Statistical analysis of ratio estimators and their estimators of variances when the auxiliary variate is measured with error

Abstract

Forest inventory relies heavily on sampling strategies. Ratio estimators use information of an auxiliary variable (x) to improve the estimation of a parameter of a target variable (y). We evaluated the effect of measurement error (ME) in the auxiliary variate on the statistical performance of three ratio estimators of the target parameter total τ _y . The analyzed estimators are: the ratio-of-means, mean-of-ratios, and an unbiased ratio estimator. Monte Carlo simulations were conducted over a population of more than 14,000 loblolly pine (Pinus taeda L.) trees, using tree volume (v) and diameter at breast height (d) as the target and auxiliary variables, respectively. In each simulation three different sample sizes were randomly selected. Based on the simulations, the effect of different types (systematic and random) and levels (low to high) of MEs in x on the bias, variance, and mean square error of three ratio estimators was assessed. We also assessed the estimators of the variance of the ratio estimators. The ratio-of-means estimator had the smallest root mean square error. The mean-of-ratios estimator was found quite biased (20%). When the MEs are random, neither the accuracy (i.e. bias) of any of the ratio estimators is greatly affected by type and level of ME nor its precision (i.e. variance). Positive systematic MEs decrease the bias but increase the variance of all the ratio estimators. Only the variance estimator of the ratio-of-means estimator is biased, being especially large for the smallest sample size, and larger for negative MEs, mainly if they are systematic.

Appendix: Expressions needed for computing the estimator of the variance of \(\widehat{\tau}_{y3}\)

We used the unbiased estimator presented by Goodman and Hartley (1958, Eq. 35). This estimator requires the computation of k ₂₂, c, c′. These statistics are computed as follows,

* k ₂₂

$$ \begin{aligned} k_{22} =& {\frac{n}{(n-1)(n-2)(n-3)}} \left\lbrace (n+1)S_{22}-{\frac{2(n+1)}{n}}\left(S_{21}S_{01}+S_{12}S_{10}\right)\right.\\ &\left.-{\frac{(n-1)}{n}}(S_{20}S_{02}+2S_{11}^2) +{\frac{2} {n}}(S_{20}S_{01}^2 + S_{02}S_{10}^2+4S_{11}S_{10}S_{01}) -{\frac{6}{n^2}}(S_{10}^2 S_{01}^2) \right\rbrace,\\ \end{aligned} $$

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where

$$ S_{tj}=\sum_{k=1}^n x_k^tr_k^j, $$

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for example $S_{22}=\sum_{k=1}^n x_k^2 r_k^2=\sum_{k=1}^n x_k^2 ({y_k^2}/{x_k^2})=\sum_{k=1}^n y_k^2.$

Note that our expression for k ₂₂ (Eq. 18) has some algebraic manipulations compared that one gave by Goodman and Hartley (1958, Eq. 30), and also considering the corrections made by Goodman and Hartley (1969).

* c. From Goodman and Hartley (1958, Eq. 36, part a)

$$ c = {\frac{1}{n(n-1)}} \left[n \sum_{k=1}^n y_k - \left(\sum_{k=1}^n x_k \sum_{k=1}^n r_k \right) \right], $$

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which is the sample covariance between x and r as at the bottom of page 497 of Goodman and Hartley (1958), as follows

$$ c = \left({\frac{1}{n-1}} \right) \sum_{i=1}^{n} (x_k - \overline{x}) (r_k - \overline{r}), $$

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* c′. From Goodman and Hartley (1958, Eq. 32)

$$ c' = \left({\frac{1}{n-1}}\right ) \sum_{k=1}^n (x_k - \overline{x})(r_k - \overline{r})^2. $$

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