## 2.2 Ratio Estimators and Matérn's Error estimators

Although a discussion of statistical estimators is provided in another section or may be obtained from the online book, Statistical Techniques for Sampling and Monitoring Natural Resources (Schreuder et al 2004), we note here the importance of selecting estimators that are consistent with the sample design in order to obtain valid variance estimates. With systematic and cluster-based sample designs it is particularly important that estimators properly account for possible spatial correlation among observations. Because of their utility with sample designs that must accommodate spatial correlation, we provide a brief discussion of Matérn estimators (Matérn 1960).

Because forest inventory estimates are frequently either means or totals for either area or volume, the relevant derived variables in forest inventory are often of the form

where *X* and *Y* are expectations of random variables, *x* and *y*. For example, consider estimation of mean forest area per land use stratum for sample plots that may intersect multiple strata, all within the category of forest land. One method for accommodating this phenomenon that is particularly useful with point sampling is to use the information from the center point only. Let *x _{i}*=1 when the center point of the plot belongs to the stratum in question and

*x*=0 otherwise, and let

_{i}*y*=1 when the center point is on forest land and

_{i}*y*=0 otherwise. Then the ratio estimator for mean area is

_{i}where n is the number of sampling units. Let E(.) denote statistical expectation; then,

means that *m* is approximately unbiased when *n* is large.

The estimation of standard errors is complicated by spatial correlation that may arise from trend-like changes in variables and either systematic or cluster sampling. Matérn (1947, 1960) suggested the error variance, *E*(*m - M*)^{2}, as a measure of the reliability of the estimator and also proposed a variance estimator. Let *i* denote field plots; let *r* denote clusters of field plots; and consider the cluster residuals *z _{r}* =

*x*-

_{r}*my*, where and Assume that the residuals form a realization of a second order stationary (weakly stationary) stochastic process. The variance of the process can be estimated by means of quadratic forms where where

_{r}*r*and

*s*both refer to clusters of field plots. Estimators of this form are unbiased if the process

*z*is spatially uncorrelated and conservative if the process is positively correlated (Matérn, 1960). This approach has been used in the Swedish and Finnish inventories (cf. Ranneby, 1981, see also Tomppo et al. 1997) and is applied by sampling strata as follows. Within each stratum, the group

*g*of four field plot clusters (r

_{1}, r

_{2}, r

_{3}, r

_{4})

**Figure 2.** Groups of clusters and clusters of sample plots.

is composed in such a way that each cluster belongs to four different groups (Figure 2). The deviance of the cluster mean, , from the stratum mean, is computed for each cluster *r*. Denote , where *n _{r}* is the number of relevant sample points in cluster

*r*(for this example,

*n*=4). The weights are often used. The quadratic forms can then be expressed as , and the resulting standard error estimators for each stratum are

_{r}*g*denotes a group of clusters in the stratum,

*i*denotes plots in the stratum, and

*k*is the number of clusters in each cluster group (for this example,

*k*=1). The standard error estimators for the entire study area can be obtained by combining the stratum-specific estimators with the usual formula for stratified sampling (Eqs. [4] and [5]). This procedure is relevant for strata having large numbers of field plots, preferably at least several hundred.