Confidence interval for the product of two parameters

Let us assume we have two parameters, p1 and p2. We also have two maximum likelihood estimators ˆp1 and ˆp2 and two confidence intervals for these parameters. Is there a way to build a confidence interval for p1p2?

Answer

You can use the Delta method to calculate the standard error of ˆp1ˆp2. The delta method states that an approximation of the variance of a function g(t) is given by:
Var(g(t))ki=1gi(θ)2Var(ti)+2i>jgi(θ)gj(θ)Cov(ti,tj).
The approximation of the expectation of g(t) on the other hand is given by:
E(g(t))g(θ)
So the expectation is simply the function. Your function g(t) is: g(p1,p2)=p1p2. The expectation of g(p1,p2)=p1p2 would simply be: p1p2. For the variance, we need the partial derivatives of g(p1,p2):
p1g(p1p2)=p2p2g(p1p2)=p1

Using the function for the variance above, we get:

Var(ˆp1ˆp2)=ˆp22Var(ˆp1)+ˆp21Var(ˆp2)+2ˆp1ˆp2Cov(ˆp1,ˆp2).
The standard error would then simply be the square root of the above expression. Once you’ve got the standard error, it is straight-forward to calculate a 95% confidence interval for ˆp1ˆp2: ˆp1ˆp2±1.96^SE(ˆp1ˆp2)

To calculate the standard error of ˆp1ˆp2, you need the variance of ˆp1 and ˆp2 which you usually can get by the variance-covariance matrix Σ which would be a 2×2-matrix in your case because you have two estimates. The diagonal elements in the variance-covariance matrix are the variances of ˆp1 and ˆp2 while the off-diagonal elements are the covariance of ˆp1 and ˆp2 (the matrix is symmetric). As @gung mentions in the comments, the variance-covariance matrix can be extracted by most statistical software packages. Sometimes, estimation algorithms provide the Hessian matrix (I won’t go into details about that here), and the variance-covariance matrix can be estimated by the inverse of the negative Hessian (but only if you maximized the log-likelihood!; see this post). Again, consult the documentation of your statistical software and/or the web on how to extract the Hessian and on how to calculate the inverse of a matrix.

Alternatively, you can get the variances of ˆp1 and ˆp2 from the confidence intervals in the following way (this is valid for a 95%-CI): SE(ˆp1)=(upper limitlower limit)/3.92. For an 100(1α)%-CI, the estimated standard error is: SE(ˆp1)=(upper limitlower limit)/(2z1α/2), where z1α/2 is the (1α/2) quantile of the standard normal distribution (for α=0.05, z0.9751.96). Then, Var(ˆp1)=SE(ˆp1)2. The same is true for the variance of ˆp2. We need the covariance of ˆp1 and ˆp2, too (see paragraph above). If ˆp1 and ˆp2 are independent, the covariance is zero and we can drop the term.

This paper might provide additional information.

Attribution
Source : Link , Question Author : guest , Answer Author : Adrian Keister

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