This paper presents a one-term approximation to the cumulative normal distribution functions. The absolute maximum error of the proposed approximation is 0.0018 less than 0.003 of Polya's approximation. Comparisons between the proposed approximation and the different approximations with one-term that stated in the literature are given.

Among different candidate parametric detection functions, it is suggested to use Akaike Information Criterion (AIC) to select the most appropriate one of them to fit line transect data. Four different detection functions are considered in this paper. Two of them are taken to satisfy the shoulder condition assumption and the other two estimators do not satisfy this condition. Once the appropriate detection function is determined, it also can be used to select the smoothing parameter of the nonparametric kernel estimator. For a wide range of target densities, a simulation results show the reasonable and good performances of the resulting estimators comparing with some existing estimator, particularly the usual kernel estimator when the half normal model is use as a reference to select the smoothing parameter.