- #1

mathmari

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For $n \in \mathbb{N}$ we consider the discrete statistical product model $(X ,(P_{\theta})_{\theta \in\Theta})$ with $X = \mathbb{N}^n$, $\Theta = (0, 1)$ and $p_{\theta}(x_i) = \theta(1 -\theta)^{x_i−1}$

for all $x_i \in \mathbb{N}, \theta \in \Theta$. So $n$ independent, identical experiments are carried out, the outcomes of which are modeled by independent, geometrically distributed random variables with an unknown probability of success $\theta$.

(a) Give the corresponding likelihood function.

(b) Determine an estimator for the parameter $\theta$ using the maximum likelihood method (for samples $x$ that do not consist only of ones). Note intermediate steps.

(c) You observe the following sample $x$: $$4 \ \ \ \ \ 2 \ \ \ \ \ 7 \ \ \ \ \ 4 \ \ \ \ \ 3 \ \ \ \ \ 1 \ \ \ \ \ 8 \ \ \ \ \ 2 \ \ \ \ \ 4 \ \ \ \ \ 5$$ Give the concrete estimated value for $\theta$ for $x$ using the estimator of part (b).I have done the following :

(a) The likelihood function is $$L_x(\theta)=\prod_{i\in \mathbb{N}}p_{\theta}(x_i)$$ or not? Can wecalculate that further or do we let that as it is?(b) We have to calculate the supremum of $L_x(\theta)$ as for $\theta$, right?

:unsure: