1. Recall the setting of Question 1 in Assignment 2. That is that we have an iidsample assumed to come from an exponential distribution where the true rateparameter is unknown i.e. X₁,..., Xn~ Exponential (Ao) such thatfao(x)= Ao exp(-Xox) for x>0You may cite any answers to anything from assignment 2 to avoid re-calculatingthings 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 ofthe true parameter Ao. State an appropriate approximation that we canmake and the resulting (approximate) asymptotic distribution of Â.(c) By using the large sample approximate sampling distribution from theprevious question A, provide an expression as a function of n and Â forPA, (|Â - Ao > 0.5)(d) Suppose that a reparametrisation is made such that µ = g(X) = 1/λ. Byusing the Delta method derive an expression for the approximate largesample distribution of in terms of û and n.û(e) Similarly to (c) but now in terms of the new reparametrisation find anexpression for the following in terms of û and nPHO (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 > 0f(x) =BnT(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 asample of size n = 100. Calculate the TRUE value ofPo (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 beable 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 forPo (μ- Ho|> 0.5)(v) It is proposed that the true value of µ is 23. Construct an appropriatehypothesis test to assess this conjecture and evaluate the correspond-ing p-value.(g) Assume that you are working again in the parametrisation of A. Supposethat the sample size is large and that the asymptotic normality of the MLEapplies. Using your working from (c) (so that you don't need to repeatcalculations) Find (as a function of A and n) the value of the constant asuch thatP(|Â A₂ > a) = 0.05

“Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

(a) Assume that the sample size n is very large. State the asymptotic distri- bution that the MLE Â takes as a function of Ao. (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 A, provide an expression as a function of n and Â for PAO (A - Ao > 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 û 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 Po (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 Bn T(n) f(x) (i) State the MLE û for u. (ii) Identify the true distribution for p 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 PHO (μ- Ho> 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 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 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 P₁(A - An> a) = 0.05