Simulation Formula Sheet Spring, 2003

 

Linear Congruential Random Number Generators. Xn+1 = aXn + c (mod m)

 

Inverse Transform Method U = F(X) or X = F-1(U)

 

Rejection Method. Pick c f(x)/g(x). Sample g(x) to get Y. Accept Y as a sample of f(x) if U f(Y)/cg(Y)

 

Squeeze Method: If g(x) = f(x)/ch(x), find gL(x) g(x) gH(x). To avoid calculating g(y) in the A-R method, test u gL(y) and u > gH(y)

Pr[Lo] = max{0,gL(x)}h(x)dx; Pr[Hi] = min{1,gH(x)}h(x)dx

 

Polar Method: v1 = 2u1 1, v2 = 2u2 1; s = v12 + v22 1

z1 = v1(-2lns/s)1/2; z2 = v2(-2lns/s)1/2

 

Choleski Method (Mulivariate Normal Distribution ). Given the covariance matrix = (sij), and the vector of means m = (m1,, mn)T:

Factor = CCT, where C is lower triangular.

Obtain a vector z = (z1,,zn)T with zi iid from standard normal

x = Cz + m samples the multivariate distribution.

 

Convolution Method: h(z) = f(z-w)g(w)dw is the pdf for Z = X + Y

 

Composition Method: f(x) = Saifi(x)

Hit or Miss. q =

Crude Monte Carlo.

Stopping Criteria. Pr( - q d) a. Use Chebyshev: Pr( - m ks/n) 1/k2. or Normal Approximation

 

Recursion formulas

k+1 = (kk + Xk+1)/(k+1)

Sk+12 = Sk2(k-1)/k + ( 1+k) (k+1 - k)2

 

Mean Square Error

MSE(F) = EF(g(X1,,Xn) - q(F))2

Bootstrap Approximation

MSE(Fe) = EFe(g(X1,,Xn) - q(Fe))2

 

Goodness of Fit

Discrete case: T = S(Ni npi)2/npi (Chi-square with m-1-k degrees of freedom)

Continuous case: D = MaxxFe(x) F(x) p-value = P(D d) is independent of F

 

Two-Sample Test

R = sum of ranks in 1st sample p-value = 2Min{P(R r), P(R r)}

Pn,m(r) = [n/(n+m)]Pn-1,m(r-n-m) + [m/(n+m)]Pn,m-1(r) , where

P1,0(k) = 0 if k 0; = 1 otherwise; P0,1(k) = 0 if k < 0, = 1 otherwise.

 

Kruskal-Wallace (multiple sample) R = (12/N(N+1))S(Ri Ni(N+1)/2)2/Ni