Exercise 1

Let $$X_{1}, X_{2}, \ldots X_{n}$$ be a random sample of size $$n$$ from a distribution with probability density function

$f(x, \theta) = \frac{1}{\theta}e^{-x/\theta}, \quad x > 0, \ \theta > 0$

Note that, the moments of this distribution are given by

$E[X^k] = \int_{0}^{\infty} \frac{x^k}{\theta}e^{-x/\theta} = k! \cdot \theta^k.$

This will be a useful fact for Exercises 2 and 3.

(a) Obtain the maximum likelihood estimator of $$\theta$$, $$\hat{\theta}$$. (This should be a function of the unobserved $$x_i$$ and the sample size $$n$$.) Calculate the estimate when

$x_{1} = 0.50, \ x_{2} = 1.50, \ x_{3} = 4.00, \ x_{4} = 3.00.$

(This should be a single number, for this dataset.)

(b) Calculate the bias of the maximum likelihood estimator of $$\theta$$, $$\hat{\theta}$$. (This will be a number.)

(c) Find the mean squared error of the maximum likelihood estimator of $$\theta$$, $$\hat{\theta}$$. (This will be an expression based on the parameter $$\theta$$ and the sample size $$n$$. Be aware of your answer to the previous part, as well as the distribution given.)

(d) Provide an estimate for $$P[X > 4]$$ when

$x_{1} = 0.50, \ x_{2} = 1.50, \ x_{3} = 4.00, \ x_{4} = 3.00.$

Exercise 2

Let $$X_{1}, X_{2}, \ldots X_{n}$$ be a random sample of size $$n$$ from a distribution with probability density function

$f(x, \alpha) = \alpha^{-2}xe^{-x/\alpha}, \quad x > 0, \ \alpha > 0$

(a) Obtain the maximum likelihood estimator of $$\alpha$$, $$\hat{\alpha}$$. Calculate the estimate when

$x_{1} = 0.25, \ x_{2} = 0.75, \ x_{3} = 1.50, \ x_{4} = 2.5, \ x_{5} = 2.0.$

(b) Obtain the method of moments estimator of $$\alpha$$, $$\tilde{\alpha}$$. Calculate the estimate when

$x_{1} = 0.25, \ x_{2} = 0.75, \ x_{3} = 1.50, \ x_{4} = 2.5, \ x_{5} = 2.0.$

Exercise 3

Let $$X_{1}, X_{2}, \ldots X_{n}$$ be a random sample of size $$n$$ from a distribution with probability density function

$f(x, \beta) = \frac{1}{2 \beta^3} x^2 e^{-x/\beta}, \quad x > 0, \ \beta > 0$

(a) Obtain the maximum likelihood estimator of $$\beta$$, $$\hat{\beta}$$. Calculate the estimate when

$x_{1} = 2.00, \ x_{2} = 4.00, \ x_{3} = 7.50, \ x_{4} = 3.00.$

(b) Obtain the method of moments estimator of $$\beta$$, $$\tilde{\beta}$$. Calculate the estimate when

$x_{1} = 2.00, \ x_{2} = 4.00, \ x_{3} = 7.50, \ x_{4} = 3.00.$

Exercise 4

Let $$X_{1}, X_{2}, \ldots X_{n}$$ be a random sample of size $$n$$ from a distribution with probability density function

$f(x, \lambda) = \lambda x^{\lambda - 1}, \quad 0 < x < 1, \lambda > 0$

(a) Obtain the maximum likelihood estimator of $$\lambda$$, $$\hat{\lambda}$$. Calculate the estimate when

$x_{1} = 0.10, \ x_{2} = 0.20, \ x_{3} = 0.30, \ x_{4} = 0.40.$

(b) Obtain the method of moments estimator of $$\lambda$$, $$\tilde{\lambda}$$. Calculate the estimate when

$x_{1} = 0.10, \ x_{2} = 0.20, \ x_{3} = 0.30, \ x_{4} = 0.40.$