Step 1: Moments Map for an Exponential Statistical Model
Let ?1,…,??∼Exp(?∗) denote a statistical experiment where ?∗ is the true, unknown parameter. You construct the associated statistical model ((0,∞),{Exp(?)}?∈(0,∞)). Since the parameter ? is one-dimensional, we only consider the first moment with moment map:
? : ℝ →ℝ
? ↦?1(?) := ?[?], (?∼Exp(?)).
What is ?(?)?
What is ?−1(?1)?
Use definition of expectation and the pdf of exponential distribution to compute the 1st moment.
(Technical detail : I believe the function defined in the
question must be from
as
and
are both positive. Otherwise, the function
is not invertible.)
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