Linear Smoothing - 5.2 Linear Smoothing In this section,...

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5.2 Linear Smoothing In this section, some of the most common smoothing methods are introduced and discussed. 5.2.1 Kernel Smoothers The simplest of smoothing methods is a kernel smoother. A point is fixed in the domain of the mean function , and a smoothing window is defined around that point. Most often, the smoothing window is simply an interval , where is a fixed parameter known as the bandwidth . The kernel estimate is a weighted average of the observations within the smoothing window: (5.2) where is a weight function. The weight function is chosen so that most weight is given to those observations close to the fitting point . One common choice is the bisquare function,
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The kernel smoother can be represented as (5.3) where the coefficients are given by A linear smoother is a smoother that can be represented in the form ( 5.3 ) for appropriately defined weights . This linear representation leads to many nice statistical and computational properties, which will be discussed later. The kernel estimate ( 5.2 ) is sometimes called the Nadaraya-Watson estimate ([ 23 , 33 ]). Its simplicity makes it easy to understand and implement, and it is available in many statistical software packages. But its simplicity leads to a number of weaknesses, the most obvious of which is boundary bias. This can be illustrated through an example. Figure 5.1: Kernel smooth of the fuel economy dataset. The bisquare kernel is used, with bandwidth pounds The fuel economy dataset consists of measurements of fuel usage (in miles per gallon) for sixty different vehicles. The predictor variable is the weight (in pounds) of the vehicle. Figure 5.1 shows a scatterplot of the sixty data points, together with a kernel smooth. The smooth is constructed using the bisquare kernel and bandwidth pounds.
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Over much of the domain of Fig. 5.1 , the smooth fit captures the main trend of the data, as required. But consider the left boundary region; in particular, vehicles weighing less than pounds. All these data points lie above the fitted curve; the fitted curve will
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Linear Smoothing - 5.2 Linear Smoothing In this section,...

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