The number of casualty insurance claims that will be made to a branch office nex
ID: 3847489 • Letter: T
Question
The number of casualty insurance claims that will be made to a branch office next week depends on an environmental factor U. If the value of this factor is U = u, then the number of claims will have a Poisson distribution with mean 15/0.5+uAssuming that U is uniformly distributed over (0, 1 ), let p denote the pro0b.5a+bui lity that there will be at least 20 clar•r ns next week . (a) Explain how to obtain the raw simulation estimator of p. (b) Develop an efficient simulatiop estimator that uses conditional expectation along with a control variable. (c) Develop an efficient simulation estimator that uses conditional expectation and antithetic variables. (d) Write a program to determine the variance of the estimators in parts (a), (b), and (c). The number of casualty insurance claims that will be made to a branch office next week depends on an environmental factor U. If the value of this factor is U = u, then the number of claims will have a Poisson distribution with mean 15/0.5+uAssuming that U is uniformly distributed over (0, 1 ), let p denote the pro0b.5a+bui lity that there will be at least 20 clar•r ns next week . (a) Explain how to obtain the raw simulation estimator of p. (b) Develop an efficient simulatiop estimator that uses conditional expectation along with a control variable. (c) Develop an efficient simulation estimator that uses conditional expectation and antithetic variables. (d) Write a program to determine the variance of the estimators in parts (a), (b), and (c). The number of casualty insurance claims that will be made to a branch office next week depends on an environmental factor U. If the value of this factor is U = u, then the number of claims will have a Poisson distribution with mean 15/0.5+uAssuming that U is uniformly distributed over (0, 1 ), let p denote the pro0b.5a+bui lity that there will be at least 20 clar•r ns next week . (a) Explain how to obtain the raw simulation estimator of p. (b) Develop an efficient simulatiop estimator that uses conditional expectation along with a control variable. (c) Develop an efficient simulation estimator that uses conditional expectation and antithetic variables. (d) Write a program to determine the variance of the estimators in parts (a), (b), and (c).Explanation / Answer
Explain how to obtain the raw simulation estimator of p.
We simulate U , and then simulate a Poisson random variable V with mean 15 base of 0.5+ u .
A raw simulation estimator of p is
I = 1 'if V 20
0 if V < 20
b)
An ecient simulation estimator that uses conditional expectation is
E ( I | U ) = P ( V 20 | U )
P ( V 20 | U )
= 1 - P ( V 19 | U)
1 - 19 k =0 exp( - 15 0.5+ U )( 15 base 0.5+ U ) k
t is known that the CDF of Poisson distribution is decreasing in terms of its parameter, i.e. its mean. Therefore an ecient simulation estimator that uses conditional expectation along with a control variable is
E ( I | U )-c ( 15 base 0.5 +U - E [ 15 base 0.5 + U ])
E ( I | U )-c ( U-0. 5)
c)
1- power 19 base k =0 exp( - 15 base 0.5+ U )( 15 base 0.5+ U ) ^k
1- power 19 base k =0 exp( - 15 base 0.5- U )( 15 base 0.5- U ) ^k
have the same distribution but are negatively correlated, an ecient simulation estimator that uses conditional expectation and antithetic variables is
1 /2 [1 - power 19 base k =0 exp( - 15 base 0.5+ U )( 15 base 0.5+ U ) ^k +1