A generalized binomial exponential 2 distribution ...

Journal of Applied Statistics

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A generalized binomial exponential 2 distribution: modeling and applications to hydrologic events A. Asgharzadeh, Hassan S. Bakouch & M. Habibi To cite this article: A. Asgharzadeh, Hassan S. Bakouch & M. Habibi (2016): A generalized binomial exponential 2 distribution: modeling and applications to hydrologic events, Journal of Applied Statistics, DOI: 10.1080/02664763.2016.1254729 To link to this article: http://dx.doi.org/10.1080/02664763.2016.1254729

Published online: 10 Nov 2016.

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Date: 11 November 2016, At: 22:13

JOURNAL OF APPLIED STATISTICS, 2016 http://dx.doi.org/10.1080/02664763.2016.1254729

A generalized binomial exponential 2 distribution: modeling and applications to hydrologic events A. Asgharzadeha , Hassan S. Bakouchb and M. Habibia a Department of Statistics, University of Mazandaran, Babolsar, Iran; b Department of Mathematics, Faculty of

Science, Tanta University, Tanta, Egypt ABSTRACT

ARTICLE HISTORY

Developing statistical methods to model hydrologic events is always interesting for both statisticians and hydrologists, because of its importance in hydraulic structures design and water resource planning. Because of this, a flexible 3-parameter generalization of the exponential distribution is introduced based on the binomial exponential 2 (BE2) distribution [2]. The proposed distribution involving the exponential, gamma and BE2 distributions as submodels; and it exhibits decreasing, increasing and bathtub-shaped hazard rates, so it turns out to be quite flexible for analyzing non-negative real life data. Some statistical properties, parameters estimation and information matrix of the distribution are investigated. The proposed distribution, Gumbel, generalized Logistic and other distributions are utilized to model and fit two hydrologic data sets. The distribution is shown to be more appropriate to the data than the compared distributions using the selection criteria: average scaled absolute error, Akaike information criterion, Bayesian information criterion and Kolmogorov–Smirnov statistics. As a result, some hydrologic parameters of the data are obtained such as return level, conditional mean, mean deviation about the return level and the rth moments of order statistics.

Received 10 June 2015 Accepted 19 October 2016 KEYWORDS

Exponential type distributions; moments; estimation; order statistics; residual life; precipitation data

1. Introduction One of the most important tasks of hydrologic data analysis is to fit the data by a statistical model. Statistical models like Gumbel, Weibull, gamma, generalized logistic and other literature distributions have been received great attention for fitting hydrologic data like rainfall data, precipitation data, flood data, stream flow and so on (see, e.g. [4,6,13,14,23]), but there is a need to develop other flexible statistical models that having important role in hydraulic structures design and water resource planning and management. Therefore, we introduce a flexible 3-parameter generalization of the exponential distribution via the binomial exponential 2 (BE2) distribution. The binomial exponential 2 (BE2) distribution has been introduced by Bakouch et al. [2] representing the distribution of a random sum of independent exponentially distributed random variables where the size of the sample has a zero truncated binomial distribution CONTACT A. Asgharzadeh

[email protected]

© 2016 Informa UK Limited, trading as Taylor & Francis Group

2

A. ASGHARZADEH ET AL.

and it has the cumulative distribution function (cdf) λθ x e−λx , H(x) = 1 − 1 + 2−θ and probability density function (pdf) (λx − 1)θ λe−λx , h(x) = 1 + 2−θ

x > 0,

(1)

x > 0,

where 0 ≤ θ ≤ 1 is the shape parameter and λ > 0 is the scale parameter. The BE2 distribution has an increasing hazard rate property. If H(x) is an arbitrary cdf (known as the baseline distribution), then we can define an exponentiated cdf as F(x) = [H(x)]α , α > 0. In the literature, this model was named the exponentiated distribution. The parameter α describes the skewness, kurtosis and distribution tails. Here, F(x) is referred to as the exponentiated H distribution and also known as Lehmann type-I distribution [17]. The exponentiated distributions have been discussed in the literature by several authors, among them we cite Gupta and Kundu [12], Nadarajah and Kotz [19], Gupta and Gupta [10], Barreto-Souza and Cribari-Neto [3], Silva et al. [21] and Lemonte and Cordeiro [18]. Following the same approach, we shall generalize the BE2 distribution to introduce a flexible distribution of interest to hydrologic modeling and applications. A random variable X has the Generalized BE2 distribution with parameters α, λ and θ, denoted as GBE2(α, λ, θ), if its cdf is α λθ x −λx F(x) = 1 − 1 + e , x > 0, (2) 2−θ and the density function is

(λx − 1)θ −λx e [H(x)]α−1 , f (x) = αλ 1 + 2−θ

x > 0,

(3)

where α > 0 and 0 ≤ θ ≤ 1 are shape parameters, λ > 0 is the scale parameter and H(.) is given by Equation (1). Obviously, if α = 1 and θ = 0, the GBE2 distribution reduces to the exponential distribution with scale parameter λ, if α = 1 and θ = 1 it reduces to the gamma distribution with shape parameter 2 and scale parameter λ, and if α = 1 it reduces to the BE2 distribution with parameters λ and θ . Let X1 , X2 , . . . , Xα be BE2 independent random variables and describe natural phenomena ( like rainfall, floods, wave height, failure of a device etc) of a system. Then, the marginal cumulative probability function for the maximum of those variables, max(X1 , X2 , . . . , Xα ), of the system is Pr(max(X1 , X2 , . . . , Xα ) ≤ x) = Pr(X1 ≤ x) Pr(X2 ≤ x) . . . Pr(Xα ≤ x) α λθ x e−λx , = 1− 1+ 2−θ that is, the max(X1 , X2 , . . . , Xα ) follows the GBE2(α, λ, θ ) distribution. By virtue of this, the GBE2 represents the distribution of the maximum of a number α of independent and

