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13. Parametric Bootstrap (Jan 24)

# 13. Parametric Bootstrap (Jan 24) - Parametric Bootstrap ^...

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Parametric Bootstrap If ˆ θ n is an estimator from an iid sample from a parametric model, it has a sampling distribution. If we know the true value of the parameter, say θ , then we can find the distribution of ˆ θ n . This is because it is a statistic, so it is a function of the data. For the purpose of our notation, we thus have ˆ θ n = h ( X 1 , . . . , X n ) for the appropriate function h . We then obtain the distribution of ˆ θ n using the methods of transformations. If the methods of transformations are not easy or useful to use, we can use a Monte Carlo simulation method to approximate the distribution of ˆ θ provided we also know θ 0 . This is the method we used to approximate the sampling distribution in one of our earlier examples for an iid exponential example. What happens if we do not know θ ? The simplest way to implement the Monte Carlo method is to use a good guess of θ 0 , in particular ˆ θ n . This method does work provided the distribution of the statistic is continuous wrt θ

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