No, (m,s) does not belong to the sample space even when it is a random variable since it cannot be “sampled”.

]]>The resulting normal distribution might be the same; but the interpretations are vastly different, depending on the notation used: the fact that {m, s} is treated as a random variable is a point that seems to have created much controversy! Specifically, treating {m, s} as a random variable then supposedly places it in the sample space instead of the parameter space, which is a distinction Aris Spanos has often emphasized. (In fact, I am currently taking a course with Spanos about the philosophical foundations of econometrics.)

]]>it is a matter of notation : N(y|m,s) is a conditional notation, meaning that {m,s} could be a random variable. While N(y;m,s) is an indexing notation, meaning that the normal distribution is indexed by the parameter {m,s}. In the end, it is the very same normal distribution with mean m and standard deviation s.

]]>For another nice example of regularity conditions mattering for MLEs, see

… simple Bayes methods should do better than MLEs, here.

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