hi carol
in kernel fitting, the optimal band width h is found so that it minimizes the sum of squared errors between the empirical density and the fitted density. I'm not sure what is the empirical density here. How do we get this, given we have a random sample to start with? Also, the fitted density means the fitted kernel function, is this right?
Thank you for your help.
helen
kernel fitting
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 Posts: 815
 Joined: Sun Sep 28, 2008 10:30 pm
Re: kernel fitting
Hiya
I see your point. You use the observed sample x_1, ..., x_n to estimate the fitted density \hat f_h(x) = (nh)^1 \sum K[(xx_i)/n] where K is the kernel and h is the bandwidth and then choose h to minimise the weighted sum of squared errors between \hat f_h(x) and f(x) where f(x) is the empirical density....but what is the empirical density? Is it a histogram, in which case it depends on a subjective choice of cell width? Or, is each observation x_i weighted equally, in which case the method will be sensitive to outliers and errors in the sample. Probably the latter in standard kernelfitting routines, but I guess you can specify which method you like if you write your own code.
Can other forum user chip in here, if they have done more than simply use the matlab automatic functions?
Maybe you get some help from here: http://www.mathworks.co.uk/help/toolbox ... #bq_u_f61 under discussion on fitting piecewise distributions. I also recommend this excellent article: http://en.wikipedia.org/wiki/Kernel_density_estimation  it is very clear and concise and has some easy to follow illustrative examples
Carol
I see your point. You use the observed sample x_1, ..., x_n to estimate the fitted density \hat f_h(x) = (nh)^1 \sum K[(xx_i)/n] where K is the kernel and h is the bandwidth and then choose h to minimise the weighted sum of squared errors between \hat f_h(x) and f(x) where f(x) is the empirical density....but what is the empirical density? Is it a histogram, in which case it depends on a subjective choice of cell width? Or, is each observation x_i weighted equally, in which case the method will be sensitive to outliers and errors in the sample. Probably the latter in standard kernelfitting routines, but I guess you can specify which method you like if you write your own code.
Can other forum user chip in here, if they have done more than simply use the matlab automatic functions?
Maybe you get some help from here: http://www.mathworks.co.uk/help/toolbox ... #bq_u_f61 under discussion on fitting piecewise distributions. I also recommend this excellent article: http://en.wikipedia.org/wiki/Kernel_density_estimation  it is very clear and concise and has some easy to follow illustrative examples
Carol
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