Simulation studies to examine bias and precision of some estimators that use auxiliary information in design-based sampling in forest inventory

West, P. W.

doi:10.1186/s40490-017-0101-7

New Zealand Journal of Forestry Science

Table 1 The sampling methods and estimators considered here

From: Simulation studies to examine bias and precision of some estimators that use auxiliary information in design-based sampling in forest inventory

Method/estimator		Equation		Equation
One-phase sampling
Simple random sampling^**			\( {\displaystyle \begin{array}{l}\overline{Y}\kern0.5em =\kern0.5em \left({\varSigma}_n\kern0.5em {y}_i\right)/n\\ {}\widehat{\sigma}\left(\overline{Y}\right)\kern0.5em =\kern0.5em {\left\{{\varSigma}_n{\left({y}_i-\overline{Y}\right)}^2/\left[n\left(n-1\right)\right]\right\}}^{1/2}\end{array}} \)	(3.2)
Two-phase (double) sampling
	Complete enumeration of N sampling units as first-phase		First-phase simple random sample of size = f (<N) ^**
Simple random sampling in second-phase ^**
Ratio of means	\( {\displaystyle \begin{array}{l}\begin{array}{l}\overline{Y}\kern0.5em =\kern0.5em \left[\left({\varSigma}_N\kern0.5em {x^f}_i\right)/N\right]\overline{R}\hfill \\ {}\overline{R}=\left({\varSigma}_n\ {y}_i\right)/\left({\varSigma}_n\ {x}_i\right)\hfill \end{array}\\ {}\widehat{\sigma}\left(\overline{Y}\right)\kern0.5em =\kern0.5em {\left\{\left[1-n/N\right]\kern0.5em \left[{\varSigma}_n{\left({y}_i-\overline{R}{x}_i\right)}^2\right]/\left[n\left(n-1\right)\right]\right\}}^{1/2}\end{array}} \) (Cochran 1977, Eqs. 6.1 and 6.9)	(4.1)	\( \overline{Y}\kern0.5em =\kern0.5em \left[\left({\varSigma}_f{x^f}_i\right)/f\right]\overline{R} \) ^‡ \( \overline{R}=\left({\varSigma}_n\ {y}_i\right)/\left({\varSigma}_n\ {x}_i\right) \)	(4.2)
Mean of ratios	\( {\displaystyle \begin{array}{l}\overline{Y}\kern0.5em =\kern0.5em \left[\left({\varSigma}_N{x^f}_i\right)/N\right]\overline{R}+\left(N-1\right)\left[\left({\varSigma}_n\kern0.5em {y}_i\right)-\left({\varSigma}_n\ {x}_i\right)\overline{R}\right]/\left[N\left(n-1\right)\right]\ \\ {}\overline{R}\kern0.5em =\kern0.5em \left({\varSigma}_n\kern0.5em {y}_i/{x}_i\right)/n\end{array}} \) (Hartley and Ross 1954)	(5.1)	\( {\displaystyle \begin{array}{l}\overline{Y}\kern0.5em =\kern0.5em \left[\left({\varSigma}_f{x^f}_i\right)/f\right]\overline{R}+\kern0.5em \left(f-1\right)\left[\left({\varSigma}_n\kern0.5em {y}_i\right)-\left({\varSigma}_n\kern0.5em {x}_i\right)\overline{R}\right]/\left[f\left(n-1\right)\right]\\ {}\overline{R}\kern0.5em =\kern0.5em \left({\varSigma}_n\ {y}_i/{x}_i\right)/n\end{array}} \)	(5.2)
Model-assisted	\( {\displaystyle \begin{array}{l}\overline{Y}=\left[\left({\varSigma}_n\kern0.5em {y}_i\right)+\left({\varSigma}_N{{\widehat{y}}^f}_i\right)-\left({\varSigma}_n\kern0.5em {\widehat{y}}_i\right)\right]/N\\ {}\mathrm{where}\kern0.5em {{\widehat{y}}^f}_i=\alpha +\beta {x^f}_i,\kern0.5em {\widehat{y}}_i\kern0.5em =\kern0.5em \alpha +\beta {x}_i\end{array}} \) (Ståhl et al. 2016)	(6.1)	\( {\displaystyle \begin{array}{l}\overline{Y}\kern0.5em =\kern0.5em \left[\left({\varSigma}_n\kern0.5em {y}_i\right)\kern0.5em +\kern0.5em \left({\varSigma}_f{{\widehat{y}}^f}_i\right)\kern0.5em -\kern0.5em \left({\varSigma}_n\kern0.5em {\widehat{y}}_i\right)\right]/f\\ {}\mathrm{where}\kern1.5em {{\widehat{y}}^f}_i\kern0.5em =\kern0.5em \alpha \kern0.5em +\kern0.5em \beta {x^f}_i,\kern0.5em {\widehat{y}}_i=\alpha +\beta {x}_i\end{array}} \)	(6.2)
Sampling with probability proportional to size in second-phase
Probability proportional to size (PPS) sampling	\( {\displaystyle \begin{array}{l}\begin{array}{l}\overline{Y}\kern0.5em =\kern0.5em \left[\left({\varSigma}_N\kern0.5em {x^f}_i\right)/N\right]\overline{R}\hfill \\ {}\overline{R}\kern0.5em =\kern0.5em \left({\varSigma}_n\ {y}_i/{x}_i\right)/n\hfill \end{array}\\ {}\widehat{\sigma}\left(\overline{Y}\right)\kern0.5em =\kern0.5em \left[\left({\varSigma}_N{x^f}_i\right)/n\right]\kern0.5em {\left\{\left[N-n\right]\left[{\varSigma}_{n\ j,k}{\left({y}_j/{x}_j-{y}_k/{x}_k\right)}^2\right]/\left[2{N}^3\left(n-1\right)\right]\right\}}^{1/2}\end{array}} \)(Schreuder et al. 1993, Eqs. 3.7, 3.9)	(7.1)
Quick probability proportional to size (QPPS) sampling	\( \overline{Y}=\kern0.5em \left[\left({\varSigma}_N\kern0.5em {x^f}_i\right)/N\right]\overline{R} \) ^† \( \overline{R}\kern0.5em =\kern0.5em \left({\varSigma}_n\kern0.5em {y}_i/{x}_i\right)/n \) (Grosenbaugh 1965, Eq. 3PSEVENTH; Furnival et al. 1987, Eq. 9; West 2011, Eq. 11)	(8.1)	\( \overline{Y}\kern0.5em =\kern0.5em \left[\left({\varSigma}_f\kern0.5em {x^f}_i\right)/f\right]\overline{R} \) ^†† \( \overline{R}\kern0.5em =\kern0.5em \left({\varSigma}_n\kern0.5em {y}_i/{x}_i\right)/n \) (Adapted from West 2011, Eq. 9; 2017, Eq. 1)	(8.2)

