I am interested in ways of reducing the bias of a point estimator when the true parameter is near the boundary of the parameter space.
Suppose g = gamma U / (N-1), where U ~ chisq(N-1), N is a known small sample size, and gamma is an unknown parameter. A priori we know that 0 <= gamma < 1. Notice that the upper inequality is strict; that is, gamma cannot have a value of 1.
One approach to estimation is to assign gamma a prior distribution that is uniform on (0,1). Then the posterior distribution of gamma is a scaled inverse chi-square, truncated on the right at 1. Now the obvious point estimators are the posterior mean and median. (I can?t use the mode because it can take a value of 1.) The trouble with the posterior mean and median is that they have large negative biases if the true value of gamma is actually close to 1.
I?d be grateful for ideas on how to reduce this bias. One idea I?ve been toying with is to use a posterior quantile greater than the median ? i.e., quantile p where p>1/2. Maybe I would use a larger p when I had a larger g. This isn?t an idea that I?ve seen discussed elsewhere. Many thanks for any references on this or other possibilities.