Let \(X_1, X_2, \ldots, X_n \stackrel{iid}{\sim} \text{Poisson}(\lambda)\). That is
\[ f(x \mid \lambda) = \frac{\lambda^xe^{-\lambda}}{x!}, \quad x = 0, 1, 2, \ldots \ \ \lambda > 0 \]
(a) Obtain a method of moments estimator for \(\lambda\), \(\tilde{\lambda}\). Calculate an estimate using this estimator when
\[ x_{1} = 1, \ x_{2} = 2, \ x_{3} = 4, \ x_{4} = 2. \]
(b) Find the maximum likelihood estimator for \(\lambda\), \(\hat{\lambda}\). Calculate an estimate using this estimator when
\[ x_{1} = 1, \ x_{2} = 2, \ x_{3} = 4, \ x_{4} = 2. \]
(c) Find the maximum likelihood estimator of \(P[X = 4]\), call it \(\hat{P}[X = 4]\). Calculate an estimate using this estimator when
\[ x_{1} = 1, \ x_{2} = 2, \ x_{3} = 4, \ x_{4} = 2. \]
Let \(X_1, X_2, \ldots, X_n \stackrel{iid}{\sim} N(\theta,\sigma^2)\).
Find a method of moments estimator for the parameter vector \(\left(\theta, \sigma^2\right)\).
Let \(X_1, X_2, \ldots, X_n \stackrel{iid}{\sim} N(1,\sigma^2)\).
Find a method of moments estimator of \(\sigma^2\), call it \(\tilde{\sigma}^2\).
Let \(X_1, X_2, \ldots, X_n\) be a random sample from a population with pdf
\[ f(x \mid \theta) = \frac{1}{\theta}x^{(1-\theta)/\theta}, \quad 0 < x < 1, \ 0 < \theta < \infty \]
(a) Find the maximum likelihood estimator of \(\theta\), call it \(\hat{\theta}\). Calculate an estimate using this estimator when
\[ x_{1} = 0.10, \ x_{2} = 0.22, \ x_{3} = 0.54, \ x_{4} = 0.36. \]
(b) Obtain a method of moments estimator for \(\theta\), \(\tilde{\theta}\). Calculate an estimate using this estimator when
\[ x_{1} = 0.10, \ x_{2} = 0.22, \ x_{3} = 0.54, \ x_{4} = 0.36. \]
Let \(X_1, X_2, \ldots, X_n\) iid from a population with pdf
\[ f(x \mid \theta) = \frac{\theta}{x^2}, \quad 0 < \theta \leq x \]
Obtain the maximum likelihood estimator for \(\theta\), \(\hat{\theta}\).
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
\[ x_{1} = 0.25, \ x_{2} = 0.75, \ x_{3} = 1.50, \ x_{4} = 2.5, \ x_{5} = 2.0. \]
Hint: Recall the probability density function of an exponential random variable.
\[ f(x \mid \theta) = \frac{1}{\theta}e^{-x/\theta}, \quad x > 0, \ \theta > 0 \]
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 hint will also be useful in the next exercise.
Let \(X_{1}, X_{2}, \ldots X_{n}\) be a random sample of size \(n\) from a distribution with probability density function
\[ f(x \mid \beta) = \frac{1}{2 \beta^3} x^2 e^{-x/\beta}, \quad x > 0, \ \beta > 0 \]
(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
\[ x_{1} = 2.00, \ x_{2} = 4.00, \ x_{3} = 7.50, \ x_{4} = 3.00. \]
Let \(Y_1, Y_2, \ldots, Y_n\) be a random sample from a distribution with pdf
\[ f(y \mid \alpha) = \frac{2}{\alpha} \cdot y \cdot \exp\left\{-\frac{y^2}{\alpha}\right\}, \ y > 0, \ \alpha > 0. \]
(a) Find the maximum likelihood estimator of \(\alpha\).
(b) Let \(Z_1 = Y_1^2\). Find the distribution of \(Z_1\). Is the MLE for \(\alpha\) an unbiased estimator of \(\alpha\)?
Let \(X\) be a single observation from a \(\text{Binom}(n, p),\) where \(p\) is an unknown parameter. (In this case, we will consider \(n\) known.)
(a) Find the maximum likelihood estimator (MLE) of \(p\).
(b) Suppose you roll a 6-sided die 40 times and observe eight rolls of a 6. What is the maximum likelihood estimate of the probability of observing a 6?
(c) Using the same observed data, suppose you now plan to perform a second experiment with the same die, and will roll the die 5 more times. What is the maximum likelihood estimate of the probability that you will observe no 6’s in this next experiment?
Suppose that a random variable \(X\) follows a discrete distribution, which is determined by a parameter \(\theta\) which can take only two values, \(\theta = 1\) or \(\theta = 2\). The parameter \(\theta\) is unknown.
Now suppose we observe \(X = 3\). Based on this data, what is the maximum likelihood estimate of \(\theta\)?
Let \(Y_1, Y_2, \ldots, Y_n\) be a random sample from a population with pdf
\[ f(y \mid \theta) = \dfrac{2\theta^2}{y^3}, \ \ \theta \le y < \infty \]
Find the maximum likelihood estimator of \(\theta.\).