JOURNAL OF APPLIED STATISTICS

3

identically (iid) BE2 distributed variables. The distribution of the maximum is of interest to hydrologic engineering applications, like a system design for the ‘α-year demand’ when the maximum demands are iid in different years (design of a building for the α-year wind, design of an offshore platform for the α-year wave and so on). Moreover, the limiting distribution of any order statistic from the GBE2 is either generalized Gamma or weighted Gumbel distribution, as it will be shown later, including the extreme ones that having great role in engineering applications of water resources planning, design and management. Other motivations of the GBE2 distribution come in its flexibility to model both monotonic and non-monotonic hazard rates, as we shall see later, and by this it provides a well alternative to several lifetime distributions for modeling non-negative real-valued data in applications. Later it will be shown that the parameter λ represents an upper bound on the hazard rate function which is an important feature of the lifetime models. Other attractive features of the GBE2 distribution are having heavy tail and closed forms for its cdf and hazard rate. The rest of this paper is established as follows. The shapes and stochastic ordering for the distribution are discussed in Section 2. Various mathematical properties of the GBE2 distribution are derived in Section 3. These include moments, conditional moments, mean deviations and order statistics. Estimation of the parameters by the maximum likelihood (ML) technique and the asymptotic properties of the estimators are discussed in Section 4. The ML estimators performance is also examined by a small Monte Carlo simulation in this section. Finally, two real hydrologic data application of the GBE2 distribution are provided in Section 5, and some hydrologic parameters of the data are obtained such as return level, conditional mean, mean deviation about the return level and the rth moments of order statistics. Finally, we hope that the distribution will be able to attract wider applications in hydrology, as many important contributions to hydrology have been investigated by collaboration between statisticians and hydrologists.

2. Functions shapes The density function takes various shapes depending on the values of the shape parameters α and θ. Particularly, the density function is decreasing or decreasing–increasing–decreasing function for 0 < α ≤ 1, while the density function becomes a skewed unimodal density for α > 1. Figure 1 shows some shapes of the density function for some different parameter values. The behavior of f (x) when x → 0 and x → ∞, respectively, is given by ⎧ ∞, α1

f (x) → 0.

The first derivative of log[f (x)] for the GBE2 distribution is d log f (x) λθ (α − 1)λ(2 − 2θ + λθ x) −λx = −λ+ e , dx 2 − 2θ + λθ x (2 − θ )H(x)

4


Figure 1. Pdf of the GBE2 distribution for some selected parameters (α, λ, θ).

and the second derivative of log f (x) is (λθ)2 (α − 1)λ2 e−λx d2 log f (x) = − + u(x), dx2 (2 − 2θ + λθ x)2 (2 − θ )[H(x)]2 where H(.) is given by Equation (1) and u(x) = (3θ − 2 − λθ x)H(x) −

(2 − 2θ + λθ x)2 −λx e . (2 − θ )

If α > 1 and θ < 23 , then we can see that u(x) < 0 and d2 log f (x)/dx2 < 0 for all x. So, f (x) is log-concave. Also d log f (x)/dx monotonically decreases from ∞ to −λ, so f (x) must attain a unique maximum at x = x0 for some x0 > 0. If α < 1 and θ < 23 , then d log f (x)/dx < 0 and so f (x) is monotonically decreasing for all x. If α < 1 and θ > 23 , f (x) attains its maximum, minimum or point of inflection according to whether d2 log f (x)/dx2 < 0, d2 log f (x)/dx2 > 0 or d2 log f (x)/dx2 = 0. Furthermore, the asymptotic of f (x) and F(x) as x → 0, ∞ are 2λ(1 − θ ) α α−1 f (x) ∼ α x , 2−θ as x → 0, f (x) ∼ as x → ∞,

αλ2 θ x exp(−λx), 2−θ

F(x) ∼

2λ(1 − θ ) 2−θ

α

xα ,

(4)

as x → 0, 1 − F(x) ∼

αλθ x exp(−λx), 2−θ

(5)

as x → ∞. So, the lower tail of f (x) is polynomial whereas its upper tail decays exponentially. Also, we note that the GBE2 distribution having a heavy tail as its survival function


5

Figure 2. Failure rate function of the GBE2 distribution for some selected parameters (α, λ, θ).

is slower by some power of x, that is 1 − F(x) → Cxn exp(−λx), where C = αλθ/(2 − θ ) and n = 1. The hazard rate function is a valuable term characterizing the phenomena life of a system, and for the GBE2 distribution it has the form (λx − 1)θ −λx r(x) = αλ 1 + e [H(x)]α−1 {1 − [H(x)]α }−1 , x > 0. (6) 2−θ The behavior of r(x) when x → 0 and x → ∞, respectively, is given by

r(x) →

⎧ ∞, ⎪ ⎪ ⎨ 2λ(1 − θ) ⎪ ⎪ ⎩ 2−θ 0,

α1

Therefore, the hazard rate at 0 is a function of both parameters for only α = 1 and independent of the parameters for other values of α. On the other hand, the parameter λ represents an upper bound on the hazard rate function which is an important feature of the lifetime models. Plots of the shapes of the hazard rate function for some selected values of the parameters are given in Figure 2. For some combinations to the parameter values, we get increasing, decreasing or bathtub-shaped hazard rate function. Therefore, the GBE2 distribution is flexible and can be used for analyzing real life data. Note that for α > 1 and θ < 23 the hazard rate function is increasing, because of the log-concavity of the density function. The quantile function of the GBE2 distribution, say Q(p), defined by F(Q(p)) = p, is the root of the equation λθ Q(p) 1+ exp{−λQ(p)} = 1 − p1/α , 2−θ for 0 < p < 1. Plots of the quartiles of the GBE2 distribution are given in Figure 3 and show an increasing behavior of quartiles. In what follows we shall use the next two lemmas where their proofs are given in the appendix.

6


Figure 3. First, second and third quartiles of the GBE2 distribution.

Lemma 2.1: Let

K(a, b, c, r, δ) =

∞ 0

2 − c + bcx −bx e x [2(1 − c) + bcx] 1 − 2−c

r

a−1 −δx

e

dx,

then we have K(a, b, c, r, δ) =

i ∞ i a−1 i=0 j=0

(−1)i bj cj (r + j)! j (2 − c)j (δ + bi)r+j+2

i

× [2(1 − c)(δ + bi) + bc(r + j + 1)], where r ∈ N ∪ {0}. Lemma 2.2:

L(a, b, c, r, δ, t) =

∞ t

2 − c + bcx −bx e x [2(1 − c) + bcx] 1 − 2−c r

a−1 −δx

e

dx,

then we have L(a, b, c, r, δ, t) =

i ∞ i a−1 i=0 j=0

i

(−1)i bj cj j (2 − c)j (δ + bi)r+j+2

× [2(1 − c)(δ + bi)(r + j + 1, (δ + bi)t) + bc(r + j + 2, (δ + bi)t)], where (a, x) =

∞ x

t a−1 e−t dt denotes the complementary incomplete gamma function.


