Generate a sample of 1,000 independent observations from a gamma distribution with scale ? = 2 and shape k = 3

and giving R code as

n = 1000 x = rgamma(n,shape = 3,scale = 2) hist(x,freq = FALSE,col = 'blue') lines(density(x),lwd =3,col = 'red')

for question

c. Construct a Kernel density estimator, f(x), assuming a rectangular kernel, from scratch (that is, without using in-built functions from R packages for constructing a Kernel density estimator). . To determine h, determine the bin-width used in the histogram from your preferred rule in b. and halve it. . The lower limit for x will be 0 and upper limit max + h. . Compare the KDE to the underlying distribution.. Sturges Rule: bw = Tmax -Tmin. where n is the sample size. log2 (n)+1 , . Scott's Rule: bw = 3.50 1/3, where o is the standard deviation of the data. . Freedman-Diaconis Rule: bw = 2(975-925) n1/3 where 975, 925 are the 75 % and 25 % quantile of the data