Implement a goodness-of-fit test for the Exponential distribution (see Example 4.2) in R. See the help topics for rexp (random exponential generator), dist (Euclidean distances) and replicate (utility to construct the loop for the replicates).

Example 4.2

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Example 4.2 (Exponential Distribution). In Example 3.1 we considered a goodness-of-fit test for a two-parameter exponential distribution with density f(z) = \e-\(z-u), , > 0. The expected distances for that example were E|x - X| = x-+ (1 2Fo(x)), - EX - X'| = I ; where Fo(r) is the cumulative distribution function (cdf) of X under Ho. Suppose that we have observed a random sample and want to test the simple hypothesis that the sampled distribution is Exponential with rate A and p = 0. Then En (X; X) Fo(x)=1-e-Ax I 0. If I,..., In is a random sample, the energy distance between the sampled distribution and the hypothesized Exponential (A) distribution Fo is n n 2 1 == - [2 + (1 - 250 (2))] - - / -. xi 2Fo(r)) 1 1 X n |Ti n i=1 i,j=1 The test statistic is n&n (X; X). Here we have a univariate sample so by Propo- sition 20.1, j-1-xj| = 2 k=1 (2k 1 n)x(k), where x(1) (n) is the ordered sample. To implement an energy goodness-of-fit test, we use a parametric simu- lation from the hypothesized distribution to obtain a reference distribution. The test procedure is much simpler by comparison than the permutation test for equal distributions in Section 4.3.