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1. Recall the setting of Question 1 in Assignment 2. That is that we have an iid sample assumed to come from an exponential distribution where the true rate parameter is unknown i.e. X₁,..., Xn~ Exponential(o) such that fo(x) = Ao exp(-x) for x>0 You may cite any answers to anything from assignment 2 to avoid re-calculating things here. (a) Assume that the sample size n is very large. State the asymptotic distri- bution that the MLE À takes as a function of Xo. (b) Notice that the Variance of this asymptotic distribution is a function of the true parameter Ao. State an appropriate approximation that we can make and the resulting (approximate) asymptotic distribution of Â. (c) By using the large sample approximate sampling distribution from the previous question for Â, provide an expression as a function of n and  for Pxo (A - Ao|> 0.5)

1. Recall the setting of Question 1 in Assignment 2. That is that we have an iid sample assumed to come from an exponential distribution where the true rate parameter is unknown i.e. X₁,..., Xn~ Exponential(o) such that fo(x) = Ao exp(-x) for x>0 You may cite any answers to anything from assignment 2 to avoid re-calculating things here. (a) Assume that the sample size n is very large. State the asymptotic distri- bution that the MLE À takes as a function of Xo. (b) Notice that the Variance of this asymptotic distribution is a function of the true parameter Ao. State an appropriate approximation that we can make and the resulting (approximate) asymptotic distribution of Â. (c) By using the large sample approximate sampling distribution from the previous question for Â, provide an expression as a function of n and  for Pxo (A - Ao|> 0.5)

MATLAB: An Introduction with Applications
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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1. Recall the setting of Question 1 in Assignment 2. That is that we have an iid sample assumed to come from an exponential distribution where the true rate parameter is unknown i.e. X₁,..., Xn~ Exponential (Ao) such that fao(x)= Ao exp(-Xox) for x>0 You may cite any answers to anything from assignment 2 to avoid re-calculating things here. (a) Assume that the sample size n is very large. State the asymptotic distri- bution that the MLE À takes as a function of Xo. (b) Notice that the Variance of this asymptotic distribution is a function of the true parameter Ao. State an appropriate approximation that we can make and the resulting (approximate) asymptotic distribution of Â. (c) By using the large sample approximate sampling distribution from the previous question A, provide an expression as a function of n and  for PA, (| - Ao > 0.5) (d) Suppose that a reparametrisation is made such that µ = g(X) = 1/λ. By using the Delta method derive an expression for the approximate large sample distribution of in terms of û and n. û (e) Similarly to (c) but now in terms of the new reparametrisation find an expression for the following in terms of û and n PHO (μ- Mol 0.5) (f) You may use without proof that the sum of n independent Exponential(3) random variables is distributed as a Gamma(n, 3) with density: xn-le-Bx for x > 0 f(x) = Bn T(n) (i) State the MLE û for u. (ii) Identify the true distribution for û in terms of μo. (iii) Suppose that an oracle tells you that po = 25 and you are seeing a sample of size n = 100. Calculate the TRUE value of Po (Hol> 0.5) You may find the command pgamma(.) in R to be useful. (iv) However, we don't know Ho in practice so you wouldn't actually be able to make that calculation in (iii). Using the result in (e) and assuming that the observed MLE was μ = 21.79 find an (approximate) value for Po (μ- Ho|> 0.5) (v) It is proposed that the true value of µ is 23. Construct an appropriate hypothesis test to assess this conjecture and evaluate the correspond- ing p-value. (g) Assume that you are working again in the parametrisation of A. Suppose that the sample size is large and that the asymptotic normality of the MLE applies. Using your working from (c) (so that you don't need to repeat calculations) Find (as a function of A and n) the value of the constant a such that P(| A₂ > a) = 0.05
Transcribed Image Text:1. Recall the setting of Question 1 in Assignment 2. That is that we have an iid sample assumed to come from an exponential distribution where the true rate parameter is unknown i.e. X₁,..., Xn~ Exponential (Ao) such that fao(x)= Ao exp(-Xox) for x>0 You may cite any answers to anything from assignment 2 to avoid re-calculating things here. (a) Assume that the sample size n is very large. State the asymptotic distri- bution that the MLE À takes as a function of Xo. (b) Notice that the Variance of this asymptotic distribution is a function of the true parameter Ao. State an appropriate approximation that we can make and the resulting (approximate) asymptotic distribution of Â. (c) By using the large sample approximate sampling distribution from the previous question A, provide an expression as a function of n and  for PA, (| - Ao > 0.5) (d) Suppose that a reparametrisation is made such that µ = g(X) = 1/λ. By using the Delta method derive an expression for the approximate large sample distribution of in terms of û and n. û (e) Similarly to (c) but now in terms of the new reparametrisation find an expression for the following in terms of û and n PHO (μ- Mol 0.5) (f) You may use without proof that the sum of n independent Exponential(3) random variables is distributed as a Gamma(n, 3) with density: xn-le-Bx for x > 0 f(x) = Bn T(n) (i) State the MLE û for u. (ii) Identify the true distribution for û in terms of μo. (iii) Suppose that an oracle tells you that po = 25 and you are seeing a sample of size n = 100. Calculate the TRUE value of Po (Hol> 0.5) You may find the command pgamma(.) in R to be useful. (iv) However, we don't know Ho in practice so you wouldn't actually be able to make that calculation in (iii). Using the result in (e) and assuming that the observed MLE was μ = 21.79 find an (approximate) value for Po (μ- Ho|> 0.5) (v) It is proposed that the true value of µ is 23. Construct an appropriate hypothesis test to assess this conjecture and evaluate the correspond- ing p-value. (g) Assume that you are working again in the parametrisation of A. Suppose that the sample size is large and that the asymptotic normality of the MLE applies. Using your working from (c) (so that you don't need to repeat calculations) Find (as a function of A and n) the value of the constant a such that P(| A₂ > a) = 0.05
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