Notation: \( \overline{\gamma} \) estimate of the population mean of target variable, \( \sigma \left(\overline{\gamma}\right) \)) estimate of the standard error of the estimate of the population mean; bootstrapping was used where no analytical estimator is shown, N population size, f first-phase sample size (≤N), x ^f _i auxiliary variable value measured in the ith member of a first-phase sample, n second-phase sample size (<f), y _i, x _i target and auxiliary variable values, respectively, measured in the ith member of a second-phase sample (the auxiliary variable value will have been measured already as part of the first-phase sample), α, β intercept and slope, respectively, of the ordinary least-squares straight-line fit between the target and auxiliary variable values in a second-phase sample, as defined by Eq. (1)
^**All first-phase simple random sampling was done with replacement in this work
^‡At their Eq. (6.95), Gregoire and Valentine (2008) show a modified version of this that can be written as \( \overline{Y}=\left\{\overline{R}\left[\left({\varSigma}_f\kern0.5em {x}_i^f\right)-\left({\varSigma}_n\kern0.5em {x}_i\right)\right]+\left[\left({\varSigma}_n\kern0.5em {y}_i\right)\right]\right\}/f \), together with an approximate analytical standard error estimator. The form here was adapted from the commonly used estimator shown by Cochran (1977, Eq. 6.1)
^†Originally called sampling with probability proportional to prediction (3P sampling) by its inventor L.R. Grosenbaugh, as discussed in the text
^††A method developed as an extension of 3P sampling by West (2011), renamed QPPS sampling by West (2017). In applying this method in the present work it was assumed that the minimum auxiliary variable value in the population was zero, as done originally by Grosenbaugh in 3P sampling. That constraint was not applied by West (2011, 2017), but was found here to yield more reliable estimates of the standard error of the estimate of the population mean

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