7

3. Some properties of the distribution In this section, we find some statistical properties of the GBE2(α, λ, θ ) distribution involving moments, conditional moments, mean deviations and order statistics. 3.1. Moments and conditional moments Let X be a random variable with the pdf (3), the rth moment (about the origin) of X is given by

∞ r

(λx − 1)θ x αλ 1 + 2−θ r

E(X ) = 0

e−λx [H(x)]α−1 dx,

where H(.) is given by Equation (1). The rth moment of X is an integral which needs to be computed numerically in software such as MAPLE and R. The moments of X can also be obtained in a closed form as a series that is finite or infinite based on the value of the parameter α. Using Lemma 2.1, it follows that μr = E(X r ) =

αλ K(α, λ, θ , r, λ). 2−θ

The above expression for α = 1 takes the form μr = E(X r ) = (r!/λr )(1 + rθ /(2 − θ )), which agrees with the moments obtained by Bakouch et al. [2]. Also, the moment generating function of X can be expressed as MX (t) = E[etX ] =

αλ K(α, λ, θ, 0, λ − t). 2−θ

The quantity of dispersion in a population can be evaluated by the deviations from the mean and median. If X has the GBE2(α, λ, θ) distribution, then the mean deviation about the mean μ = E(X) and about the median M can be obtained by

∞ ξ1 (X) = |x − μ|f (x) dx = 2I(μ) + 2μF(μ) − 2μ, (7) 0

and

ξ2 (X) =

respectively, where I(u) =

∞

ξ1 (X) =

u

∞ 0

|x − M|f (x) dx = 2I(M) − μ,

xf (x) dx. So, by using Lemma 2.2 we have

2αλ L(α, λ, θ, 1, λ, μ) + 2μF(μ) − 2μ, (2 − θ)

and ξ2 (X) =

2αλ L(α, λ, θ, 1, λ, M) − μ, (2 − θ)

where F(.) is given by Equation (2).

(8)

8


For lifetime distributions, also it is important to know the conditional moments E(X r |X > x), r = 1, 2, . . ., which are of interest in prediction. Lemma 2.2 implies that E(X r | X > x) =

αλ L(α, λ, θ , r, λ, x), (2 − θ){1 − H α (x)}

(9)

where H(.) is given by Equation (1). 3.2. Residual life with some properties Residual (or remaining) life random variable is used extensively for the analysis of risks in actuarial sciences, so it is of interest to verify some of its properties for the GBE2 distribution. For survival time t ≥ 0, the period from time t until the time of failure is the residual life and defined as X(t) := X − t | X > t. The survival function of the residual life X(t) , t ≥ 0, for the GBE2 distribution is SX(t) (x) =

1 − H α (x + t) , 1 − H α (t)

x>0

and its pdf is fX(t) (x) =

αλ 1 +

(λ(x+t)−1)θ 2−θ

e−λ(x+t) [H(x + t)]α−1

1 − H α (t)

.

Consequently, the hazard rate function of X(t) is

e−λ(x+t) [H(x + t)]α−1 αλ 1 + (λ(x+t)−1)θ 2−θ . rX(t) (x) = 1 − H α (x + t) Further, the mean residual life is obtained as follows

∞ 1 K(t) = E(X(t) ) = xf (x) dx − t ¯ F(t) t αλ = L(α, λ, θ , 1, λ, t) − t, (2 − θ){1 − H α (t)}

t ≥ 0,

¯ is the where H(.) and f (.) are defined by Equations (1) and (3), respectively; and F(.) survival function of the GBE2 distribution. The variance residual life has considerable interest in the recent years [11] and for the GBE2 distribution it is obtained as follows

∞ 2 ¯ V(t) = Var(X(t) ) = xF(x) dx − 2tK(t) − [K(t)]2 ¯ F(t) t =

αλ L(α, λ, θ, 2, λ, t) − t 2 − 2tK(t) − [K(t)]2 , (2 − θ){1 − H α (t)}

where L(α, λ, θ, 2, λ, t) is defined by Lemma 2.2 at r = 2.


9

3.3. Order statistics Let X1 , X2 , . . . , Xn be a random sample of the GBE2(α, λ, θ ) distribution and X1:n ≤ X2:n ≤ · · · ≤ Xn:n denote its order statistics. Let fi:n (x) and Fi:n (x) denote, respectively, the pdf and the cdf of the ith order statistic Xi:n . Using concepts of the order statistics, we get Fi:n (x) =

n−j n n−j n j=i l=0

and

l

j

(−1)l [H(x)]α(j+l) ,

(λx − 1)θ −λx αλn! 1+ e fi:n (x) = (i − 1)!(n − i)! 2−θ n−i n−i (−1)l [H(x)]α(l+i)−1 , × l l=0

where H(.) is given by Equation (1). The rth moment of the ith order statistic Xi:n is

∞ r E(Xi:n ) = xr fi:n (x) dx, (10) 0

by using Lemma 2.1, it can be obtained as r )= E(Xi:n

αλn! (2 − θ)(i − 1)!(n − i)! n−i n−i (−1)l K(α(i + l), λ, θ , r, λ). × l

(11)

l=0

In the following propositions, we investigate the asymptotic distributions of the extreme values X1:n and Xn:n , and show that they are Weibull and Gumbel distributions, which adapt the results of Fisher and Tippett [8], where they showed that the asymptotic distribution of extremes must be either Gumbel, Fréchet or Weibull distribution. Moreover, we get the asymptotic distribution of any order statistics. Proposition 3.1: Let X1:n and Xn:n be the minimum and maximum of a random sample X1 , X2 , . . . , Xn from GBE2(α, λ, θ), respectively. Then α

(a) limn→∞ P{(X1:n − a∗n )/b∗n ≤ x} = 1 − e−x , x > 0, (b) limn→∞ P{(Xn:n − an )/bn ≤ t} = exp(−e−t ), −∞ ≤ t ≤ ∞,where 1 1 ∗ ∗ −1 1 −1 , an = F , bn = . 1− an = 0, bn = F n n nf (an ) Proof: From Equations (4) and (5), we have F(tx)/F(t) → xα as t → 0, (1 − F(t + xg(t)))/ (1 − F(t)) → exp(−x) as t → ∞ and where g(t) = 1/λ. Hence the proof follows from Theorem 1.6.2 in Leadbetter et al. [16].

