- #1

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We know that ##\hat{\vec{\beta}}##, the vector of ##\hat{\beta_{0}}## and ##\hat{\beta_{1}}##, is distributed asmptotically as ##N( \vec{\beta}, \hat{{Var{\hat{\beta}}}})## for large sample sizes where ##\hat{{Var{\hat{\beta}}}}## is the estimated variance covariance matrix.( similar to how ##\sigma^{2}## is estimated by ##s^{2}## for the basic t test).

So solving for x using ##log (5)=\beta_{0}+\beta_{1}x## I get: ##x( \beta_{0},\beta_{1}) =\frac{ log(5)-\beta_{0}} {\beta_{1} }## a multivariate function of both parameters.

So it seems that I would use the delta method but the one problem is the variance covariance matrix is an estimated one, so it doesn't have any actual parameters in it. Would delta method still work for this? Or would I need to try another method?