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.

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SUMMARYThis article proposes a simple approach for reducing the bias of the traditional histogram estimator using line transect sampling. The approach uses the bias correction technique, which produces a new estimator for density of objects D. The proposed estimator reduces the bias from Oðh 2 Þ to Oðh 3 Þ as h ! 0 under the shoulder condition assumption. The asymptotic properties of the proposed estimator are derived under some mild assumptions, and the optimal formula for the bin width is given. Small-sample properties of the proposed estimator are studied and compared with some other existence estimators by using a simulation technique. The results show that improvements over the traditional histogram estimator often can be realized even at small or moderate sample size.