Exercise 2

Question 1.

Let y1,...,yn be a random sample from the following distributions with the unknown parameter(s). Estimate them by maximum likelihood and by the method of moments.
  1. double exponential (Laplace) distribution fθ(y)=(θ/2)e-θ|y|, θ>0
  2. Pareto distribution fα,β(y)=(α/β) (β/y)α+1, y > β; α, β > 0

Question 2.

Consider a population with three kinds of individuals labeled 1, 2 and 3 occuring in the following proportions: p1=p2, p2=2p(1-p), p3=(1-p)2, 0 < p < 1 (can you give an example from genetics where such a model may be relevant?). In a random sample of size n chosen from this population, n1 individuals have been labeled 1, n2 - 2 and n3 - 3 respectively (n1+n2+n3=n). Write down the likelihood function and find the MLE of p.

Question 3.

Consider the following model:
yi=ayi-1+εi, i=1,...,n,
where y0=0, εi~N(0,σ2) and i.i.d., known as the first order autoregressive process AR(1). Write down the likelihood function and find the MLEs of a and σ2.

Question 4.

The number of cars passing a certain cross-roads during the day is a Poisson random variable with the known average number of λ cars per day. An electronic counter was fixed on the cross-roads to count the number of cars passing it every day. Every midnight the data of that day, yi, was automatically read by computer and the counter was set to zero for a new day data. Unfortunately, as it usually happens, the counter was not perfect and with some (unknown) probability p did not renew counting after reseting and simply showed "0" at the end of the following day. Given the counts of n days, y1,...,yn,
  1. Write down the likelihood function.
  2. Find the sufficient statistic for p.
  3. Find the MLE for p.
  4. Estimate p by the method of moments.
  5. Assume now that λ is also unknown. Find the MLEs for λ and p or give the form of the iteration procedure by Newton-Raphson and Fisher scoring algorithms if the closed solutions are unavailable.