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Figure 1: A scatterplot with density and regression smoothing using rectangular kernels.
(1) The data is generated as y = x + sin (8x) + N(0,0.49), with 51 equally spaced x values. Three bandwidths (h) are used for the regression estimate at x*, which for this kernel, is just the average of the y values that fall in . Note how the bias increases and variance decreases as h increases. The estimate with shows a good balance between bias and variance. The bias increases with curvature of the target function.

 
Figure 1: Caption for Figure One continued:

(2) A rectangular kernel with is chosen for the bivariate density estimate. The locations of the points A, B, and C are arbitrary. The density estimate f(.), is proportional to the number of data points (1, 0, 6 for A, B, C, respectively) per unit area of the box. The marginal density of x is the U(0,1), and the conditional density f(y|x) is N(x + sin (8x), 0.49).