$(30$pts$)$ The $k-$nearest neighbor decision rule is one of the simplest decision rules to implement and has a good performance in practice. Answer the following questions.

$($a$)$ Suppose the Bayes error rate for a $c=3$ category classification problem

is $5\%.$ What are the bounds on the error rate of a nearest$-$neighbor

classifier trained with an "infinitely large" training set?

$($b$)$ How should we select the value of $k?$

$($c$)$ One of the well$-$known methods to reduce the complexity of the nearest

neighbor classifier is called editing, pruning or condensing. $($i$)$ Briefly describe

this nearest$-$ neighbor editing procedure. $($ii$)$ Illustrate the editing procedure

with a simple $2-$class $2-$dimensional data set.