pset 4.1 Consider the $\operatorname{rv} X \sim \operatorname{Gamma}(\alpha, p)$, where $\alpha>0$ is the scale parameter and $p>0$ is the shape parameter, and with$$\begin{aligned}\mathrm{E}(X) & =\frac{p}{\alpha} \\\mathrm{E}\left(X^{2}\right) & =\frac{p(p+1)}{\alpha^{2}}\end{aligned}$$and assume you have the random sample $X_{1}, \ldots, X_{n}$.1. Find the method of moment estimators of $\alpha$ and $p$.2. By noticing that$$\begin{aligned}\mathrm{E}\left(\frac{1}{X}\right) & =\frac{\alpha}{p-1} \\\mathrm{E}(\log (X)) & =\Psi(p)-\log (\alpha)\end{aligned}$$write the criterion function for the two-step GMM estimator of $\boldsymbol{\theta}=(\alpha, p)^{\prime}$.

pset 3.3 Consider the regression model$$y_{i}=\boldsymbol{x}_{i}^{\prime} \boldsymbol{\beta}+\varepsilon_{i}, \quad \text { with } \quad \varepsilon_{i} \mid \boldsymbol{x}_{i}, \boldsymbol{z}_{i} \sim N\left(0, \sigma^{2} \exp \left(\boldsymbol{z}_{i}^{\prime} \boldsymbol{\gamma}\right)\right)$$where $\boldsymbol{z}_{i}$ is a vector of regressors not including an intercept and $\boldsymbol{\theta}=\left(\boldsymbol{\beta}^{\prime}, \boldsymbol{\gamma}^{\prime}, \sigma^{2}\right)^{\prime}$ is th vector of unknown parameters.1. Write the first-order conditions for the maximum likelihood estimation of $\boldsymbol{\theta}$.2. Identify the set of restrictions that makes $V\left(\varepsilon_{i} \mid \boldsymbol{x}_{i}, \boldsymbol{z}_{i}\right)$ constant across observations.3. Write down the formulation of the LM test statistic for the null hypothesis corresponding to the above set of restrictions.

Suppose X = (X1, . . . , Xn) is a random sample from the Gamma distribution withshape α and rate θ, i.e.,fX (x) = θαΓ(α)xα−1e−xθ, x > 0, α > 0, θ > 0.Each Xi has expectation E(X) = α/θ, variance Var(X) = α/θ2, and momentgenerating function MX (t) = (1 − t/θ)−α.(a) Assuming both α and θ to be unknown, write down the log likelihood functionℓ(θ, α; X) and the corresponding score functions ∂ℓ∂θ and ∂ℓ∂α .(b) Verify that each score function has zero expectation.(c) Assuming α to be known:(i) Calculate the Cramer Rao Lower Bound (CRLB) for the variance of anunbiased estimator of θ.(ii) Is there any unbiased estimator of θ whose variance attain the CRLB?(iii) Show thatS = α − 1nnXi=1 1Xiis an unbiased estimator for θ. What is the MVU estimator for θ?(iv) Identify a change of parameter η = η(θ) for which there exists an unbiasedestimator with variance attained the CRLB.(v) For the parameter in the previous part, identify the MVU estimator whosevariance attains the CRLB. Compute this variance

1. Suppose X = (X1, . . . , Xn) is a random sample from the Gamma distribution withshape α and rate θ, i.e.,fX (x) = θαΓ(α)xα−1e−xθ, x > 0, α > 0, θ > 0.Each Xi has expectation E(X) = α/θ, variance Var(X) = α/θ2, and momentgenerating function MX (t) = (1 − t/θ)−α.(a) Assuming both α and θ to be unknown, write down the log likelihood functionℓ(θ, α; X) and the corresponding score functions ∂ℓ∂θ and ∂ℓ∂α .(b) Verify that each score function has zero expectation.(c) Assuming α to be known:(i) Calculate the Cramer Rao Lower Bound (CRLB) for the variance of anunbiased estimator of θ.(ii) Is there any unbiased estimator of θ whose variance attain the CRLB?(iii) Show thatS = α − 1nnXi=1 1Xiis an unbiased estimator for θ. What is the MVU estimator for θ?(iv) Identify a change of parameter η = η(θ) for which there exists an unbiasedestimator with variance attained the CRLB.(v) For the parameter in the previous part, identify the MVU estimator whosevariance attains the CRLB. Compute this variance

The expected value of a discrete random variable ‘x’ is given byReview LaterP(x)∑ x P(x)∫ x P(x) dx∑ P(x)

This determines the height and width of the graph of the normal distribution.