10


Remark 3.1: The minimal domain of attraction of GBE2(α, λ, θ ) is the Weibull distribution and the maximal domain of attraction of GBE2(α, λ, θ ) is the Gumbel distribution. Now, we give the limiting distribution of any order statistic. Proposition 3.2: Assume that X1:n ≤ X2:n ≤ · · · ≤ Xn:n is the order statistics of a random sample X1 , X2 , . . . , Xn from GBE2(α, λ, θ). Then, for i = 1, 2, . . . , n, (a) limn→∞ P{(Xi:n − a∗n )/b∗n ≤ x} = 1 − (b) limn→∞ P{(Xn−i+1:n − an )/bn ≤ t} =

i−1

−xα (xjα /j!), x > 0, j=0 e i−1 −e−t −jt (e /j!), −∞ j=0 e

≤ t ≤ ∞.

Proof: The proof follows from Equations (8.4.2) and (8.4.3) of Arnold et al. [1].

Remark 3.2: (i) The limiting distribution of (Xi:n − a∗n )/b∗n has the generalized Gamma distribution α u(x) = (α/(i))xiα−1 e−x , x > 0. (ii) The limiting distribution of (Xn−i+1:n − an )/bn has the weighted Gumbel distribution υ(x) = (1/(i))e−it exp(−e−t ), −∞ ≤ t ≤ ∞, with weight function w(t) = e−(i−1)t .

4. Estimation and inference with simulation Estimating the parameters of a distribution is an essential topic because many of its features depend on the parameters. In this section, we investigate the ML estimation of the unknown parameters of the GBE2(α, λ, θ ) distribution using a random sample X1 , . . . , Xn of this distribution with inference on the parameters and conduct a small Monte Carlo simulations in terms of their mean and standard deviations to gain an idea on the estimators. 4.1. Estimation and inference Consider the parameter vector θ = (α, λ, θ)T , the log-likelihood function of θ based on the taken random sample is

αλ ln f (X, θ ) = n ln 2−θ + (α − 1)

n

−λ

n

xi +

i=1

ln(H(xi )),

n

ln(2 − 2θ + λθ xi )

i=1

(12)

i=1

where H(.) is given by Equation (1). The ML estimators of the unknown parameters are obtained by maximizing the log-likelihood function in Equation (12) with respect to θ.


11

The partial derivatives of ln f (X, θ ) are n ∂ ln f (X, θ ) = + ln(H(xi )), ∂α α i=1 n

n θ xi ∂ ln f (X, θ ) = − xi + ∂λ λ i=1 2 − 2θ + λθ xi i=1 n

n

+ (α − 1)

n xi e−λxi i=1

H(xi )

(λxi − 1)θ 1+ 2−θ

,

n n n 2 − λxi 2λxi e−λxi ∂ ln f (X, θ ) = − . + (1 − α) ∂θ 2−θ 2 − 2θ + λθ xi (2 − θ )2 H(xi ) i=1 i=1

ˆ θˆ )T of θ = (α, λ, θ )T can be gained by solving simultaneThe ML estimator θˆ = (α, ˆ λ, ously the likelihood equations ∂ ln f (X, θ ) ∂ ln f (X, θ ) ∂ ln f (X, θ ) = = = 0. ∂α ∂λ ∂θ Nonlinear optimization algorithms can be applied to solve the likelihood equations and get the estimator θˆ numerically. The asymptotic inference of the vector θ = (α, λ, θ )T ˆ θˆ )T . Under some regular conditions, depends on the normal approximation of θˆ = (α, ˆ λ, we have √ n(θˆ − θ ) → N3 (0, I−1 (θ )), where I−1 (θ ) is the inverse of the expected Fisher information matrix of θ, ⎡ Iαα ∂2 ln f (X, θ ) = ⎣Iαλ I(θ) = [Iij (θ)] = E − ∂θi ∂θj Iαθ

Iαλ Iλλ Iλθ

⎤ Iαθ Iλθ ⎦ , Iθ θ

whose elements are given in the appendix. Thus, the asymptotic 100(1 − γ )% confi ˆ and α), ˆ λˆ ± zγ /2 se( λ) dence intervals of α, λ and θ are, respectively, given by αˆ ± zγ /2 se(

is the square root of the diagonal element of (1/n)I−1 (θ ) corθˆ ), where se(.) θˆ ± zγ /2 se( responding to each parameter, and zγ /2 is the quantile 1 − γ /2 of the standard normal distribution.

4.1.1. Simulation. Here, the performance of the ML estimators is examined by a small Monte Carlo simulation. In this simulation, we generated 1000 random samples of size n from the ˆ θˆ ). GBE2(α, λ, θ) distribution. We then computed the ML estimates of α, λ and θ , say (α, ˆ λ, In our simulation, we used different sets of parameter values (α, θ ). Since λ is a scale parameter, only one case λ = 1 is considered. For different sample sizes n = 25,50,100,500, we computed the bias and root mean squared errors (RMSEs) over 1000 replications from the

12


Table 1. Bias and RMSEs (in brackets) of the estimates for arbitrary values of α, θ, and λ = 1. α

θ

n

bias (RMSE) α

bias (RMSE) λ

bias (RMSE) θ

0.8

0.3

0.8

0.6

0.8

0.9

1

0.3

1

0.6

1

0.9

3

0.3

3

0.6

3

0.9

25 50 100 500 25 50 100 500 25 50 100 500 25 50 100 500 25 50 100 500 25 50 100 500 25 50 100 500 25 50 100 500 25 50 100 500

0.023 (0.273) −0.022 (0.183) −0.027 ( 0.156) −0.009 (0.059) 0.078 (0.311) 0.011 (0.208) −0.002 (0.152) 0.001 (0.085) 0.403 (0.644) 0.259 (0.420) 0.200 (0.325) 0.088 (0.179) 0.022 (0.342) −0.025 (0.234) −0.039 (0.185) −0.018 (0.090) 0.121 (0.444) 0.036 (0.272) 0.0001 (0.221) −0.0112 (0.132) 0.577 (0.861) 0.405 (0.623) 0.326 (0.483) 0.160 (0.282) −0.009 (0.941) −0.131 (0.786) −0.154 (0.693) −0.123 (0.444) 0.127 (0.955) 0.060 (0.816) 0.054 (0.756) −0.009 (0.630) 0.744 (1.781) 0.382 (1.066) 0.114 (0.740) 0.009 (0.548)

