Math Problems Vault

Problem 2.7.1

Give an example of a parametric family of distributions $\mathcal{P}$ and a nontrivial, incomplete sufficient statistic $S(X_1, \ldots , X_n)$ , where $X_1, \ldots , X_n$ is a sample from an unknown distribution $\mathbb {P} \in \mathcal{P}$ , such that the dimension of the statistic $S$ equals 1

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Problem 2.7.2

Prove that in the model

N(\theta , \gamma \theta ^2), \quad (\theta , \gamma ) \in \mathbb {R} \times (0, \infty ),

the statistic $(\overline{X}, s^2)$ is complete but not sufficient.

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Problem 2.7.3

Find sufficient statistics for the following parametric distributions:

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(a)

$N(a, \sigma^2)$ ;

(b)

$\Gamma (\alpha , \lambda )$ ;

(c)

$R(a,b)$ ;

(d)

$\mathrm{Pois}(\lambda )$ ;

(e)

$\mathrm{Ber}(p)$ ;

(f)

$\mathrm{Geom}(p)$ .

Problem 2.7.4

(a)(b)(c)(d)

Find the optimal estimator of $\theta > 0$ from a sample of the distribution

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(a)

$N(\theta , 1)$ ;

(b)

$R(0, \theta )$ ;

(c)

$\mathrm{Pois}(\theta )$ ;

(d)

$\mathrm{Bern}(\theta ), \; \theta \in (0,1)$ .

Problem 2.7.5

Let $X_1, \ldots , X_n$ be a sample from a normal distribution with parameters $(a,\sigma^2), \; a \in \mathbb {R}, \; \sigma > 0$ . Find the optimal estimator of the parameter $\theta = (a, \sigma^2)$ .

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Problem 2.7.6

Let $X_1, X_2, \dots , X_n$ be a sample from $\mathrm{Exp}(\theta )$ . Find the optimal estimators for $\theta$ and for $\sqrt{\theta }$ .

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Problem 2.7.7

Let $X_1, X_2, \dots , X_n$ be a sample from $N(0, \theta^2)$ . Find the optimal estimator for $\theta$ .

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Problem 2.7.8

Let $X_1, \dots , X_n$ be a sample from a Poisson distribution with parameter $\theta > 0$ . Find

\mathbb {E}\left[X_1^2 \; \mid \; \sum _{i=1}^n X_i\right].

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