Mathematical Statistics I · Parametric models and sampling distributions
A parametric model is a collection of probability laws indexed by a parameter
Problem
A parametric model is a collection of probability laws indexed by a parameter. Consider independent observations \(X_1,\ldots,X_n\), each normally distributed with mean \(\theta^2\) and variance 1, where initially \(\theta\in\mathbb R\). The normal density with mean \(m\) and variance 1 is \((2\pi)^{-1/2}e^{-(x-m)^2/2}\). Write the joint density and determine when two parameter values give the same sampling law. A model is identifiable when equality of its probability laws implies equality of its parameters. Give a restriction of the parameter space that makes this model identifiable without removing any of its probability laws. A statistic is a specified function of the observed sample that does not require an unknown parameter to compute. Classify \(\bar X\), \(\bar X-\theta^2\), and \(\max(0,\bar X)\) under this definition. Explain why observing a negative sample mean does not contradict the model.
Hint
Which expression involving \(\theta\) actually appears in a single-observation density?
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