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 fo(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 for Â, provide an expression as a function of n and Â for PA(Â - Xol> 0.5) (d) Suppose that a reparametrisation is made such that μ = g(X) = 1/A. By using the Delta method derive an expression for the approximate large sample distribution of in terms of and n. (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(Hol> 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: f(x) = Bn I(n)` (i) State the MLE û for u. n-¹e² e-Ba for x > 0 (ii) Identify the true distribution for in terms of po. (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 Pμo (μ- Hol > 0.5) You may find the command pgamma(.) in R to be useful. (iv) However, we don't know o 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 21.79 find an (approximate) value for Pμo (Hol> 0.5) (v) It is proposed that the true value of u 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 X. 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 PA(Â - Ao > a) = 0.05

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 fo(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 for Â, provide an expression as a function of n and Â for PA(Â - Xol> 0.5) (d) Suppose that a reparametrisation is made such that μ = g(X) = 1/A. By using the Delta method derive an expression for the approximate large sample distribution of in terms of and n. (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(Hol> 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: f(x) = Bn I(n)` (i) State the MLE û for u. n-¹e² e-Ba for x > 0 (ii) Identify the true distribution for in terms of po. (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 Pμo (μ- Hol > 0.5) You may find the command pgamma(.) in R to be useful. (iv) However, we don't know o 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 21.79 find an (approximate) value for Pμo (Hol> 0.5) (v) It is proposed that the true value of u 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 X. 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 PA(Â - Ao > a) = 0.05

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter6: Exponential And Logarithmic Functions

Section6.1: Exponential Functions

Problem 60SE: The formula for the amount A in an investmentaccount with a nominal interest rate r at any timet is...

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