Mathematische Statistik - Übung Section 1.1 Beispiel 3

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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) \,