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lecturenotes08 - MS&E 223 Simulation Brad Null Lecture...

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MS&E 223 Lecture Notes #8 Simulation Estimating Non-linear Functions of Means Brad Null Spring Quarter 2008-09 Page 1 of 7 Estimating Nonlinear Functions of Means Back at the beginning of the course, we discussed how to obtain point estimates and confidence intervals for quantities of the form μ = E[X]. (E.g., X is the random reward from a play of the gambling game.) In many applications, we need to estimate quantities of the form 1 2 d g( , , , ) α = μ μ μ K , where g is a nonlinear function and (i) i E[X ] μ = for 1 i d. To keep notation simple, we will focus on the case k = 2, and look at quantities of the form X Y g( , ) α = μ μ with X E[X] μ = and Y E[Y] μ = . 1. Examples Example 1 : Suppose that one is running a retail outlet in which one wishes to compute the long-run average revenue generated per customer. Let i R denote the revenue generated on day i, and set i i X R = i Y = total number of customers on day i. It is often reasonable to assume that the i i (X ,Y ) pairs are i.i.d. as a random pair (X,Y). Then, the average revenue over n days is equal to n n n 1 n 1 Y X = Y + ... Y X + ... X + + where = = n 1 i i n X n 1 X and = = n 1 i i n Y n 1 Y . Observe that Y X n n Y X μ μ with probability 1 as n by the SLLN (applied to both the numerator and demoninator). Thus, the long-run average revenue per customer, α, takes the form X Y g( , ) α = μ μ , where g(x, y) = x/y. Here, the (X i , Y i ) replicates can be simulated by repeatedly and independently simulating one day’s operation of the retail outlet. Example 2 : Suppose that one wishes to estimate the variance of the revenue generated per day. As above, suppose that { } i R :i 1 are i.i.d. as a random variable R. Set X = 2 R and Y = R Then we wish to compute ( 29 2 2 X Y E[R ] E[R] g( , ) α = - = μ μ , where 2 g(x,y) x y = - .
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MS&E 223 Lecture Notes #8 Simulation Estimating Non-linear Functions of Means Brad Null Spring Quarter 2008-09 Page 2 of 7 2. Taylor-Series Approach (Delta-Method) We need to assume here that g is continuously differentiable in a neighborhood of the point X Y ( , ) μ μ , i.e., g is continuous and its partial derivatives exist and are continuous. Point Estimate : To estimate α , simulate n i.i.d. replicates 1 1 n n (X ,Y ), ,(X ,Y ) K of (X, Y). Use the natural estimator ( n n n g X ,Y α = . Observe that, in general, n n n n n X Y E[ ] E[g(X ,Y )] g(E[X ],E[Y ]) g( , ) α = = μ μ = α , because g is nonlinear.
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