# Estimate Density Values by Gamma kernel This function provide the estimated Kernel density values by using Gamma Kernel.The Gamma kernel is developed by Chen (2000). He was first to introduce asymetrical kernels to control boundary Bias. Gamma Kernel is [REMOVE_ME]$$ K_{Gam1( \frac{x}{h}+1, h)}(y) = \frac{y^ \frac{x}{h} exp(-\frac{y}{h})}{ \Gamma (\frac{x}{h}+1)h^{ \frac{x}{h}+1}} [REMOVE_ME_2]$$ ```r Gamma(x = NULL, y, k = NULL, h = NULL) ``` ## Arguments - `x`: scheme for generating grid points - `y`: a numeric vector of positive values - `k`: number of gird points - `h`: the bandwidth ## Returns - **x**: grid points - **y**: estimated values of density ## Description This function provide the estimated Kernel density values by using Gamma Kernel.The Gamma kernel is developed by Chen (2000). He was first to introduce asymetrical kernels to control boundary Bias. Gamma Kernel is $$ K_{Gam1( \frac{x}{h}+1, h)}(y) = \frac{y^ \frac{x}{h} exp(-\frac{y}{h})}{ \Gamma (\frac{x}{h}+1)h^{ \frac{x}{h}+1}} $$ ## Details see the details in the `BS`. ## Examples ```r ##Number of grid points "k" should be at least equal to the data size. ###If user defines the generating scheme of grid points then length ####of grid points should be equal or greater than "k". Otherwise NA will be produced. y <- rexp(100, 1) xx <- seq(min(y) + 0.05, max(y), length = 500) h <- 2 den <- Gamma(x = xx, y = y, k = 200, h = h) ##If scheme for generating grid points is unknown y <- rexp(200, 1) h <- 3 Gamma(y = y, k = 90, h = h) ## Not run: y <- data(TUNA) xx <- seq(min(y) + 0.05, max(y), length = 500) h <- 2 den <- Gamma(x = xx, y = y, k = 200, h = h) ## End(Not run) ## Not run: ##If user do not mention the number of grid points y <- rexp(1000, 1) xx <- seq(0.001, 1000, length = 1000) #any bandwidth can be used require(KernSmooth) h <- dpik(y) Gamma(x = xx, y = y, h = h) ## End(Not run) ## Not run: #if generating scheme and number of grid points are missing then function generate NA y <- rexp(1000, 1) band = 3 Gamma(y = y, h = band) ## End(Not run) #if bandwidth is missing y <- rexp(100,1) xx <- seq(0.001, max(y), length = 100) Gamma(x = xx, y = y, k = 90) ``` ## References Chen, S. X. 2000. Probability density function estimation using Gamma kernels. **Annals of the Institute of Statistical Mathematics** 52 (3), 471-480. Silverman, B. W. 1986. **Density Estimation**. Chapman & Hall/ CRC, London. ## See Also For further kernels see `Erlang`, `BS`, `Beta`and `LogN`. To plot its density see `plot.Gamma` and to calculate MSE `mse`. ## Author(s) Javaria Ahmad Khan, Atif Akbar.

Gamma function