0.166 (0.416) 0.099 (0.281) 0.052 (0.210) 0.007(0.133) 0.023 (0.286) −0.012 (0.216) −0.036 (0.173) −0.016 (0.101) −0.024 (0.242) −0.033(0.179) −0.051 (0.143) −0.023 (0.067) 0.117 (0.360) 0.085 (0.268) 0.044 (0.201) 0.001 (0.129) 0.0318 (0.280) −0.020 (0.206) −0.026 (0.160) −0.019 (0.101) −0.046 (0.237) −0.0385 (0.183) −0.053 (0.144) −0.025(0.070) 0.077 (0.266) 0.046 (0.202) 0.015 (0.152) −0.001 (0.1099) 0.027 (0.241) −0.023 (0.177) −0.038 (0.144) −0.031 (0.092) 0.036 (0.215) 0.009 (0.150) 0.003 (0.096) −0.003 (0.0388)

0.107 (0.371) 0.100 (0.361) 0.060 (0.331) −0.007(0.246) −0.121 (0.373) −0.091 (0.363) −0.104 (0.352) −0.051 (0.221) −0.367 ( 0.507) −0.286(0.440) −0.244 (0.398) −0.098 (0.192) 0.086 (0.368) 0.088 (0.359) 0.065 (0.341) −0.0103 (0.254) −0.133 (0.376) −0.108(0.366) −0.079 (0.343) −0.037 (0.238) −0.421 (0.553) −0.329 (0.478) −0.297 (0.442) −0.138(0.247) 0.105 (0.387) 0.092 (0.362) 0.057 (0.347) 0.016 (0.286) −0.035 (0.384) −0.080 (0.381) −0.094 (0.369) −0.056 (0.298) −0.076 (0.294) −0.064 (0.263) −0.018 (0.182) −0.0003 (0.107)

following equations 1 ˆ (hi − h), 1000 i=1 1 1000 RMSEh (n) = (hˆ i − h)2 , 1000 i=1 1000

biash (n) =

where h = α, λ, θ. Table 1 shows the bias and RMSEs over 1000 replications. From Table 1, we observe that the root mean squared errors are decreasing when the sample size n increasing.

5. Applications to hydrologic data In this section, we investigate a set of possible applications for the GBE2 distribution using two real hydrologic data. Fit and modeling such data have been investigated with obtaining


13

Table 2. Descriptive statistics for data sets. Data

Mean

Median

SD

Kurtosis

Skewness

MD(mean)

MD(median)

Precipitation Flood

175.67 51.49

158 40.4

83.17 32.38

4.81 7.95

1.34 2.07

62.87 22.94

61.17 20.96

some hydrologic parameters for the data under the selected GBE2 distribution, like return level, conditional mean, mean deviation about the return level and the rth moments of order statistics. Description of the data is as follows: (i) Precipitation data The data represent the annual maximum precipitation (inches) for one rain gauge in Fort Collins, Colorado from 1900 through 1999 [15]. The data are 239, 232, 434, 85, 302, 174, 170, 121, 193, 168, 148, 116, 132, 132, 144, 183, 223, 96, 298, 97, 116, 146, 84, 230, 138, 170, 117, 115, 132, 125, 156, 124, 189, 193, 71, 176, 105, 93, 354, 60, 151, 160, 219, 142, 117, 87, 223, 215, 108, 354, 213, 306, 169, 184, 71, 98, 96, 218, 176, 121, 161, 321, 102, 269, 98, 271, 95, 212, 151, 136, 240, 162, 71, 110, 285, 215, 103, 443, 185, 199, 115, 134, 297, 187, 203, 146, 94, 129, 162, 112, 348, 95, 249, 103, 181, 152, 135, 463, 183, 241. (ii) Flood data The following data set represents the maximum annual flood discharges, in units of 1000 cubic feet per second, of the North Saskachevan River at Edmonton, over a period of 48 years [22]: 19.885, 20.940, 21.820, 23.700, 24.888, 25.460, 25.760, 26.720, 27.500, 28.100, 28.600, 30.200, 30.380, 31.500, 32.600, 32.680, 34.400, 35.347, 35.700, 38.100, 39.020, 39.200, 40.000, 40.400, 40.400, 42.250, 44.020, 44.730, 44.900, 46.300, 50.330, 51.442, 57.220, 58.700, 58.800, 61.200, 61.740, 65.440, 65.597, 66.000, 74.100, 75.800, 84.100, 106.600, 109.700, 121.970, 121.970, 185.560. The descriptive statistics for both data sets are summarized in Table 2. 5.1. Fit of hydrologic data and model selection We compare the fits of the GBE2 distribution to the fits of other models including the exponential, gamma, BE2, Weibull, gumbel and generalized logistic distributions using the hydrologic data above. The cdf of the gumbel distribution [9] is given by

x−μ F(x) = exp − exp − , σ

∞ < x < ∞,

where σ > 0 and −∞ < μ < ∞, and the cdf of the generalized logistic distribution [7] is given by F(x) = [1 + exp(−β(x − μ))]−α ,

−∞ < x < ∞,

where α > 0 and β > 0, and −∞ < μ < ∞. To verify which distribution fits better to the data, we apply some goodness-of-fit tests, that are Kolmogorov–Smirnov (K–S) test statistic, the p-value of K–S test, Akaike information criterion (AIC), Bayesian information criterion (BIC) and the average scaled absolute

14


Table 3. Estimates, log-likelihood value (log(L)), ASAE, AIC, BIC, K–S statistics and their p -values, precipitation data. Distribution

Estimates

log(L)

ASAE

AIC

BIC

K–S

p-Value

Exponential BE2 Weibull Gamma Gumbel Generalized logistic GBE2

λˆ = 0.006 λˆ = 0.011, θˆ = 1 αˆ = 2.25, βˆ = 0.005 αˆ = 5.27, βˆ = 0.03 μˆ = 139.89,σˆ = 57.85 μˆ = −169.45, αˆ = 231.18, βˆ = 0.017

