Stratified Sampling: Let U1, . . . , Un be independent random numbers and set ¯

Ui = (Ui + i − 1), i = 1, . . . , n. Hence, ¯Ui, i ≥ 1, is uniform on

((i − 1), i).

n i=1 g( ¯Ui) is called the stratified sampling estimator of 1 0 g(x) dx.

(a) Show that E[

n i=1 g( ¯Ui)] =

1 0 g(x) dx.

(b) Show that Var[

n i=1 g( ¯Ui)] ≤ Var[

n i=1 g(Ui)].

Hint: Let U be uniform (0, 1) and define N by N = i if (i − 1) i, i = 1, . . . , n. Now use the conditional variance formula to obtain

Transcribed Image Text:

Var(g(U)) =E[Var(g (U)|N)] + Var(E[g(U)|N]) = E[Var(g(U)|N)] n Var(g(U)|N=i) n Var[g (;)] = n n i=1 i=1