1. 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 \(\theta\).

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

the moment estimator of θ;• the MLE of θ

For each of the following pdfs let X1, . . . , Xn be iid observations. Find a complete sufficient statistic,or show that one does not exist.(a) f (x; θ) = 2xθ2 , 0 < x < θ, θ > 0(b) f (x; θ) = e−(x−θ) exp (−e−(x−θ)) , −∞ < x < ∞, −∞ < θ < ∞(c) f (x; θ) = (2x)θx(1 − θ)2−x, x = 0, 1, 2, 0 ≤ θ ≤ 1