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Question
I want to improve my skills at finding the distribution of functions of real-valued functions of real-valued random variables. I’m familiar with the basics of the algebra of random variables as far as manipulating expectations, and also understanding the expectation as an integral. I also think I understand that a sum of two random variables is a convolution, and a product of two random variables has a distribution that can be derived from their cumulative distribution function. But, let me formalize what I am interested in.
Let there exist a collection of real-valued random variables \(\{X_j \sim f_j\}_{j=1}^{n}\) where the univariate distributions \(f_i\) are known, and let there be a population parameter \(\theta\) that is estimated by \(\hat{\theta}\) which is a function \(g\) of the random variables:
\[\hat{\theta} = g\left( X_1, \cdots, X_n \right)\]
Let’s also assume that \(g\) and \(f_i\) are sufficiently smooth. But let us not assume that any subset of \(\{X_j \sim f_j\}_{j=1}^{n}\) is necessarily independent.
While this description is not sufficient to provide an exact algorithm, I would like to know if there are some common approaches or procedures to deriving the distribution of \(\hat{\theta}\) (Casella and Berger 2002).