(Solved): Kernel density estimation is a nonparametric method which can be used to estimate a true probabilit ...

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 $\hat{f}(x)$, the kernel density estimate of $f(x)$. Explain all notation in the formula. [5 marks ii. The triangular kernel can be written as $w(t)=(1-|t|),|t|\le 1$ 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 $\hat{f}(9)$. Show your workings by hand.