On Thursday, October 18, 2012 8:13:11 AM UTC-5, David Jones wrote: > "Paul" wrote in message > > news:firstname.lastname@example.org... > > > > > i.e., > > > > > > > > > Many of the usual frequentist techniques for bias adjustment would lead > > > to estimates not obeying your bound. If you were not insisting on not > > > allowing a value of gamma exactly equal to one then the obvious thing to > > > start from > > > would be a simple truncation at 1. You might consider truncating at a > > > value slightly smaller than one if an exact-1 causes problems. Otherwise > > > you > > > might be able to base something on he following.... > > > > > > (a) let T be the non-truncated version of the estimated for gamma; > > > > > > (b) the transformed estimate S=T/(1+T) should have good properties if the > > > true value of gamma is small, and is forced to lie in the required > > > interval; > > > > > > (c) consider the extended set of estimates S=T/(1+T^a)^(1/a), which are > > > again forced to lie in 0 to 1, and choose a for good properties > > > > > > Other families of transformed estimates might be considered. > > > > This is an intriguing possibility except that what I need is an estimator > > that performs well if the true value of gamma is large (close to 1) not > > small. > > > > -------------------------------------------------------------------------------- > > > > The basis here is that such an approach starts from an estimating function > > that has good properties both for small gamma, and for large (since the > > estimators are constrained to be close to 1 if the raw sample estimate is > > large). It is not clear how much can be achieved analytically, but a > > minimum might be to use an estimator S=bT/(1+bT), and choose b to that the > > bias is correct to a low order in a power series expansion in terms of > > gamma. This would notionally spread the good property of the estimator near > > zero, away from zero to some unknown extent ... you might hope to get an > > explicit formula for a value of b in terms of your sample size N. > > Alternatively, you might allow a clipped estimator S= min(1, bT) and see > > what arises ... you might get some formulae in terms of incomplete gamma > > functions, valid for any gamma, and then see if you are happy with the > > result for gamma=1. You were not clear whether a value for the estimator > > exactly equal to one causes problems (for your confidence intervals)... > > which is a different question from whether gamma=1 is valid within the > > model.
Thank you. I haven't seen estimators of the form S=bT/(1+bT) before. Is there a name for that class of estimators? Perhaps I should read more about it.
An estimate of 1 for gamma is unfortunately not usable, though I have experimented with estimators of the form S= min(1-e, bT) where e is small.