Question

Problem 2 We will practice finding kernels and relating them to known distributions. The gamma function and the beta function will come up as well. (a) easyFind the kernel of the negative binomial PMF. (b) Jeasy Find the kernel of the beta PDF. (c) (easy) Find the kernel of the beta binomial PMF. -1,>o how would you know if the r.v. x was an Erlang (k. A) (d) Jeasy If kr) = or a Gamma (k, X)? (e) harder] If k(x) = xe +1.>, how is X distributed?

Answer 1

b) Let X1, ..., Xn be a random sample from a univariate distribution with density f having support I ≡ [0, 1]. Suppose it is desired to estimate f at each point 0 < x < 1 by means of a kernel-type estimator f_hat(x) = inv(n)*(sum(K(x, Xi ; h))) ,where, i =1(1)n

Xi in the summands of the beta kernel estimator:

fC1(x) = [1 / nB((x/h^2 ) + 1, {(1 − x)/h^2} + 1)]* sum( Xi^( x/h^2) * (1 − Xi)^( (1−x)/h^2)) ,where, i =1(1)n

where B(·, ·) is the beta function.

d) The gamma distribution generalizes the Erlang distribution by allowing k to be any positive real number, using the gamma function instead of the factorial function.

That is: if k is an integer and ${\displaystyle X\sim \operatorname {Gamma} (k,\lambda ),}$ then ${\displaystyle X\sim \operatorname {Erlang} (k,\lambda )}$

so the given k(x) is gamma.