To investigate the properties and features of data or in anomaly detection, density estimation performs a vital role. For this purpose, the nonparametric kernel density estimation or curve estimation is a famous technique. Nonparametric estimation has certain advantages over parametric estimation, e.g. the problem of priori distribution choice, possibility of using non-homogenous data, no functional form and the most important is allocation of weights, etc. [1]. In nonparametric density estimation, boundary bias is a very serious issue. It affects the performance of the estimator at boundary points due to boundary effects, then from the interior points. Such problem is happened; when smoothing is carried out near the boundary and fixed symmetric kernel allocate weights outside the density support. That’s why in some cases, parametric method for curve estimation performs better than nonparametric estimation [2]. Such problem is happened when variables represent some sort of physical measure such as time or length. These variables thus have a natural lower boundary, e.g. time of birth, etc. So, when smoothing is carried out near the boundary and then fixed, symmetric kernels allocate weights outside the density support and due to this, boundary bias arise [3].

There is a vast literature on removing boundary effects in nonparametric method. As yet there appears to be no single dominating solution that corrects the boundary problem for all shapes of densities. Some common techniques are reflection of data, introduced by Schuster [4]. Similarly, Silverman [5] proposed negative-reflection. Eubank and Speckman [6] suggested semi-parametric model. Chen [7] suggested the solution of this problem by replacing the symmetric kernels by the asymmetric Bata kernel which never assigns weight outside the support. Many others used Chen’s idea and proposed other kernels, i.e. Gamma [8], lognormal [9], Inverse Gaussian [10], Weibull [11], etc.

By following Chen [7], we are going to propose a new class of density estimator named as a Gumbel kernel estimator along with its bias, variance and optimal bandwidth, which will be the keen addition in category of asymmetrical kernel(s) that solve the problem of boundary bias. The Gumbel distribution is a particular case of the Generalized Extreme Value (GEV) distribution, and also known as the log Weibull or Fisher-Tippett distribution. GEV is a family of continuous probability distributions which combines the Gumbel, Frechet and Weibull families and also known as type I, II and III extreme value distributions. The common functional form for all 3 distributions was discovered by McFadden [12].

This paper is organized as follows. In the first Section, we present some information about kernel smoothing and in Section 2 we presented the proposed kernel. In third Section, we investigated the bias, variance and optimal bandwidth of the Gumbel kernel estimator. The performance of the proposed estimator will be tested via real and simulated data sets in Section 4, while Section 5 concludes.

Let X 1 , ⋯ , X n be a random sample from a distribution with an unknown probability density function f which has bounded support on [0, ∞). Representation of pdf of Gumbel (µ, β) is

f ( j ) = 1 β e − ( z + e − z ) , j > 0 , (1)

where z = j − μ β and β > 0 . The mean and variance of J are equal to μ + β γ and β 2 π 2 6 , where γ ≈ 0.5772 is the Euler-Mascheroni constant.

As μ = x and s = h 1 2 , the class of Gumbel kernel considered is:

K Gumbel ( x , h 1 2 ) ( j ) = 1 h e − ( j − x h + e − ( j − x h ) ) . (2)

where h is bandwidth satisfying the condition that h → 0 and n h → ∞ as n → ∞ . If a random variable X has a pdf K G u m b e l ( x , h 1 2 ) ( x ) , then E ( X ) = x + h γ and the variance is var ( X ) = h π 2 6 .

The corresponding estimator of pdf is

f ^ Gumbel ( X ) = n − 1 ∑ i = 1 n K Gumbel ( x , h 1 2 ) ( X i ) . (3)

This estimator is easy to use and similar to following kernels for comparison:

Gamma 1 and Gamma 2 kernels by Chen [7] are;

K G a m 1 ( x / b + 1 , b ) ( y ) = y x / b exp { − y / b } Γ { x / b + 1 } b x / b + 1 , (4)

and

K G a m 2 ( ρ b , b ) ( y ) = y ρ b ( x ) − 1 exp { − y / b } Γ { ρ b ( x ) } b ρ b ( x ) . (5)

where

ρ b ( x ) = { x / b                             if   x ∈ [ 2 b , ∞ ) ; 1 4 ( x / b ) 2 + 1         if   x ∈ [ 0 , 2 b ) . (6)

Beta kernel by Chen [7] is;

K B ( x / b + 1 , 1 − x / b + 1 ) ( y ) = y x / b ( 1 − y ) ( 1 − x ) / b B { x / b + 1 , ( 1 − x ) / b + 1 } , (7)

where, B is Beta function.

