# Estimate Density Values by Erlang kernel This function provide the estimated values for density by using Erlang Kernel. Erlang kernel is developed by Salha et al. (2014). They developed this asymmetrical kernal with its hazard function and also proved its asymtotic normality. [REMOVE_ME]$$ K_{E(x,\frac{1}{h})} (y)=\frac{1}{\Gamma (1+\frac{1}{h})} \left[\frac{1}{x} (1+\frac{1}{h}) \right]^\frac{h+1}{h} y^\frac{1}{h} exp\left(-\frac{y}{x} (1+\frac{1}{h}) \right) [REMOVE_ME_2]$$ ```r Erlang(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 ## Returns - **x**: grid points - **y**: estimated values of density ## Description This function provide the estimated values for density by using Erlang Kernel. Erlang kernel is developed by Salha et al. (2014). They developed this asymmetrical kernal with its hazard function and also proved its asymtotic normality. $$ K_{E(x,\frac{1}{h})} (y)=\frac{1}{\Gamma (1+\frac{1}{h})} \left[\frac{1}{x} (1+\frac{1}{h}) \right]^\frac{h+1}{h} y^\frac{1}{h} exp\left(-\frac{y}{x} (1+\frac{1}{h}) \right) $$ ## Details see the details in the `BS`. ## Examples ```r ## Data: Simulated or real data can be used ## 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 <- rlnorm(100, meanlog = 0, sdlog = 1) xx <- seq(min(y) + 0.05, max(y), length = 500) h <-2 den <- Erlang(x = xx, y = y, k = 200, h = h) ##If scheme for generating grid points is unknown y <- rlnorm(1000, meanlog = 0, sdlog = 1) h <- 3 Erlang(y = y, k = 90, h = h) ## Not run: ##If user do not mention the number of grid points y <- rlnorm(100, meanlog = 0, sdlog = 1) xx <- seq(0.001, 1000, length = 1000) #any bandwidth can be used require(kedd) h <- h.ucv(y) #Unbaised cross validation bandwidth Erlang(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 <- rlnorm(100, meanlog = 0, sdlog = 1) band = 3 Erlang(y = y, h = band) ## End(Not run) #if bandwidth is missing y <- rlnorm(100, meanlog = 0, sdlog = 1) xx <- seq(0.001, 100, length = 100) Erlang(x = xx, y = y, k = 90) ``` ## References Salha, R. B.; Ahmed, E. S.; Alhoubi, I. M. 2014. Hazard rate function estimation using Erlang Kernel. **Pure Mathematical Sciences** 3 (4), 141-152. ## See Also For further MSE by using other kernels see `Beta`, `BS`, `Gamma` and `LogN`. For plotting these estimated values `plot.Erlang` and for calculating MSE use `mse`. ## Author(s) Javaria Ahmad Khan, Atif Akbar.

Erlang function