4 (MLE of ARCH) Suppose rt follows ARCH(2) with E(rt) = 0, rt = ut, ut = στει, σε where {+} is a sequence of independent and identically distributed (iid) standard normal random variables. With observations r₁,...,, write down the log-likelihood function for the model esti- mation.
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- Now, suppose you run an Auto Arima and you find R gives you the following model: Identify the ARIMA (p,d,q) i.e how many p terms are there? How many q terms? Coefficients: ar1 ma1 ma2 -0.7921 -0.0970 -0.3945 s.e. 0.1396 0.1802 0.1580 sigma^2 estimated as 0.1834: log likelihood=-44.07 AIC=96.14 AICc=96.68 BIC=105.61Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s^2is known.Find the maximum likelihood estimator of b from first principles.Find the Score function, the estimating equation and the information matrix using GLMThe number of ice creams sold per hour from Mr Fishy’s van is observed to be a Poisson random variable with parameter λ = 8. Each ice cream costs two pounds but Mr Fishy has to pay five pounds per hour for the pitch. What is the variance of his hourly profit (measured in pounds and ignoring all other income and expenses)?
- Let X1, ..., Xn be a sample from an exponential population with parameter λ.(a) Find the maximum likelihood estimator for λ. (b) Is the estimator unbiased?(c) Is the estimator consistent?The useful life of a concrete built on the seashore has a Weibull distribution of parameter scale δ> 0 year and power β = 2. (δ=21). Determine a) The average life of concrete EXb) The variance VXc) The probability that the concrete lasts more than 10 years; P (X> 10)Q3) Let X1,X2,. X, be a random sample from the truncated exponential distribution with pdf: f(x,0) = Please else (a) Derive the method of moment estimator of 0; (b) Derive the MLE of 0. (c) Are the MOME and the MLE unbiased estimators for 0? (d) Compare the MSE of the MOME and the MLE for 0. Which one is a better estimator for 0
- The number of offspring of an organism is a discrete random variable withmean μ and variance σ^2. Each of its offspring reproduces in the same manner. Find the expected number of offspring in the third generation and its variance.Is it maximal model, logistic regression or null model?If X ∼ Gamma(2, 5), find E(X^2 ) andE(X^4 ) Do I simply use the pdf for gamma square it, then find the expected value of that?
- Suppose Y1,.,Yn are independent random variables with probability mass function f(y) = p"(1 – p)*-", y = 0,1. Here, p E (0, 1), but unknown. Suppose Y = Lisi is the sample mean. Then, show that the Maximum Likelihood Estimator (MLE) of p is p = Y. What is the V(Y)? Now, give the MLE of V(Y).A jar contains 7 gold, 5 silver, 4 blue, 3 red, and 1 green marbles. Two marbles are to be randomly drawn from the jar. What is the probability a sliver marble is drawn, not returned to the jar, and then a blue marble is drawn? Given the power regression model y = 25x^1.2, which is the linear regression model after transforming to a log-log graph?Find the least square estimate for beta (the slope) given that its y-intercept is 0. Y_i = BetaX_i + e_i where e_i are independent and identically distributed N(0, variance_e) randon variables i = 1,...,2.

