Elizabeth E. Holmes and William F. Fagan. 2002. Validating population viability analysis for corrupted data sets. Ecology 83:2379-2386.

A pdf version of Appendix C is also available for viewing.

Appendix C. Derivation of the distributions for cross-validating parameters.

Difference between the estimates, Eq. 5.

If the number of years in the parameterization and evaluation periods are set equal, n_e=n_p=n, the variance of the estimates from the two periods will be the same, = , if _p and _np remain constant within the two periods. Using this assumption, we can derive the t distribution of the t statistic in Eq. 5.

This has the distribution which has the distribution

The degrees of freedom, df_slp, are given in Appendix B.

Ratio of the estimates

If the number of years in the parameterization and evaluation periods are set equal, n_e=n_p=n, and if we assume that the non-process and process error variance remain constant within the two periods, the expected values of and are the same and the degrees of freedom for the ² distributions that approximate and are the same. Thus,

.