Problem set 7: Methods to compute Estimators

TipProblem 1

Find the method of moments estimators for the parameters of the following distributions:

1. \(\operatorname{Pois}(\lambda)\);

2. \(\operatorname{Geom}(p)\);

3. \(\operatorname{Beta}(\alpha,\beta)\);

4. \(\operatorname{U}(a,b)\);

5. \(\Gamma(\alpha,\beta)\);

6. \(\operatorname{Bin}(m,p)\);

7. \(\mathcal N(a,\sigma^2)\);

8. Pareto \(P(\gamma)\);

9. Cauchy \(C(\theta)\).

TipProblem 2

Find the maximum likelihood estimators for the parameters of the following distributions:

1. \(\operatorname{Geom}(p)\);

2. \(\operatorname{U}(0,a)\);

3. \(\mathcal N(a,\sigma^2)\) in each of the following three cases: when only one of the parameters is unknown, and when both parameters are unknown;

4. the parameter \(\lambda\) of the \(\operatorname{Gamma}(\alpha,\lambda)\) distribution, assuming that \(\alpha\) is known;

5. \(\operatorname{Bin}(n,p)\).

TipProblem 3

Find the maximum likelihood estimator for the location parameter in the Cauchy distribution model \[ p_\theta(x) = \frac{1}{\pi\bigl(1+(x-\theta)^2\bigr)}, \] when the sample consists of

1. one observation;

2. two observations (i.e., \(n=1,2\)).

TipProblem 4

Construct an asymptotically normal estimator for the scale parameter in the Cauchy distribution model \[ p_\theta(x) = \frac{\theta}{\pi(\theta^2+x^2)}. \]

TipProblem 5

Propose an asymptotically normal estimator of the parameter \(\theta^2\) in the location Cauchy distribution model (see Problem 3.4 from the problem set), and find its asymptotic variance.