Kernel density estimation of 100 normally distributed random numbers using different smoothing bandwidths.
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Let \((x1, x2, …, xn)\) be an independent and identically distributed sample drawn from some distribution with an unknown density \(ƒ\). We are interested in estimating the shape of this function \(ƒ\). Its kernel density estimator is
\[\hat{f}_h(x) = \frac{1}{n}\sum_{i=1}^n K_h (x - x_i) = \frac{1}{nh} \sum_{i=1}^n K\Big(\frac{x-x_i}{h}\Big), \]
Kernel density estimation of 100 normally distributed random numbers using different smoothing bandwidths.
where \(K(•)\) is the kernel — a non-negative function that integrates to one and has mean zero — and h > 0 is a smoothing parameter called the bandwidth. A kernel with subscript h is called the scaled kernel and defined as \(Kh(x) = 1/h K(x/h)\). Intuitively one wants to choose h as small as the data allow, however there is always a trade-off between the bias of the estimator and its variance; more on the choice of bandwidth below.
Comparison of the histogram (left) and kernel density estimate (right) constructed using the same data. The 6 individual kernels are the red dashed curves, the kernel density estimate the blue curves. The data points are the rug plot on the horizontal axis.
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