Nonparametric Estimation of f(0) Applying Line Transect Data with and without the Shoulder Condition

The problem of estimating f(0); the probability density function of observed distances at the left boundary x = 0 using line transect data is considered. A general nonparametric histogram estimator (f) over cap (0) for f(0) is proposed and investigated. The proposed estimator is formulated as linear of indicator functions with corresponding weights. These weights can be determined in a variety of different ways to ensure desirable properties for the proposed estimator and to deal with the presence or absence of the shoulder condition assumption. Theoretical motivation is provided for the existence of weights that prove to be effective in reducing the bias without increasing variance. We consider a direct implementation of the proposed estimator in which the weights are optimized subject to some constraints. The optimization problem converts into the problem of solving a system of linear equations. Some comparison performances are obtained, which demonstrate the good results of the proposed estimator at least for adequately large samples.