Distribution of reciprocal of regression coefficient

Suppose that we have a linear model yi=β0+β1xi+ϵi that meets all the standard regression (Gauss-Markov) assumptions. We are interested in θ=1/β1.

Question 1: What assumptions are necessary for the distribution of ˆθ to be well defined? β10 would be important—any others?

Question 2: Add the assumption that the errors follow a normal distribution. We know that, if ˆβ1 is the MLE and g() is a monotonic function, then g(ˆβ1) is the MLE for g(β1). Is monotonicity only necessary in the neighborhood of β1? In other words, is ˆθ=1/ˆβ the MLE? The continuous mapping theorem at least tells us that this parameter is consistent.

Question 3: Are both the Delta Method and the bootstrap both appropriate means for finding the distribution of ˆθ?

Question 4: How do these answer changes for the parameter γ=β0/β1?

Aside: We might consider rearranging the problem to give
xi=β0β1+1β1yi+1β1ϵi=γ+θyi+1β1ϵi
to estimate the parameters directly. This doesn’t seem to work to me as the Gauss-Markov assumptions no longer make sense here; we can’t talk about E[ϵy], for example. Is this interpretation correct?

Answer

Q1. If ˆβ1 is the MLE of β1, then ˆθ is the MLE of θ and β10 is a sufficient condition for this estimator to be well-defined.

Q2. ˆθ=1/ˆβ is the MLE of θ by invariance property of the MLE. In addition, you do not need monotonicity of g if you do not need to obtain its inverse. There is only need for g to be well-defined at each point. You can check this in Theorem 7.2.1 pp. 350 of “Probability and Statistical Inference” by Nitis Mukhopadhyay.

Q3. Yes, you can use both methods, I would also check the profile likelihood of θ.

Q4. Here, you can reparameterise the model in terms of the parameters of interest (θ,γ). For instance, the MLE of γ is ˆγ=ˆβ0/ˆβ1 and you can calculate the profile likelihood of this parameter or its bootstrap distribution as usual.

The approach you mention at the end is incorrect, you are actually considering a “calibration model” which you can check in the literature. The only thing you need is to reparameterise in terms of the parameters of interest.

I hope this helps.

Kind regards.

Attribution
Source : Link , Question Author : Charlie , Answer Author : XMX

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