Length biased density estimation of fibres

Cox (1969) discussed several procedures used in sampling of textile fibres. One such procedure is called "length biased" or weighted sampling and occurs when the chance of selection is proportional to fibre length. Cox considered the problem of estimating the unweighted distribution function F at a fixed x > 0 and compared the asymptotic variance of estimators based on length biased samples with those based on unweighted samples. Consideration here is devoted to estimating the probability density function f at a fixed x > 0 based on length biased samples. It is shown, under suitable regularity conditions, that the square of the bias of the weighted estimator is less (greater) than the square of the bias of the Parzen (l962)-Rosenblatt (1956) kernel estimator of f(x) based on unweighted observations when and n is sufficiently large. Moreover, the variance of the length biased estimator is less (greater) than that of the unweighted estimator when x > μ (x < μ) for all n sufficiently large, where m denotes the mean with respect to f. An optimal window width h n (x) is given which makes the asymptotic mean square error of the length biased estimator a minimum. Under regularity assumptions, it is shown that the optimal asymptotic mean square error of the length biased estimator at x is less than that for the unweighted estimator exactly when Moreover, simulations are undertaken to compare the two estimators for several sample sizes.