John R. Fieberg. 2007. Kernel density estimators of home range: smoothing and the autocorrelation red herring. Ecology 88:1059–1066.

Appendix B. Estimation of goodness-of-fit measures.

Integrated squared error (ISE), integrated squared bias (ISB), integrated variance (IV), and mean integrated squared error (MISE) were used to measure the accuracy and precision of the UD estimators, with UD_true given by Eq. 3 serving as the target distribution [f(x,y) below]. All of these statistics were calculated numerically using a 150 × 150 size grid, approximating using a sum over grid cells, :

1. Integrated squared error (ISE):
   
2. Integrated squared bias (ISB):

1. Integrated variance (IV):

,

where n gives the number of possible samples, estimates at (x_j, y_k), and x and y give the distance between points at which the density estimates were calculated. Mean integrated squared error (MISE) was then calculated as: MISE = ISB + IV = .