−616.9 −588 −576.1 −569 −567.6 −567.7

0.155 0.073 0.038 0.026 0.023 0.024

1235.72 1180.01 1156.19 1141.94 1139.29 1141.43

1238.33 1185.22 1161.41 1147.15 1144.50 1149.24

0.341 0.21 0.095 0.06 0.064 0.065

8.55e−011 0.0001 0.312 0.84 0.80 0.774

αˆ = 8.94, λˆ = 0.016,θˆ = 0.0007

−566.4

0.019

1138.90

1146.72

0.057

0.89

Table 4. Estimates, log-likelihood value (log(L)), ASAE, AIC, BIC, K–S statistics and their p -values, flood data. Distribution Exponential BE2 Weibull Gamma Gumbel Generalized Logistic GBE2

Estimates

log(L)

ASAE

AIC

BIC

K–S

p-Value

λˆ = 0.019 λˆ = 0.04, θˆ = 1 αˆ = 1.77, βˆ = 0.017 αˆ = 3.65, βˆ = 0.07 μˆ = 38.88,σˆ = 18.84 μˆ = −48.27, αˆ = 103.82, βˆ = 0.053 αˆ = 5.85, λˆ = 0.048,θˆ = 0.0001

−237.2 −225.5 −225.7 −221.5 −221 −221.1 −219.4

0.078 0.041 0.047 0.039 0.039 0.039 0.035

476.38 455.03 455.41 447.03 446.06 448.24 444.89

478.25 458.77 459.15 450.77 449.80 453.86 450.51

0.320 0.181 0.14 0.135 0.123 0.123 0.117

7.13e−05 0.075 0.279 0.318 0.432 0.431 0.499

error (ASAE). The ASAE is defined as follows (see [5]) 1 |x(i) − xˆ (i) | , n i=1 x(n) − x(1) n

ASAE =

where x(i) are the ascendingly ordered observations, and xˆ (i) is obtained from the quantile function with the estimates plugged in the equation for pi = i/(n + 1). This ASAE is useful to measure the predictability of the fitted model. The parameter estimates and goodnessof-fit statistics of those data sets are given in Tables 3 and 4, respectively. Estimating the parameters is obtained via the method of ML. These tables show that the smallest values of ASAE, AIC and K–S statistic, and the largest values of p -value are obtained for GBE2 distribution, which indicate that the GBE2 distribution is the best fit for the given data sets among all these distributions. Moreover, the likelihood ratio test (LRT) can be used to compare a distribution having additional parameters with some of its special cases. Therefore, we apply LRT to the GBE2 distribution with the Exponential (E) and BE2 distributions to check which distribution can give a better fit to the data. The statistic of the LRT is written as likelihood under null hypothesis 2 χ = −2 ln , likelihood under the whole parameter space under the null hypothesis H0 , where χ 2 follows Chi-square distribution (asymptotically) with d degrees of freedom which equals to the number of additional parameters in the model with extra parameters. Using this and standard statistical tables, we can get the critical value of the test statistic. Table 5 displays the LR statistics and their corresponding


15

Table 5. LR statistics and their p-values. Hypotheses H0 H1 H0 H1

:α :α :α :α

= 1 (BE2) = 1 (GBE2) = 1, θ = 0 (E) = 1, θ = 0 (GBE2)

Precipitation data

Flood data

43.11 (5.17e−11)

12.12 (4.99e−04)

100.82 (0)

35.49(1.97e−08)

Table 6. The 95% confidence intervals for the parameters α, λ and θ. Data set Precipitation data Flood data

α

λ

θ

[5.136, 12.750] [2.453, 9.255]

[0.010, 0.022] [0.029, 0.067]

[0, 0.699] [0, 0.570]

Table 7. Some moments of the GBE2 distribution for data sets. Data

Mean

Median

Kurtosis

Skewness

SD(σ )

MD-mean(ξ1 )

MD-median(ξ2 )

Precipitation Flood

176.48 50.56

162.28 45.66

5.74 5.93

1.25 1.31

77.56 25.41

59.29 19.39

58.16 18.98

p-values of the two data sets to test the hypotheses there. From the values of the LR statistics of both data and comparing with the critical value of the test statistics in both cases, we do not accept the null hypotheses in favor of the GBE2 distribution. Therefore, the GBE2 distribution gives significantly a better representation of these data sets than the Exponential and BE2 distributions. The variance covariance matrix of the MLEs under GBE2 distribution for the precipitation and flood data sets are respectively, given as ⎤ ⎤ ⎡ ⎡ 3.0117 0.0106 −0.0002 3.7778 0.0026 −0.0018 A = ⎣ 0.0026 1.05e-05 0.0010 ⎦ , B = ⎣ 0.0106 9.78e-05 0.0020 ⎦ . −0.0002 0.0020 0.0846 −0.0018 0.0010 0.1258 Note that the diagonal entries in the above variance covariance matrices for both data sets are the variances of the MLEs of the parameters α, λ and θ, respectively. Using this and the results of Section 4.1, then the 95% confidence intervals of the parameters α, λ and θ are listed in Table 6 for the two data sets. Further, some moments of the GBE2 distribution for the data sets are summarized in Table 7, where the theoretical definitions of mean, median, variance and mean deviation about the mean (and also median), at Section 3, are used for computations, noting that the parameters of the GBE2 distribution are replaced by their corresponding ML estimates for each data set under the GBE2 distribution. From Tables 2 and 7, we conclude that the considered moments of the GBE2 distribution are closed to the sample moments of both data sets. 5.2. Some hydrologic parameters In the preceding subsection, it is shown that the GBE2 distribution is the best model for both data sets, therefore we get the following hydrologic parameters for the data under this distribution.

16


Figure 4. Estimated return level (ˆxT ) and corresponding 95% confidence interval of the precipitation and flood data for the GBE2 distribution.

