Basic Idea Of Smoothing : Basic Concepts Of Setting

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2. Basic idea of smoothing If is believed to be smooth, then the observation at , near should contain information about the value of at . Thus it should be possible to use something like local average of data near to construct an estimator of . Smoothing of a data set , involves the approximation of the mean response curve in the regression relationship. The function of interest could be the regression curve itself, certain derivatives of it or functions of derivatives such as extrema or inflection points. In the trivial case in which is a constant, estimation of reduces to the point of location, since an average over the response variables yields an estimate of . In practical studies though it is unlikely that the…show more content…
This smoothing parameter regulates the size of the neighborhood around meaning a local average over too large a neighborhood would cast away the good with the bad. In this situation an extremely over smooth curve would be produced, resulting in a biased estimate . On the other hand defining the smoothing parameter so that it corresponds to a very small neighborhood would not shift the chaff from the wheat .Only a small number of observations would contribute none negligibly to the estimate at making it very rough and wiggly. In this case the variability of would be inflated. Finding the choice of smoothing parameter that balances the trade – off between over smoothing and under smoothing is called the smoothing parameter selection problem. 3. Choosing the smoother Some of the smoothing techniques include Kernel, Spline, Locally weighted regression, Recursive Regressogram, Convolution, Median, Split linear fit and K-Nearest Neighbor among others. One of the most active research areas in Statistics in the last 20 years has been the search for a method to find the "optimal" bandwidth for a smoother. There are now a great number of methods to do this. Unfortunately none of them is fully satisfactory. Here a comparative study of the two mostly used and easy to implement smoothers is presented. The Kernel and the cubic spline smoothers .The comparison is preformed on a simulated data set. Looking at
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