The MKLE Package. July 24, Kdensity... 1 Kernels... 3 MKLE-package... 3 checkparms... 4 Klik... 5 MKLE... 6 mklewarp... 7 state... 8.

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1 The MKLE Package July 24, 2006 Type Package Title Maximum kernel likelihood estimation Version 0.02 Date Author Maintainer Package to compute the maximum kernel likelihood estimator (MKLE) for mu. License GNU R topics documented: Kdensity Kernels MKLE-package checkparms Klik MKLE mklewarp state Index 9 Kdensity Kernel density estimator Evaluates the shifted kernel density estimator Kdensity(x, data, Kernel = dnorm, bw = 2*sd(data), theta = mean(data)) 1

2 2 Kdensity x data Kernel bw theta point at which the kernel density estimator is evaluated. the data from which the estimate is to be computed. a R function to be used as the kernel function. the smoothing bandwidth to be used. the location parameter used. The location parameter theta shifts the kernel density estimator. Instead of centering the individual kernels on top of each datapoint, they will be shifted by theta-mean(data). Setting theta=mean(data) therefore gives the usual kernel density estimator. 1 nh n i=1 K( y X i X + θ ). h The value of the kernel density estimator Silverman, B. W. (1986) Density Estimation for Statistics and Data Analysis. Chapman & Hall density ## plots the kernel density estimator attach(state) x<-seq(min(crime)-10,max(crime)+10,0.1) plot(x,kdensity(x,crime,theta=mean(crime)),type='l',ylab='kernel Density',xlab='',lwd=2)

3 MKLE-package 3 Kernels Kernel functions Evaluates the finite support kernels for a given x biweight(x) triweight(x) triangle(x) x Point at which the kernel is evaluated The biweight kernel is defined as 15/16 (1 x 2 ) 2 for x <1 The triweight kernel is defined as 35/32 (1 x 2 ) 3 for x <1 The triangle kernel is defined as 1 abs(x) for x <1 of the kernel function. Silverman, B. W. (1986) Density Estimation for Statistics and Data Analysis. Chapman & Hall triweight and triangle MKLE-package Maximum kernel likelihood estimation Computes the kernel density estimator for mu Package: MKLE Type: Package Version: 0.02 Date: License: GNU

4 4 checkparms Maintainer: Silverman, B. W. (1986) Density Estimation for Statistics and Data Analysis. Chapman & Hall\ more to come checkparms Checks if the kernel function is proper. Checks if the kernel function is nonnegative and integrates to 1. checkparms(kernel) Kernel a R function to be used as the kernel function. will return a warning if the kernel is improper. Kdensity

5 Klik 5 Klik Kernel log likelihood The function computes the kernel log likelihood for a given theta klik(theta = 0, data, Kernel = dnorm, bw = 2*sd(data)) theta data Kernel bw the parameter value for which the log likelihood will be computed. the data for which the log likelihood will be computed. a R function to be used as the kernel function. the smoothing bandwidth to be used. The log likelihood of theta for a given bandwidth. The log likelihood based on the shifted kernel density estimator. in preperation Kdensity and mkle ## plots the kernel log likelihood attach(state) tv<-seq(min(crime),max(crime)) lik<-sapply(tv,klik,data=crime) plot(tv,lik,type='l',xlab='theta',ylab='kernel Likelihood') abline(v=mean(crime),col='red')

6 6 MKLE MKLE Maximum kernel likelihood estimation Computes the maximum kernel likelihood estimator for a given dataset and bandwidth. mkle(data, Kernel = dnorm, bw = 2*sd(data), small = TRUE) data Kernel bw small the data for which the log likelihood will be computed. a R function to be used as the kernel function. the smoothing bandwidth to be used. logical; if TRUE, only the value of the estimator is returned. Otherwise the full optimization history will be included. The underlying shifted kernel density estimator is defined as 1 nh n i=1 K( y X i X + θ ). h The default for the bandwidth is 2*sigma, which is the optimal value if a Gaussian kernel is used. The MKLE or a list with components: par value counts convergence message The best set of parameters found. The value of klik corresponding to par. A two-element integer vector giving the number of calls to Klik. This excludes those calls needed to compute the Hessian, if requested, and any calls to fn to compute a finite-difference approximation to the gradient. An integer code. 0 indicates successful convergence. Error codes are 1 indicates that the iteration limit maxit had been reached. 10 indicates degeneracy of the Nelder-Mead simplex. 51 indicates a warning from the "L-BFGS-B" method; see component message for further details. 52 indicates an error from the "L-BFGS-B" method; see component message for further details. A character string giving any additional information returned by the optimizer, or NULL. Note The optim with the method BFGS is used for the optimization.

7 mklewarp 7 not yet optim and klik attach(state) mkle(crime) mklewarp Maximum kernel likelihood estimation Computes the maximum kernel likelihood estimator for a given dataset and bandwidth using fast fourier transforms. mklewarp(data, bw = 1, gs = 2^11, K="gaussian") data bw gs the data for which the log likelihood will be computed. the smoothing bandwidth to be used. the number of gridpoints to be used for the fourier transform K a character string giving the kernel function to be used. This must be one of "gaussian", "rectangular", "triangular", "epanechnikov", "biweight", "cosine" or "optcosine", with default "gaussian". The underlying shifted kernel density estimator is defined as 1 nh n i=1 K( y X i X + θ ) h. The default for the bandwidth is 2*sigma, which is the optimal value if a Gaussian kernel is used.

8 8 state the MKLE. coming soon mkle and klik ## compares the MKLE and the warped MKLE. attach(state) mkle(crime) mklewarp(crime) state Violent crimes in the USA Format Source The dataset gives the number of violent crimes per 100,000 population per state A data frame with 50 observations on the following 2 variables. state a factor with state abbreviations as levels crime a numeric vector Shapiro, Robert J. (1998) Statistical Abstract of the United States. 118 edn. U.S. Bureau of the Census. hist(state$crime)

9 Index Topic datasets state, 8 Topic distribution Kdensity, 1 Topic misc checkparms, 4 Kernels, 2 MKLE-package, 3 Topic univar Klik, 4 MKLE, 5 mklewarp, 7 biweight (Kernels), 2 checkparms, 4 density, 2 Kdensity, 1, 4, 5 Kernels, 2 Klik, 4 klik, 6, 7 klik (Klik), 4 MKLE, 5 MKLE (MKLE-package), 3 mkle, 5, 7 mkle (MKLE), 5 MKLE-package, 3 mklewarp, 7 optim, 6 state, 8 triangle, 3 triangle (Kernels), 2 triweight, 3 triweight (Kernels), 2 9