5.2.1. Return level. Once the best model of the data has been selected, the interest is to get the return level parameter of the data for a given return period (i.e. a high quantile of the distribution). The T-year return level, say xT , is the level exceeded on average only once in every Tyears. As the GBE2 is the best model, then its return level is obtained by inverting the cdf F(xT ) = 1 − 1/T, and hence xˆ T is the root of the equation 1 1/α λθ xT exp{−λxT } = 1 − 1 − , 1+ 2−θ T

(13)

where T ≥ 1. Also, the confidence interval for the return level is obtained by the delta method as [20] ˆ ≤ xT ≤ xˆ T + δα/2 D). ˆ (ˆxT − δα/2 D Here ˆ θ) ˆ T Iˆ−1 (θ )∇xT (α, ˆ θˆ ), ˆ 2 = ∇xT (α, ˆ λ, ˆ λ, D where the gradient vector is ˆ θ) ˆ = ∇xT (α, ˆ λ,

∂ xˆ T ∂ xˆ T ∂ xˆ T , , , ∂α ∂λ ∂θ

and the derivatives of the T-year return level with respect to the parameters are given in ˆ θˆ are the ML estimates given in Tables 3 and 4. Estimated return levels Appendix, and α, ˆ λ, and corresponding 95% confidence intervals of the precipitation and flood data for the GBE2 distribution are plotted in Figure 4.


17

ˆ and mean deviation about the return level (η) for data sets. Table 8. Estimated return periods (T) Precipitation data xT 463 443 434 354 348

Flood data

Tˆ

η

184.43 134.06 116.15 32.63 29.69

287.20 267.46 258.60 181.38 175.77

xT 185.56 121.97 109.70 106.60 84.10

Tˆ

η

1269.97 59.98 33.47 28.90 10.10

135.02 72.09 60.38 57.48 37.74

5.2.2. Return period. The return period of data can be estimated using T = 1/S(x), where S(x) = 1 − F(x). The ˆ for some largest values represented the return level xT of these estimated return periods, T, two data sets, are reported in Table 8 and have been computed using Tˆ = 1/S(xT ), where αˆ λˆ θˆ xT ˆ S(xT ) = 1 − 1 − 1 + e−λxT . 2 − θˆ 5.2.3. Mean deviation about the return level. The mean deviation about the return level is the mean of the distances of each value from their return level. Using the definition of the mean deviation about mean in Equation (7), the mean deviation about return level is

∞ η = 2xT F(xT ) − xT − μ + 2 xf (x) dx, xT

where f (x) is the density of the GBE2 distribution with parameters α, λ and θ . Table 8 provides mean deviation about some largest values represented the return level xT of the precipitation and flood data sets. 5.2.4. Conditional mean of a hydrologic data value. Conditional mean of the event data is defined based on Equation (9) as E(X | X > Q) =

αλ L(α, λ, θ, 1, λ, Q), (2 − θ){1 − H α (Q)}

where {1 − H α (Q)} is the survival function of GBE2 distribution, Q is a deterministic quantity of the event and L(α, λ, θ , 1, λ, Q) is defined by Lemma 2.2. For example, for the precipitation and flood data , we find that E(X | X > 463 ) = 525.597 and E(X | X > 185.56 ) = 206.398, respectively, noting that in the computations, the parameters α, λ and θ of the GBE2 are replaced by their corresponding ML estimates for each data set. 5.2.5. The rth moments of order statistics of the data. For planning to future emergencies to hydrologic events, it is of interest to have the largest observations and this is the case in both data sets we considered here. Order statistics is the tool for this purpose, so we get the rth moments of some order statistics of both data sets using Equation (10) under the GBE2 distribution, where the parameters of the GBE2 distribution are replaced by their corresponding MLEs for each data set. Those order statistics are given in Table 9 and they increase by increasing the rth order.

18


r ) for data sets. Table 9. Some numerical values of E(Xi:n

Precipitation data

Flood data

i

r

r E(Xi:100 )

i

r

r ) E(Xi:48

1

1 2 3

52.99 2921.717 166,750.5

1

1 2 3

13.85 205.87 3246.78

20

1 2 3

112.165 12,630.97 1,428,003

15

1 2 3

35.35 1261.35 45,430.48

70

1 2 3

1.01e−40 4.71e−38 2.20e−35

30

1 2 3

52.82 2811.90 150,873.3

100

1 2 3

461.0037 218,950.1 107,448,853

48

1 2 3

129.53 17,491.53 2,472,069

Acknowledgements The authors would like to express thanks to the two anonymous referees for useful suggestions and comments which have improved the first version of the manuscript.

Disclosure statement No potential conflict of interest was reported by the authors.

References [1] B.C. Arnold, N. Balakrishnan, and H.N. Nagaraja, A First Course in Order Statistics, Wiley, New York, 1992. [2] H.S. Bakouch, M. Aghababaei Jazi, S. Nadarajah, A. Dolati, and R. Roozegar, A lifetime model with increasing failure rate, Appl. Math. Model. 38 (2014), pp. 5392–5406. [3] W. Barreto-Souza and F. Cribari-Neto, A generalization of the exponential-Poisson distribution, Statist. Probab. Lett. 79 (2009), pp. 2493–2500. [4] P.K. Bhunya, R.D. Singh, R. Berndtsson, and S.N. Panda, Flood analysis using generalized logistic models in partial duration series, J. Hydrol. 420 (2012), pp. 59–71. [5] E. Castillo, A.S. Hadi, N. Balakrishnan, and J.M. Sarabia, Extreme Value and Related Models with Applications in Engineering and Science, Wiley, Hoboken, 2005. [6] A. Chadwick, J. Morfett, and M. Borthwick, Hydraulics in Civil and Environmental Engineering, Spon Press, London, 2004. [7] S.D. Dubey, A new derivation of the logistic distribution, Naval Res. Logist. 16 (1969), pp. 37–40. [8] R.A. Fisher and L.H.C. Tippett, Limiting forms of the frequency distribution of the largest or smallest member of a sample, Proc. Cambridge Philos. Soc. 24 (1928), pp. 180–290. [9] EJ Gumbel, Statistics of Extremes, Columbia University Press, New York, 1958. [10] R.D. Gupta and R.C. Gupta, Analyzing skewed data by power normal model, Test 17 (2008), pp. 197–210. [11] R.C. Gupta and S.N.U.A. Kirmani, Residual coefficient of variation and some characterization results, J. Statist. Plann. Inference 91 (2000), pp. 23–31. [12] R.D. Gupta and D. Kundu, Generalized exponential distributions, Aust. N. Z. J. Statist. 41 (1999), pp. 173–188. [13] J Heo and D.C. Boes, Regional flood frequency analysis based on a Weibull model: Part 2. Simulations and applications, J. Hydrol. 242 (2001), pp. 171–182. [14] D. Kang, K. Ko, and J. Huh, Determination of extreme wind values using the Gumbel distribution, Energy 86 (2015), pp. 51–58.


