# Gumbel: Estimate Density Values by Gumbel kernel In DEEVD: Density Estimation by Extreme Value Distributions

## Description

The Gumbel kernel is developed by Khan and Akbar (2020). They provided evidence that performance of their proposed is better then Weibull kernel especially when data belongs to family of extreme distributions. Gumbel Kernel is

K_{Gumbel(x, √{h})}(j)=\frac{1}{√{h}}exp-≤ft( \frac{j-x}{√{h}} +exp≤ft( \frac{j-x}{√{h}}\right) \right)

## Usage

 1 Gumbel(x = NULL, y, k = NULL, h = NULL) 

## Arguments

 x scheme for generating grid points y a numeric vector of positive values. k gird points h the bandwidth

## Details

In this function, choice of bandwidth, number of grid points and scheme that how these grid points are generated are user based. If any parameter(s) is missing then function used default parameters. But at least x or k should be specified otherwise NA will be produced. If x is missing then function will generate k grid points between minimum and maximum values of vector. Similarly, if k is missing then function consider it same to length of main vector. In case if h is missing then function used normal scale rule bandwidth for non-normal data and described in Silverman (1986).

## Value

 x grid points y estimated values of density

## References

Khan, J. A., & Akbar, A. 2021. Density Estimation Using Gumbel Kernel Estimator. Open Journal of Statistics 11 (2), 319-328.

For other kernel see Weibull. To plot the density by using Gumbel kernel plot.Gumbel and to calculate MSE use mse.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 #Data can be simulated or real data ## Number of grid points "k" should be at least equal to the data size. ### If user define the generating scheme of gridpoints than number of gridpoints should ####be equal or greater than "k" ###### otherwise NA will be produduced. y <- rexp(100, 1) xx <- seq(min(y) + 0.05, max(y), length = 100) h <- 2 den <- Gumbel(x = xx, y = y, k = 200, h = h) ##If scheme for generating gridpoints is unknown y <- rexp(50, 1) h <- 3 den <- Gumbel(y = y, k = 90, h = h) ##If user do not mention the number of grid points y <- rexp(23, 1) xx <- seq(min(y) + 0.05, max(y), length = 90) ## Not run: #any bandwidth can be used require(KernSmooth) h <- dpik(y) #Direct Plug-In Bandwidth den <- Gumbel(x = xx, y = y, h = h) ## End(Not run) #if bandwidth is missing y <- rexp(100, 1) xx <- seq(min(y) + 0.05, max(y), length = 100) den <- Gumbel(x = xx, y = y, k = 90)