Birnbaum-Saunders and Log-Normal kernels by Jin and Kawczak [9] are;

K B S ( b 1 / 2 , x ) ( y ) = 1 2 π b 2 ( 1 x y + x y 3 ) exp [ − 1 2 b ( y x − 2 + x y ) ] , (8)

and

K L N ( ln x , 4 ln ( 1 + b ) ) ( y ) = 1 8 π ln ( 1 + b ) y exp [ − ( ln y − ln x ) 2 8 ln ( 1 + b ) ] (9)

Inverse Gaussian and Reciprocal Inverse Gaussian kernels by Scaillet [10] are;

K I G ( x , 1 / b ) ( y ) = 1 2 π b y 3 exp [ − 1 2 b x ( y x − 2 + x y ) ] , (10)

and

K R I G ( ln x , 4 ln ( 1 / b ) ) ( y ) = 1 2 π b y exp [ − x − b 2 b ( y x − b − 2 + x − b y ) ] (11)

Erlang kernel by Salha, et al. [13] is;

K E ( x , 1 b ) ( y ) = 1 Γ ( 1 + 1 b ) [ 1 x ( 1 + 1 b ) ] b + 1 b y 1 h exp ( − y x ( 1 + 1 b ) ) (12)

Weibull kernel by Salha, et al. [11] is;

K w ( x , 1 b ) ( y ) = Γ ( 1 + b ) b x [ y Γ ( 1 + b ) x ] 1 b − 1 exp ( − ( y Γ ( 1 + b ) x ) 1 b ) (13)

Theorem 1 (Bias)

The bias of proposed estimator is given by;

Bias { f ^ Gumbel ( x ) } = h ( γ f ′ ( x ) h + 1 2 f ″ ( x ) π 2 6 ) + o ( 1 ) (14)

Proof:

E ( f ^ Gumbel ( x ) ) = ∫ 0 ∞     K Gumbel ( x , h 1 2 ) ( X i ) f ( x ) d x = E ( f ( ξ x ) ) ,

where ξ x follows a Gumbel distribution with scale parameter h 1 2 and shape parameter x.

The Taylor expansion about μ x for f ( ξ x ) is:

f ( ξ x ) = f ( μ x ) + f ′ ( μ x ) ( ξ x − μ x ) + 1 2 f ″ ( μ x ) ( ξ x − μ x ) 2 + o ( 1 ) .

So, E ( f ( ξ x ) ) = f ( x ) + 1 2 f ″ ( x ) v a r ( ξ x ) + o ( 1 ) .

E ( f ( ξ x ) ) = f ( x ) + h ( γ f ′ ( x ) + h π 2 12 f ″ ( x ) ) + o ( 1 ) .

Hence,

Bias ( f ^ Gumbel ( x ) ) = h ( γ f ′ ( x ) + h π 2 12 f ″ ( x ) ) + o ( 1 )

Theorem 2 (Variance)

The variance of the proposed estimator is given by:

var ( f ^ Gumbel ( x ) ) = 1 2 h 1 2 ( x + h γ 2 f ( x + h γ 2 ) ) (15)

Proof:

var ( f ^ Gumbel ( x ) ) = 1 n var [ K Gumbel ( x , h 1 2 ) ( X i ) ] = n − 1 [ E ( K Gumbel ( x , h 1 2 ) 2 ( X i ) ) ] + o ( n − 1 )

Let η x be a Gumbel ( x , h 1 2 2 ) distributed random variable. Hence μ x = E [ η x ] = x + h γ 2 and V x = var [ η x ] = h π 2 24 . We have

E ( K Gumbel ( x , h 1 2 ) 2 ( X i ) ) = J h E [ η x f ( η x ) ] ,

where, J h = 1 2 h 1 2 . By Taylor expansion of η x f ( η x ) we get:

η x f ( η x ) = η x f ( η x ) + [ ( η x f ′ ( η x ) + f ( η x ) ) ] ( η x − μ x )   + 1 2 [ η x f ″ ( η x ) + 2 f ′ ( η x ) ] ( η x − μ x ) 2 + o ( h − 2 )