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[15] R.W. Katz, M.B. Parlange, and P. Naveau, Statistics of extremes in hydrology, Adv. Water Resour. 25 (2002), pp. 1287–1304. [16] M.R. Leadbetter, G. Lindgren, and H. Rootzn, Extremes and Related Properties of Random Sequences and Processes, Springer Verlag, New York, 1987. [17] E.L. Lehmann, The power of rank tests, Ann. Math. Statist. 24 (1953), pp. 23–43. [18] A.J. Lemonte and G.M. Cordeiro, The exponentiated generalized inverse Gaussian distribution, Statist. Probab. Lett. 81 (2011), pp. 506–517. [19] S. Nadarajah and S. Kotz, The exponentiated type distributions, Acta Appl. Math. 92 (2006), pp. 97–111. [20] K. Persson and J. Ryden, Exponentiated Gumbel distribution for estimation of return levels of significant wave height, J. Environ. Statist. 1 (2010), pp. 1–12. [21] R.B. Silva, W. Barreto-Souza, and G.M. Cordeiro, A new distribution with decreasing, increasing and upside-down bathtub failure rate, Comput. Statist. Data Anal. 54 (2010), pp. 935–944. [22] M.A.J. Van Montfort, On testing that the distribution is of type I when type II is the alternative, J. Hydrol. 11 (1970), pp. 421–427. [23] E.F. Zelenhasic, Theoretical probability distributions for flood peaks, Hydrology papers (Colorado State University); no. 42, 1970.

Appendix 1 Proof of Lemma 2.1.: For a real non-integer a > 1, the representation of (1 − z)a−1 is ∞ a−1 a−1 = (−1)i zi , (1 − z) i i=0

where |z| < 1. By using the expression above, we find that

∞ ∞ a − 1 (−1)i xr [2(1 − c) + bcx][2 − c + bcx]i e−(δ+bi)x dx K(a, b, c, r, δ) = i (2 − c)i 0 i=0

=

∞ i a−1 i (−1)i bj cj (2 − c)i−j i=0 j=0

× =

∞

0

i

j

(2 − c)i

xr+j [2(1 − c) + bcx]e−(δ+bi)x dx

∞ i a−1 i i=0 j=0

i

(−1)i bj cj (r + j)! j (2 − c)j (δ + bi)r+j+2

× [2(1 − c)(δ + bi) + bc(r + j + 1)].

Proof of Lemma 2.2.: Analogously to the proof of Lemma 2.1 and using the complementary incomplete gamma function, we get the required proof. Entries of I(θ) of GBE2 distribution. The entries of I(θ) of the GBE2 distribution are ∂2 1 Iαα = E − 2 ln f (X, θ) = 2 , ∂α α ∂2 1 αλθ 2 Iλλ = E − 2 ln f (X, θ) = 2 + K1 (α, λ, θ, 2, λ) ∂λ λ (2 − θ)2 −

αλ(α − 1)(3θ − 2) K(α − 1, λ, θ, 2, 2λ) (2 − θ)2

+

αθ λ2 (α − 1) K(α − 1, λ, θ, 3, 2λ) (2 − θ)2

20


−

4αλ(α − 1)(1 − θ)2 K(α − 2, λ, θ, 2, 3λ) (2 − θ)3

−

4αθλ2 (α − 1)(1 − θ) K(α − 2, λ, θ, 3, 3λ) (2 − θ)3

αθ 2 λ3 (α − 1) K(α − 2, λ, θ, 4, 3λ), (2 − θ)3 ∂2 1 4αλ = E − 2 ln f (X, θ) = − + K1 (α, λ, θ, 0, λ) 2 ∂θ (2 − θ) (2 − θ)2 −

Iθθ

−

2αλ2 αλ3 K (α, λ, θ , 1, λ) + K1 (α, λ, θ, 2, λ) 1 (2 − θ)2 (2 − θ)2

4αλ2 (α − 1) 4αλ3 (α − 1) K(α − 1, λ, θ, 1, 2λ) + K(α − 2, λ, θ, 2, 3λ), (2 − θ)4 (2 − θ)5 2αλ(1 − θ) ∂2 ln f (X, θ ) = − =E − K(α − 1, λ, θ, 1, 2λ) ∂α∂λ (2 − θ)2 −

Iαλ

αλ2 θ K(α − 1, λ, θ, 2, 2λ), (2 − θ)2 ∂2 2αλ2 =E − K(α − 1, λ, θ , 1, 2λ), ln f (X, θ) = ∂α∂θ (2 − θ)3 ∂2 2αλ =E − K1 (α, λ, θ, 1, λ) ln f (X, θ) = − ∂λ∂θ (2 − θ)2 −

Iαθ Iλθ

−

2αλ2 (α − 1) 2αλ(α − 1) K(α − 1, λ, θ, 2, 2λ) + K(α − 1, λ, θ, 1, 2λ) (2 − θ)3 (2 − θ)3

2αθλ3 (α − 1) 4αλ2 (α − 1)(1 − θ) K(α − 2, λ, θ, 2, 3λ) − K(α − 2, λ, θ , 3, 3λ), (2 − θ)4 (2 − θ)4 ∞ where K1 (a, b, c, r, δ) = 0 xr [1 + (bx − 1)c/(2 − c)]−1 [1 − ((2 − c + bcx)/(2 − c))e−bx ]a−1 e−δx dx and K(a, b, c, r, δ) is defined in Lemma 2.1. −

ˆ of GBE2 distribution. Let Entries of ∇xT (α, ˆ λˆ , θ) 1 1/α λθ xT exp{−λxT } + 1 − − 1, G= 1+ 2−θ T we have

where

∂xT Gα , =− ∂α GxT

∂xT Gλ , =− ∂λ GxT

∂xT Gθ , =− ∂θ GxT

∂G 1 1/α 1 1 , ln 1 − =− 2 1− Gα = ∂α α T T (λx − 1)θ ∂G =− 1+ xe−λx , Gλ = ∂λ 2−θ ∂G 2λx e−λx , = ∂θ (2 − θ)2 ∂G (λx − 1)θ λe−λx . = =− 1+ ∂xT 2−θ

Gθ = GxT