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In this article, our proposed kernel estimator, named as Gumbel kernel, which broadened the class of non-negative, asymmetric kernel density estimators. Such kernel estimator can be used in nonparametric estimation of the probability density function ( pdf ). When the density functions have limited bounded support on [0, ∞) and they are liberated of boundary bias, always non-negative and obtain the optimal rate of convergence for the mean integrated squared error (MISE). The bias, variance and the optimal bandwidth of the proposed estimators are investigated on theoretical grounds as well as on simulation basis. Further, the applicability of the proposed estimator is compared to Weibul l kernel estimator, where performance of newly proposed kernel is outstanding.

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. [

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 [

By following Chen [

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 [

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 [

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 [

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 [

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. [

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. [

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.

We take suicide data given in Silverman [

We used the logarithm of the data to draw

h N S R = 0.79 R n − 1 5 , (18)

where R is inter-quartile range, which results in 0.4894. It can be observed that Gumbel kernel performed very well, especially near end points and free of boundary bias.

Further, we used the flood data given in Gumbel [

Here fixed bandwidth which is 1,000,000, is used.

In this section we wish to investigate the finite sample properties of the two asymmetric kernel estimators; Gumbel and Weibull, which belong to family of extreme value distributions. The experiments are based on 1000 random samples of length n = 3 5 = 243 , n = 486 and n = 972 . For each simulated sample and each estimator considered, mean squared errors (MSE) are reported in

Gumbel | Weibull | |
---|---|---|

n = 243 | ||

Gumbel (3, 1) | 0.0017 | 0.0054 |

Gumbel (10, 1) | 0.0026 | 0.0213 |

Gumbel (100, 1) | 0.0013 | 0.0279 |

Frechet (3, 1, 1) | 0.0009 | 0.0086 |

Frechet (5, 1, 1) | 0.0015 | 0.0140 |

Frechet (10, 1, 1) | 0.0006 | 0.0098 |

Weibull (25.713, 1) | 7.0497e−05 | 0.0001 |

Weibull (76.318, 1) | 8.9387e−06 | 6.8821e−06 |

Weibull (350.832, 1) | 1.0607e−06 | 2.0332e−06 |

n = 486 | ||

Gumbel (3, 1) | 0.0007 | 0.0034 |

Gumbel (10, 1) | 0.0015 | 0.0127 |

Gumbel (100, 1) | 0.0009 | 0.0342 |

Frechet (3, 1, 1) | 0.0006 | 0.0071 |

Frechet (5, 1, 1) | 0.0008 | 0.0074 |

Frechet (10, 1, 1) | 0.0011 | 0.0126 |

Weibull (25.713, 1) | 9.7541e−05 | 0.0006 |

Weibull (76.318, 1) | 8.4365e−06 | 7.2863e−06 |

Weibull (350.832, 1) | 4.7844e−07 | 5.0882e−07 |

n = 972 | ||

Gumbel (3, 1) | 0.0010 | 0.0021 |

Gumbel (10, 1) | 0.0014 | 0.0137 |

Gumbel (100, 1) | 0.0012 | 0.0241 |

Frechet (3, 1, 1) | 0.0006 | 0.0056 |

Frechet (5, 1, 1) | 0.0005 | 0.0052 |

Frechet (10, 1, 1) | 0.0005 | 0.0074 |

Weibull (25.713, 1) | 8.5902e−05 | 0.0007 |

Weibull (76.318, 1) | 8.1771e−06 | 5.4514e−06 |

Weibull (350.832, 1) | 4.15146e−07 | 3.1866e−08 |

Here in

the Gumbel estimator is acceptable at the boundary near the zero with different densities. In the interior, the behavior of the pdf estimator becomes more similar as we get away from zero in any extreme value distribution case.

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. [

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