A1: Find the MLE of the unknown parameter θ ∈ (0,1) based on one observation of X ∼ Bin(n,θ). Are there particular values of the data x for which the MLE does not exist?
A2: Find the MLE of the unknown positive parameter θ from a random sample of size n from the continuous uniform distribution on [0,θ].
A3: Consider independent random variables with:
Xi ∼ N(µ1,σ2), i = 1,···,n1, Yj ∼ N(µ2,σ2), j = 1,···,n2,
Formulate the statistical model and derive the MLE of the unknown parameters.
A4: The Lognormal distribution is sometimes used to model positive quantities and it has density
Compute MLEs of µ and σ based on the leukemia survival data described in class. Compare the fit of the Lognormal model with that of the Gamma and Weibull models: which model appears to give the best fit for the survival data?
B1: Consider a random sample from the family of distributions defined by the pdf
where x0 is a known positive value ans α is an unknown positive parameter. Find the MLE of α
B2: In R, get hold of the speed-of-light measurements by typing: sp = morley[["Speed"]]. Use the method of maximum likelihood to fit both a normal and a Gamma model to the data. Which model gives the best fit?