So,

E [ η x f ( η x ) ] = x + h γ 2 f ( x + h γ 2 )     + 1 2 [ x + h γ 2 f ' ' ( x + h γ 2 ) + 2 f ′ ( x + h γ 2 ) ] h π 2 24 + o ( h − 2 ) = x + h γ 2 f ( x + h γ 2 ) + h [ x + h γ 2 f ' ' ( x + h γ 2 ) π 2 24 + f ′ ( x + h γ 2 ) 12 ]     + o ( h − 2 x − 2 ) = x + h γ 2 f ( x + h γ 2 ) + o ( h − 2 x − 2 )

Therefore,

var ( f ^ Gumbel ( x ) ) = 1 2 h 1 2 [ x + h γ 2 f ( x + h γ 2 ) ] + o ( h − 2 x − 2 )

Optimal Bandwidth

To proceed for optimal bandwidth, initially Mean Squared Error (MSE) and Mean Integrated Squared Error (MISE) are derived as;

As we know Mean square errors for Gumbel kernel estimator is

MSE [ f ^ Gumbel ( x ) ] = Bias 2 [ f ^ Gumbel ( x ) ] + var [ f ^ Gumbel ( x ) ] = h ( ( γ f ′ ( x ) + h π 2 12 f ″ ( x ) ) ) 2 + 1 2 h [ x + h γ 2 f ( x + h γ 2 ) ]

We can approximate MISE to be:

MISE [ f ^ Gumbel ( x ) ] = h A + 1 2 h B (16)

where, A = ∫ ​ ( γ f ′ ( x ) + h π 2 12 f ″ ( x ) ) 2 d x and B = ∫ ​     x + h γ 2 f ( x + h γ 2 ) d x .

To find optimal bandwidth, now we minimize Equation (16) with respect to h, so we have

dMISE [ f ^ G u m b e l ( x ) ] d h = A − h − 3 2 4 B (17)

d 2 MISE [ f ^ G u m b e l ( x ) ] d h 2 = 3 8 h 5 2 B > 0

Setting (17) equal zero yields an optimal bandwidth h o p t for the given pdf and kernel:

h o p t = [ 4 ∫ ​ ( γ f ′ ( x ) + h π 2 12 f ″ ( x ) ) 2 d x ∫ ​     x + h γ 2 f ( x + h γ 2 ) d x ] − 2 3

In this section, the performance of the proposed estimators in estimating the pdf is observed through real life data as well as by simulation study.

In this paper, we have proposed a new kernel estimator for probability density functions for (iid) data [0, ∞), namely Gumbel kernel. Such densities are encountered in a wide variety of applications to describe extreme wind speeds, sea wave heights, floods, rainfall, age at death, minimum temperature, rainfall during droughts, electrical strength of materials, air pollution problems, geological problems, naval engineering etc. [4]. Gumbel kernel is free of boundary bias, non-negative, with natural varying shape. We showed that the bias depends on the smoothing parameter h and the estimated point x, and it goes to zero as h → 0, also it gets smaller for the values of x closed to zero. The variance of the new proposed kernel estimator was investigated, and we noticed that it depends also on h and x. On the other hand, it goes to zero as h → 0, and gets large at the values of x close to zero.

In addition, the performance of the proposed estimators is tested in three applications. In a simulation study, we used different densities of GEV distribution and compared it with Weibull (Extreme value distribution III) kernel estimator on basis of MSE. We observed that the performance of the proposed estimator is excellent, and gives a smaller MSE. Additionally, by using real data examples, we exhibited the practical performance of the new estimator.

From the above discussion, it can be concluded that one of the reason for adaptation of nonparametric method was to control the allocation of weights at boundary points. But boundary bias is still present if symmetrical kernels are used for curve estimation. In this situation, best alternative is to use asymmetrical kernel and Gumbel kernel is finest selection than Weibull kernel, comparatively.

The authors declare no conflicts of interest regarding the publication of this paper.

Khan, J.A. and Akbar, A. (2021) Density Estimation Using Gumbel Kernel Estimator. Open Journal of Statistics, 11, 319-328. https://doi.org/10.4236/ojs.2021.112018