statistical modelling team that is developing models based on independent non identically distributed random variables. You consider the following model - Gamma(awi, B), i = {1, ..., N} where a, B, and w1,.. ,wN are positive scalar parameters (the gamma distribution has several common parametrisations, here we define the p.d.f. of a random variable X ~ Gamma(a, B) as f(x) https://mathworld.wolfram.com/GammaFunction.html). a a-exp{-Bx} for x > 0, where r(-) denotes the gamma function, see You have been asked to explore the statistical properties of the maximum likelihood estimator of B assuming that a and wi,...,WN are known. Assume that a > 2 andE,wi > 1. Let X = X1, ..., XN. You decide to perform the following analyses: 1. State the likelihood function L(B; X) = f(X; B). 2. Derive the maximum likelihood estimator for 3, denoted by B(X). 3. Show that the bias of B(X) is given by B(B) = B/(a Ewi - 1). 4. Derive the expression for the Cramer Rao Lower Bound for biased estimator of 3 with bias B(B) = B/(aE1wi - 1). %D 5. Let a = 4 and w; = i for i = {1,. , N}. Use simulation in R to calculate the variance of B(X) for B = {1,2, 4, 8, 16} and for different sample sizes N E [1, 20]. Briefly discuss your results and present appropriate graphical summaries. How close are the variances of B(X) to the theoretical lower bound found in Part 4?
statistical modelling team that is developing models based on independent non identically distributed random variables. You consider the following model - Gamma(awi, B), i = {1, ..., N} where a, B, and w1,.. ,wN are positive scalar parameters (the gamma distribution has several common parametrisations, here we define the p.d.f. of a random variable X ~ Gamma(a, B) as f(x) https://mathworld.wolfram.com/GammaFunction.html). a a-exp{-Bx} for x > 0, where r(-) denotes the gamma function, see You have been asked to explore the statistical properties of the maximum likelihood estimator of B assuming that a and wi,...,WN are known. Assume that a > 2 andE,wi > 1. Let X = X1, ..., XN. You decide to perform the following analyses: 1. State the likelihood function L(B; X) = f(X; B). 2. Derive the maximum likelihood estimator for 3, denoted by B(X). 3. Show that the bias of B(X) is given by B(B) = B/(a Ewi - 1). 4. Derive the expression for the Cramer Rao Lower Bound for biased estimator of 3 with bias B(B) = B/(aE1wi - 1). %D 5. Let a = 4 and w; = i for i = {1,. , N}. Use simulation in R to calculate the variance of B(X) for B = {1,2, 4, 8, 16} and for different sample sizes N E [1, 20]. Briefly discuss your results and present appropriate graphical summaries. How close are the variances of B(X) to the theoretical lower bound found in Part 4?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question
![You are a statistical trainee actuary working for an insurance company. You are part of a
statistical modelling team that is developing models based on independent non identically
distributed random variables. You consider the following model
X; ~ Gamma(awi, B), i= {1,..., N}
where a, B, and w1,..,WN are positive scalar parameters (the gamma distribution has several
common parametrisations, here we define the p.d.f. of a random variable X ~
as f(x)
https://mathworld.wolfram.com/GammaFunction.html).
Gamma(a, B)
Hora-1 exp{-Bx} for x > 0, where r(-) denotes the gamma function, see
You have been asked to explore the statistical properties of the maximum likelihood estimator
of B assuming that a and wi,... ,WN are known. Assume that a > 2 andE,wi > 1.
Let X = X1,..., XN. You decide to perform the following analyses:
1. State the likelihood function L(B; X) = f(X; B).
2. Derive the maximum likelihood estimator for 3, denoted by B(X).
3. Show that the bias of 3(X) is given by B(B) = B/(a Ewi – 1).
4. Derive the expression for the Cramer Rao Lower Bound for biased estimator of 3 with
bias B(B) = B/(a Ewi – 1).
5. Let a = 4 and w; = i for i = {1,…, N}. Use simulation in R to calculate the variance
of B(X) for 3 = {1,2,4,8, 16} and for different sample sizes N E [1,20].
Briefly discuss your results and present appropriate graphical summaries. How close
are the variances of B(X) to the theoretical lower bound found in Part 4?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab68ef6f-9760-4c1f-88a8-6e5ceb2168b2%2F04f949b4-bd0f-4788-a786-e7dfa4fbf1ee%2Fxnp9ch_processed.jpeg&w=3840&q=75)
Transcribed Image Text:You are a statistical trainee actuary working for an insurance company. You are part of a
statistical modelling team that is developing models based on independent non identically
distributed random variables. You consider the following model
X; ~ Gamma(awi, B), i= {1,..., N}
where a, B, and w1,..,WN are positive scalar parameters (the gamma distribution has several
common parametrisations, here we define the p.d.f. of a random variable X ~
as f(x)
https://mathworld.wolfram.com/GammaFunction.html).
Gamma(a, B)
Hora-1 exp{-Bx} for x > 0, where r(-) denotes the gamma function, see
You have been asked to explore the statistical properties of the maximum likelihood estimator
of B assuming that a and wi,... ,WN are known. Assume that a > 2 andE,wi > 1.
Let X = X1,..., XN. You decide to perform the following analyses:
1. State the likelihood function L(B; X) = f(X; B).
2. Derive the maximum likelihood estimator for 3, denoted by B(X).
3. Show that the bias of 3(X) is given by B(B) = B/(a Ewi – 1).
4. Derive the expression for the Cramer Rao Lower Bound for biased estimator of 3 with
bias B(B) = B/(a Ewi – 1).
5. Let a = 4 and w; = i for i = {1,…, N}. Use simulation in R to calculate the variance
of B(X) for 3 = {1,2,4,8, 16} and for different sample sizes N E [1,20].
Briefly discuss your results and present appropriate graphical summaries. How close
are the variances of B(X) to the theoretical lower bound found in Part 4?
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