# Mathematische Statistik - Übung Section 1.1 Beispiel 3

Which of the following parametrization are identifiable?

# (a)

$X_1, \ldots, X_p \,$ are independent with $X_i \sim N(\alpha_i + \nu, \sigma^2) \,$

$\theta = (\alpha_1, \alpha_2, \ldots, \alpha_p, \nu, \sigma^2) \,$

$P_\theta \,$ is the distribution of $X=(X_1, \ldots, X_p) \,$

Gegenbeispiel:

$\theta_1 = (1,2,3,0,5) \,$

$\theta_2 = (0,1,2,1,5) \,$

Daher nicht identifiable.

# (b)

same as (a) with the restriciton $\sum_{i=1}^p \alpha_i = 1 \,$

Gegenbeispiel:

$\theta_1 = \left(\frac 1 2, - \frac 1 2, 0, 1 \right) \,$

$\theta_2 = \left(-\frac 1 2, \frac 1 2, 0, 1 \right) \,$

oder

gemeinsame Verteilung: $P_\theta = \prod_{i=1}^n \varphi \left( \frac{x_i - \alpha_i - \mu}{\sigma^2} \right) \,$ mit $\mu=0, \sigma^2 = 1 \,$

# (c)

X and Y are independet $N(\mu_1, \sigma^2) \,$ and $N(\mu_2, \sigma^2) \,$ with $\theta = (\mu_1, \mu_2) \,$. Observe $Y - X \,$.

$Y - X = N(\mu_1 - \mu_2, 2 \sigma^2) \,$

Gegenbeispiel: $\theta_1 = (1,1) ; \theta_2 = (0,0) \,$