Estimation of the binomial parameter n using a linex loss function

Suppose that X has a binomial distribution B(n,p), with known p and unknown n∈ {0,1,...}. We consider estimation of n when the loss function is Varian's asymmetric LINEX loss function , where δ is the estimation error and b> 0 and a ≠ 0 are the parameters of the loss function. This loss function is useful when overestimation of n is more serious than its underestimation, or vice versa: The sign of the shape parameter a reflects the direction of the asymmetry and its magnitude reflects the degree of the asymmetry. Results concerning the admissibility of linear estimators ṋ=cX+d are presented, including the interesting result that the admissibility of the usual estimator ṋ=X/p depends on the sign of the shape parameter a, i.e. on whether underestimation of n is regarded as more, or less, serious than its overestimation : if a