Question

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)?

Answer 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.)