mercury in given in Fig. 3. Figure 3 and also values of defined test micromercurialism: bimodal distribution and diagnostic impli- statistics in the previous section that are shown in Table 3 cations. Bull. Environ. Contam. Toxicol. 63(5), 553–559 (1999) confirm that the new Burr distribution provides a signifi- 6. Galton, F.: Enquiries into Human Faculty and its Development. Macmillan and Company, London (1883) cantly better fit than Burr X, Burr XII, Burr II, and gen- 7. Garbos´, S., S´wiecicka, D.: Application of bimodal distribution to eralized Burr II distributions. The required numerical the detection of changes in uranium concentration in drinking evaluations are implemented using Matlab (version 2013) water collected by random daytime sampling method from a large and R software (version 3.3.1). water supply zone. Chemosphere 138, 377–382 (2015) 8. Khodabin, M., Ahmadabadi, A.: Some properties of generalized gamma distribution. Math. Sci. 4, 9–28 (2010) 9. Knapp, T.R.: Bimodality revisited. J. Modern Appl. Stat. Meth- ods 6, 8–20 (2007) Conclusions 10. Lee, E.T., Wang, J.W.: Statistical Methods for Survival Data Analysist, 3rd edn. Wiley, New York (2003) 11. Lewis, A.W.: The Burr distribution as a general parameteric We introduced a new family of Burr-type distributions as family in survivorship and reliability theory applications. Ph.D. new Burr distribution. Various properties of the distribu- Thesis, University of North Carolina at Chapel Hill, USA (1981) tion are investigated. The distribution is found to be uni- 12. Moors, J.J.: A quantile alternative for kurtosis. Statistician 37, 25–32 (1988) modal and bimodal. This new distribution with one 13. Okasha, M.K., Matter, M.Y.: On the three-parameter Burr type parameter and simple form of cdf has modified unimodal XII distribution and its application to heavy tailed lifetime data. (unimodal followed by increasing) hazard function. Hence, J. Adv. Math. 10, 3429–3442 (2015) this new distribution can be used quite effectively in ana- 14. Pa´l, D., Szepesva´ri, Cs, Poćzos, B.: Estimation of Reńyi entropy and mutual information based on generalized nearest-neighbor lyzing lifetime data with non-monotonic hazard function. graphs (2010). arXiv:1003.1954

123 54 Math Sci (2017) 11:47–54

15. Pearson, E.S., Hartley, H.O.: Biometrika Tables for Statisticians, 18. Surles, J.G., Padgett, W.J.: Inference for reliability and stress- vol. I, 3rd edn. Cambridge University Press, Cambridge (1970) strength for a scaled Burr Type X distribution. Lifetime Data 16. Sewell, M.A., Young, C.M.: Are echinoderm egg size distribu- Anal. 7, 187–200 (2001) tions bimodal? Biol. Bull. 193, 297–305 (1997) 19. Zhang, C., Mapes, B.E., Soden, B.J.: Bimodality in tropical water 17. Shannon, C.E.: The mathematical theory of communication. Bell vapor. Q. J. R. Meteorol. Soc. 129, 2847–2886 (2003) Syst. Tech. J. 27, 249–428 (1948)

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