Bandwidth of kernel estimator of DBH distribution in black alder (Alnus glutinosa (L.) Gaertn.) stands from west part of the Sandomierz Basin

Set of `nonparametric' methods, that don't make a priori assumption about functional form of empirical distribution was developed as an alternative to the parametric distribution modeling. The kernel estimators are one of such methods, that can be used to describe the frequency of data representing for example DBH records. Kernel smoothing requires the choice of weighting function and bandwidth also called as smoothing parameter or window. The lack of comprehensive analysis on the applicability of particular bandwidth selection methods to model DBH structure gave an impulse to present investigation aimed at determining value and variability of smoothing parameter in black alder stands. The optimal bandwidth was obtained according to six different variants of plug-in method proposed by Altman and Leger. Presented investigations were based on DBH measurements collected in 163 managed black alder stands aged from 6 to 89 years, growing in the west part of the Sandomierz Basin (S Poland). We measured in total 22,530 black alders, from 48 to 359 in individual stand. Stands were characterized by: age, quadratic mean diameter, basal area, mean height, Reineke's stand density index and standard deviation of DBH. Smoothing parameter was obtained by means of plug-in method with the pilot bandwidth selected by: Silverman's rule of thumb (nrd0), Scott's method (nrd), unbiased cross-validation (ucv), biased cross-validation (bcv), method of Sheather and Jones (sj) and one-stage method of Wand and Jones (onestage). The bandwidth was first obtained to real data, then to 100 bootstrap samples of 5, 10, 15 . . . and 100 trees from each stand. Smoothing parameters were characterized by mean and variance. Relationship between values of smoothing parameter and stand characteristics was determined. Finally the influence of sample size on value and variability of bandwidth was assessed. Value and variability of smoothing parameter in black alder stands are determined by stand age, sample size and method of bandwidth choice. There is a close relationship between bandwidth and the mean height (r from 0.75 to 0.83), quadratic mean diameter (r from 0.79 to 0.88) and standard deviation of DBH (r from 0.84 to 0.93). Potentially these stand features can be used to predict smoothing parameter values. Minor changes of bandwidth for samples containing above 50 trees together with persistence of standard error give an objective grounds for defining optimal number of diameters, that are necessary to kernel estimation of DBH distribution.