Combining double sampling for stratification and cluster sampling to a three-level sampling design for continuous forest inventories

Abstract

We extend the well-known double sampling for stratification sampling scheme by cluster subsampling to a three-level design and present corresponding estimators based on the infinite population approach in the first phase. After stratification of the sample points (phase I), a second-phase sample is drawn independently among the first-phase points within each stratum. On level III, clusters are formed of those phase II points and a sample of clusters is finally drawn without replacement. We used the forest planning units compartment and subdistrict as clusters and moreover formed clusters with a heuristic for the vehicle routing problem. The precision of the new estimator was compared to that achieved with classical double sampling for stratification in a case study. The results indicate that the expected increase in sampling errors caused by clustering cannot be compensated by the reduced inventory costs under the conditions given in the case study.

Appendices

Appendix

Proofs

To derive the variance of the new estimator, we decompose $\hat{\bar{y}}_h$ as follows

$$ \hat{\bar{y}}_h=\bar{y}_h+\left(\hat{\bar{y}}_h-\bar{y}_h\right). $$

(21)

The variance of $\hat{\bar{y}}_h$ is then given as the sum of the variances of the two components, because both components are not correlated (see (12.6) in Cochran 1977).

$$ \hbox{Var}\hat{\bar{Y}}_{cl}=\hbox{Var}\sum_{h=1}^{L}\frac{n'_h}{n'}\bar{y}_h+\hbox{Var}\sum_{h=1}^{L}\frac{n'_h}{n'}\left(\hat{\bar{y}}_h-\bar{y}_h\right) $$

(22)

In this equation, the first variance is the variance from 2st (Eq. 2). Due to the fact that $E_3(\hat{\bar{y}}_h-\bar{y}_h)=0,$

$$ \hbox{Var}\sum_{h=1}^{L}\frac{n'_h}{n'}\left(\hat{\bar{y}}_h-\bar{y}_h\right)=E \hbox{Var}_3\sum_{h=1}^{L}\frac{n'_h}{n'}\left(\hat{\bar{y}}_h-\bar{y}_h\right) $$

(23)

holds for the second variance. Assuming simple random sampling with drawing without replacement for the clusters, the variance and the covariance can be calculated as

$$ \hbox{Var}_3\left(\hat{\bar{y}}_h-\bar{y}_h\right)=\hbox{Var}_3\hat{\bar{y}}_h=\frac{1}{n_h^2}\frac{K^2}{k}\left(1-\frac{k}{K}\right)\breve{S}_h^2 $$

(24)

and

$$ Cov_3 \left(\hat{\bar{y}}_h-\bar{y}_h,\hat{\bar{y}}_{h'}-\bar{y}_{h'}\right)=Cov_3\left(\hat{\bar{y}}_{h},\hat{\bar{y}}_{h'}\right)=\frac{1}{n_h n_{h'}}\frac{K^2}{k}\left(1-\frac{k}{K}\right)\breve{S}_{hh'} $$

(25)

respectively. Substituting (24) and (25) in (23) yields

$$ \begin{aligned} Var\sum_{h=1}^{L}\frac{n'_h}{n'}\left(\hat{\bar{y}}_h-\bar{y}_h\right)=&E\sum_{h=1}^{L}\left(\frac{n'_h}{n'}\right)^2 \frac{1}{n_h^2}\frac{K^2}{k}\left(1-\frac{k}{K}\right)\breve{S}_h^2\\ &+E\sum_{h \neq {h'}}^{L}\frac{n'_h n'_{h'}}{{n'}^2}\frac{1}{n_h n_{h'}}\frac{K^2}{k}\left(1-\frac{k}{K}\right)\breve{S}_{hh'}. \end{aligned} $$

(26)