Let Y1, . . . , Yniid∼ f (y; θ). For the following densities f , find:• the Fisher information I(θ) = −nE( d2d2θ log f (Yi; θ));• the Cramer-Rao lower bound for an unbiased estimator of θ.(a) Poisson: f (y; θ) = θy exp{−θ}y! for y ∈ {0, 1, 2, . . . } and θ > 0

Let Y1, . . . , Yn iid∼ f (y; θ). For the following densities f , find:• the moment estimator of θ;• the MLE of θ;(a) Poisson: f (y; θ) = θy exp{−θ}y! for y ∈ {0, 1, 2, . . . } and θ > 0

The Cramér-Rao Lower Bound (CRLB) provides a lower bound on the variance of unbiased estimators of a parameter. To find the CRLB for the variance of an unbiased estimator of the parameter \(\theta\), we need to calculate the Fisher information \(I(\theta)\) for a single observation from the given probability density function (pdf) \(f(x;\theta)\). The Fisher information is given by: \[ I(\theta) = -E\left[\frac{\partial^2}{\partial \theta^2} \ln f(X;\theta)\right] \] Given the pdf: \[ f(x;\theta) = \begin{cases} 3a\theta^2 e^{-\theta x^3}, & 0 < x < \infty \\ 0, & \text{otherwise} \end{cases} \] First, we need to compute the natural logarithm of the pdf: \[ \ln f(x;\theta) = \ln(3a) + 2\ln(\theta) - \theta x^3 \] Now, we take the first derivative with respect to \(\theta\): \[ \frac{\partial}{\partial \theta} \ln f(x;\theta) = \frac{2}{\theta} - x^3 \] Next, we take the second derivative with respect to \(\theta\): \[ \frac{\partial^2}{\partial \theta^2} \ln f(x;\theta) = -\frac{2}{\theta^2} \] Now, we calculate the expected value of the second derivative. Since the second derivative does not depend on \(x\), the expected value is simply the negative of the second derivative itself: \[ E\left[-\frac{\partial^2}{\partial \theta^2} \ln f(X;\theta)\right] = E\left[\frac{2}{\theta^2}\right] = \frac{2}{\theta^2} \] Therefore, the Fisher information for a single observation is: \[ I(\theta) = \frac{2}{\theta^2} \] For a sample of size \(n\), the Fisher information is \(n\) times the information for a single observation, since the observations are independent: \[ I_n(\theta) = nI(\theta) = \frac{2n}{\theta^2} \] The Cramér-Rao Lower Bound for the variance of any unbiased estimator \(\hat{\theta}\) of \(\theta\) is then given by the reciprocal of the Fisher information for the sample: \[ \text{Var}(\hat{\theta}) \geq \frac{1}{I_n(\theta)} = \frac{\theta^2}{2n} \] This is the Cramér-Rao Lower Bound for the variance of unbiased estimators of the parameter \(\theta\).

The Poisson distribution truncated at zero has probability function given by f (x; θ) = exp(−θ)θx A(θ)x! , x = 1, 2, ... (θ > 0), where A(θ) = 1 − exp(−θ). Let X1, . . . , Xn be a random sample of size n from this truncated Poisson distribution. (i) Show that this distribution belongs to the regular exponential distri- bution and give its canonical parameter c(θ) and the corresponding sufficient statistic T (X1, ..., Xn)

the moment estimator of θ;• the MLE of θ

Let X1, . . . , Xn denote a random sample from a N(μ, σ2) distribution, where the mean μ and the variance σ2 are both unknown so that the param- eter vector is given by θ = (μ, σ2)T . (i) Show that the joint p.d.f. of X1, . . . , Xn belongs to the two-parameter regular exponential family, stating what the canonical parameter is equal to in terms of θ. (ii) Use the result in (i) to derive the maximum likelihood (ML) estimate of θ, θˆ = (μˆ, σˆ2)T , without the need to differentiate the log likelihood function with respect to θ.