Kernel Estimators

You can compute and display kernel estimators to your data using the Kernel tab. There are five parameters and one button on the Kernel tab:

• Estimator: You can set the class of estimators to consider. The default choice is the evaluation-weighted estimator, also known as a Nadaraya-Watson estimator (Nadaraya, 1964; Watson, 1964). The other choice is a convolution-weighted estimator, also called a Gasser-Muller estimator (Gasser and Muller, 1979). For a general reference, see Chu and Marron, 1991.
• Kernel: You can set the shape of the kernel used to locally weight the curve. The kernels are truncated to zero outside of [–1,1] except for the Normal kernel, which, for computational simplicity, is truncated to zero outside of [–6,6]. The kernel choices are:

  Uniform

  

  Triangular

  

  Quadratic

  

  Biweight

  

  Triweight

  

  Normal

  

• Bandwidth Criterion: You can use this field to determine the choice of bandwidth for the kernel. The bandwidth, h, is the half-width that defines the scaling of the kernel, as illustrated below. In general, small values of h result in an "undersmoothed" curve that wiggles quite a bit in order to model small variations in the data, whereas a large value of h results in an "oversmoothed" curve that averages out variation in the data. For more information on choosing bandwidths, see Hurvich, Simonoff, and Tsai, 1998.

  

  The bandwidth criteria are:

  • Manual: You can type in a bandwidth, provided that the value is greater than twice the width of an interior bin. For example, if the X variable of your data ranges in [0,5] and you specify 51 bins, then the width of each interior bin is 0.1, and you can choose any bandwidth greater than or equal to 0.2. See the section on bins for more information about bin widths.
  • AIC: You can choose this criterion to produce a bandwidth that minimizes the Akaike information criterion (Akaike, 1973). This criterion may undersmooth the data.
  • Corrected AIC: You can choose this criterion to produce a bandwidth that minimizes the corrected Akaike information criterion as defined in Hurvich, Simonoff, and Tsai, 1998. The authors argue that this criterion tends to result in bandwidths that neither undersmooth nor oversmooth the data. This is the default criterion.
  • GCV: You can choose this criterion to produce a bandwidth that minimizes the generalized cross-validation criterion. This criterion may undersmooth the data.
  • Rice’s T: You can choose this criterion to produce a bandwidth that minimizes Rice’s T criterion (Rice, 1984). This criterion may oversmooth the data.
• Bandwidth Value: You can set the value of the bandwidth used to compute a kernel estimator. The value of this field must be greater than twice the width of a bin. If the bandwidth criterion is not "Manual," then this field is not editable.
• Bins: For large data sets, kernel methods can be slow. For data with non-uniformly spaced ordinates (random designs) and for data with replications among the ordinates, the convolution estimator has several undesirable qualities. For both of these reasons, the kernel estimators are not applied to raw data. Instead, data is pooled and averaged into bins, and the estimator is applied to the average values of the data in each bin. You can use this field to specify the number of bins used to pool the data. The default value for this field is 50.

  The two end bins are half the width of the uniformly spaced interior bins. For example, if the range of the horizontal data is [0,4], and you specify five bins, then the data is pooled into bins centered at the integers 0 – 4. The two end bins have width 0.5, while the interior bins have unit width. This scheme may result in "boundary effects" that underweight the data points near the ends of the data range.

  

  Since the number of bins determines the smallest allowed bandwidth parameter, choosing a small number of bins may result in oversmoothing the data. When using an automatic bandwidth selection criterion, you should choose enough bins so that the bandwidth selection algorithm can examine small and large bandwidths while trying to determine an optimal bandwidth.

• Add/Modify Curve: You can use this button either to create a new curve with the specified properties or to modify the computational parameters for an existing curve.