3. Kernel density estimation is a nonparametric method which can be used to estimate a true probability density function f(x). (a) i. State the formula for f(x), the kernel density estimate of f(x). Explain all notation in the formula. [5 marks] ii. The triangular kernel can be written as w (t) = (1 |t|), |t|1. - List any three of the properties this triangular kernel function shares with the Epanechnikov kernel. [6 marks] iii. For the data given below, show that the estimated density using the triangular kernel with a bandwidth of 2 evaluates to 1/13 for f(9). Show your workings by hand. 5 6 7 8 9 1 2 3 4 10 11 12 13 7 10 2.5 7.5 6.5 8.5 6.5 7.5 7.5 7 4.5 4.5 4 Xi