Maximum kernel likelihood estimation

Description

Computes the maximum kernel likelihood estimator using fast fourier transforms.

Details

The maximum kernel likelihood estimator is defined to be the value \hat \theta that maximizes the estimated kernel likelihood based on the general location model,

f(x|\theta) = f_{0}(x - \theta).

This model assumes that the mean associated with $f_0$ is zero which of course implies that the mean of X_i is \theta. The kernel likelihood is the estimated likelihood based on the above model using a kernel density estimate, \hat f(.|h,X_1,\dots,X_n), and is defined as

\hat L(\theta|X_1,\dots,X_n) = \prod_{i=1}^n \hat f(X_{i}-(\bar{X}-\theta)|h,X_1,\dots,X_n).

The resulting estimator therefore is an estimator of the mean of X_i.

Author(s)

Thomas Jaki

Maintainer: Thomas Jaki <jaki.thomas@gmail.com>

References

Jaki T., West R. W. (2008) Maximum kernel likelihood estimation. Journal of Computational and Graphical Statistics Vol. 17(No 4), 976-993.

Silverman, B. W. (1986), Density Estimation for Statistics and Data Analysis, Chapman & Hall, 2nd ed.

Examples

data(state)
mkle(state$CRIME)

Kernel log likelihood

Description

The function computes the kernel log likelihood for a given \hat \theta .

Usage

klik(delta , data, kde, grid, min)

Arguments

Details

This function is intended to be called through the function mkle and is optimized for fast computation.

Value

The log likelihood based on the shifted kernel density estimator.

Author(s)

Thomas Jaki

References

Jaki T., West R. W. (2008) Maximum kernel likelihood estimation. Journal of Computational and Graphical Statistics Vol. 17(No 4), 976-993.

See Also

mkle

Examples

data(state)
attach(state)
bw<-2*sd(CRIME)
kdensity<-density(CRIME,bw=bw,kernel="biweight",
          from=min(CRIME)-2*bw,to=max(CRIME)+2*bw,n=2^12)
min<-kdensity$x[1]
grid<-kdensity$x[2]-min

# finds the kernel log likelihood at the sample mean
klik(0,CRIME, kdensity, grid, min)

Maximum kernel likelihood estimation

Description

Computes the maximum kernel likelihood estimator for a given dataset and bandwidth.

Usage

mkle(data,bw=2*sd(data),kernel=c("gaussian", "epanechnikov", "rectangular", "triangular", 
     "biweight", "cosine", "optcosine"),gridsize=2^14)

Arguments

Details

The default for the bandwidth is 2s, which is the near-optimal value if a Gaussian kernel is used. If the bandwidth is zero, the sample mean will be returned.

Larger gridsize results in more acurate estimates but also longer computation times. The use of gridsizes between 2^{11} and 2^{20} is recommended.

Value

The maximum kernel likelihood estimator.

Note

optimize is used for the optimization and density is used to estimate the kernel density.

Author(s)

Thomas Jaki

References

Jaki T., West R. W. (2008) Maximum kernel likelihood estimation. Journal of Computational and Graphical Statistics Vol. 17(No 4), 976-993.

See Also

klik

Examples

data(state)
plot(density(state$CRIME))
abline(v=mean(state$CRIME),col='red')
abline(v=mkle(state$CRIME),col='blue')

Confidence intervals for the maximum kernel likelihood estimator

Description

Computes different confidence intervals for the maximum kernel likelihood estimator for a given dataset and bandwidth.

Usage

mkle.ci(data, bw=2*sd(data), alpha=0.1, kernel=c("gaussian", "epanechnikov", 
        "rectangular", "triangular", "biweight", "cosine", "optcosine"), 
        method=c("percentile", "wald","boott"), B=1000, gridsize=2^14)

Arguments

Details

The method can be a vector of strings containing the possible choices.

The bootstrap-t-interval can be very slow for large datasets and a large number of resamples as a two layered resampling is necessary.

Value

A dataframe with the requested intervals.

Author(s)

Thomas Jaki

References

Jaki T., West R. W. (2008) Maximum kernel likelihood estimation. Journal of Computational and Graphical Statistics Vol. 17(No 4), 976-993.

Davison, A. C. and Hinkley, D. V. (1997), Bootstrap Methods and their Applications, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press.

See Also

mkle

Examples

data(state)
mkle.ci(state$CRIME,method=c('wald','percentile'),B=100,gridsize=2^11)

Optimal bandwidth for the maximum kernel likelihood estimator

Description

Estimates the optimal bandwidth for the maximum kernel likelihood estimator using a Gaussian kernel for a given dataset using the bootstrap.

Usage

opt.bw(data, bws=c(sd(data),4*sd(data)), B=1000, gridsize=2^14)

Arguments

Details

The bandwidth considered fall between one and 4 standard deviations. In addition the mse of the mkle for a bandwidth of zero will also be included.

The estimation of the optimal bandwidth might take several minutes depending on the number of bootstrap resamples and the gridsize used.

Value

The estimated optimal bandwidth.

Note

The optimize is used for the optimization.

Author(s)

Thomas Jaki

References

Jaki T., West R. W. (2008) Maximum kernel likelihood estimation. Submitted to Journal of Computational and Graphical Statistics Vol. 17(No 4), 976-993.

Davison, A. C. and Hinkley, D. V. (1997), Bootstrap Methods and their Applications, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press.

See Also

mkle

Examples

data(state)
opt.bw(state$CRIME,B=10)

Violent death in the USA

Description

The dataset gives the number of violent death per 100,000 population per state

Usage

data(state)

Format

A data frame with 50 observations on the following 2 variables.

STATE

a factor with levels AK AL AR AZ CA CO CT DE FL GA HI IA ID IL IN KS KY LA MA MD ME MI MN MO MS MT NC ND NE NH NJ NM NV NY OH OK OR PA RI SC SD TN TX UT VA VT WA WI WV WY

CRIME

a numeric vector

Source

Shapiro, Robert~J. 1998. Statistical Abstract of the United States. 118 edn. U.S. Bureau of the Census.

Examples

data(state)
hist(state$CRIME)
mkle(state$CRIME)