Given
, i.i.d. with mean
and variance
, suppose that at stage
n one wishes to estimate
by the sample mean
, subject to the loss function
. If
is known, the optimal fixed sample size
can be used, with corresponding risk
, but if
is unknown there is no
... [Show full abstract] fixed sample size procedure that will achieve the risk . For the sequential estimation procedure with stopping rule , the second order approximation of Woodroofe (1977) to the risk for normal is extended to the distribution-free case. Specifically, if the have finite moments of order greater than eight and are non-lattice, under certain conditions on the delay it is shown that the regret as , where c depends on the first four moments of the distribution of the . For the lattice case, bounds of the form are obtained, where the are . It follows from these approximations that the regret can take arbitrarily large negative values as the distribution of the varies, in contrast to previous results for normal and gamma cases.