Towards a sampling strategy for the assessment of forest condition at European level: combining country estimates

Abstract

A correct characterization of the status and trend of forest condition is essential to support reporting processes at national and international level. An international forest condition monitoring has been implemented in Europe since 1987 under the auspices of the International Co-operative Programme on Assessment and Monitoring of Air Pollution Effects on Forests (ICP Forests). The monitoring is based on harmonized methodologies, with individual countries being responsible for its implementation. Due to inconsistencies and problems in sampling design, however, the ICP Forests network is not able to produce reliable quantitative estimates of forest condition at European and sometimes at country level. This paper proposes (1) a set of requirements for status and change assessment and (2) a harmonized sampling strategy able to provide unbiased and consistent estimators of forest condition parameters and of their changes at both country and European level. Under the assumption that a common definition of forest holds among European countries, monitoring objectives, parameters of concern and accuracy indexes are stated. On the basis of fixed-area plot sampling performed independently in each country, an unbiased and consistent estimator of forest defoliation indexes is obtained at both country and European level, together with conservative estimators of their sampling variance and power in the detection of changes. The strategy adopts a probabilistic sampling scheme based on fixed-area plots selected by means of systematic or stratified schemes. Operative guidelines for its application are provided.

Appendices

Appendix 1. Theoretical results on estimation under uniform random sampling

When a site is randomly selected onto a study area, the inclusion zone for the jth tree in the population is the locus of sites giving rise to the selection of that tree. It is well known (Gregoire and Valentine 2008, Sect. 7.4) that the inclusion zone of the jth tree is centred at the tree with the same size b and the same shape of the sample plot. Accordingly, the inclusion probability of a tree whose inclusion zone completely lies within the study area is b/A where A is the size of the study region. However, for trees sufficiently close to the edge of the study region, a part of their nominal inclusion zones may lay outside the study region, giving rise to inclusion probabilities smaller than b/A. Accordingly, the determination of the actual inclusion probabilities for edge trees would require the measurement in the field of the truncated inclusion zones, which represents a very labour-intensive task. In order to avoid these problems, usually referred to as edge problems, a very simple solution is to enlarge the area in which sites are selected in such a way that the inclusion zones for all the trees in the population completely lie within the enlarged area and hence have sizes invariably equal to b (Gregoire and Valentine 2008, Sect. 7.5.1).

Accordingly, consider an area ${\cal G}$ of size G covering the study area in such a way as to eliminate any edge effect; then, a point (site) is randomly selected onto ${\cal G}$, and the sampled trees are those lying within the plot of pre-fixed shape and pre-fixed size b centred at the random site. By construction, the probability of any trees to enter the sample (inclusion probability) is invariably equal to b/G. Then, the Horvitz–Thompson (HT) estimator (Särndal et al. 1992, Sect. 2.8) of N _k turns out to be

$$ {\widehat{N}_k} = \frac{G}{b}{n_k},k = 5(5)100 $$

where n _k denotes the number of sampled trees whose defoliation level equals k. Accordingly, the HT estimate of the whole abundance vector N can be written as

$$ \widehat{{{\bf N}}} = \frac{G}{b}{{\bf n}} $$

where ${{\bf n}} = {\left[ {{n_0}, \ldots, {n_{{100}}}} \right]^{\text{T}}}$ is the vector of the counts of sampled trees belonging the 21 defoliation classes. Extending the results on plot sampling (e.g. Gregoire and Valentine 2008, Sect. 7.4) to the multivariate case, $\widehat{{{\bf N}}}$ is an unbiased estimator of N with a variance–covariance matrix, say ${\text{Va}}{{\text{r}}_{\text{URS}}}\left( {\widehat{{{\bf N}}}} \right)$, where V_URS denotes variances and covariances arising from URS, i.e. the complete random placement of sites onto ${\cal G}$. The variances of the ${\widehat{N}_k}$s strictly depend on the spatial distribution of trees within the study area: a distribution of trees evenly scattered throughout the study area generally provides a more accurate estimator than a clumped one.

As no country can be adequately sampled by means of one site, R sites are randomly and independently thrown. Hence, the replication procedure gives rise to R independent samples, say S ₁,…,S _R , which in turn give rise to R estimates, say ${\widehat{{{\bf N}}}_1}, \ldots, {\widehat{{{\bf N}}}_R}$, which constitute R independent realizations of the HT estimator $\widehat{{{\bf N}}}$. Accordingly, on the basis of the very standard results on independently and identically distributed random vectors (e.g. Mardia et al. 1979, Sect. 2.8 and Theorem 2.9.1), the arithmetic mean vector

$$ \widehat{{\overline {{\bf N}} }} = \frac{1}{R}\sum\limits_{{i = 1}}^R {{{\widehat{{{\bf N}}}}_i}} $$

provides an estimator for N which is unbiased, consistent and asymptotically ($R \to \infty$) normal with variance–covariance matrix which is unbiasedly and consistently estimated by ${{{\bf V}}_N} = {{{{\bf S}}} \left/ {R} \right.}$, where

$$ {{\bf S}} = \frac{1}{{R - 1}}{\sum\limits_{{i = 1}}^R {\left( {{{{\bf N}}_i} - \widehat{{\overline {{\bf N}} }}} \right)\left( {{{{\bf N}}_i} - \widehat{{\overline {{\bf N}} }}} \right)}^{\text{T}}} $$

is the empirical variance–covariance matrix of ${\widehat{{{\bf N}}}_i}{\text{s}}$.

In accordance with these results, an obvious estimator for P is given by $\widehat{{\overline {{\bf P}} }} = {{{\widehat{{\overline {{\bf N}} }}}} \left/ {{\left( {{{{\bf 1}}^{\text{T}}}\widehat{{\overline {{\bf N}} }}} \right)}} \right.}$. From the most familiar version of the Delta Method (see, e.g. Mardia et al. 1979, Theorem 2.9.2) it follows that $\widehat{{\overline {{\bf P}} }}$ constitutes a consistent and asymptotically normal estimator for P with variance–covariance matrix which is consistently estimated by ${{{\bf V}}_P} = \left( {{{\bf I}} - \widehat{{\overline {{\bf P}} }}{{{\bf 1}}^{\text{T}}}} \right){{{\bf V}}_N}\left( {{{\bf I}} - {{\bf 1}}{{\widehat{{\overline {{\bf P}} }}}^{\text{T}}}} \right)$ where I denotes the identity matrix of appropriate order. Finally, any C parameter can be simply estimated by $\widehat{{\overline C }} = {{{\bf c}}^{\text{T}}}\widehat{{\overline {{\bf P}} }}$. Thus, from the Delta Method, $\widehat{{\overline C }}$ constitutes a consistent and asymptotically normal estimator for C with variance which is consistently estimated by $V_C^2 = {{{\bf c}}^{\text{T}}}{{{\bf V}}_P}{{\bf c}}$. Similarly, denote by ${\widehat{{{\bf N}}}_t}$ the HT estimators of N _t based on a unique plot randomly selected onto ${\cal G}$ and then visited at period t (t = 1,2). From the previous considerations on HT estimators, ${\widehat{{{\bf N}}}_t}$ is unbiased with variance–covariance matrix ${\text{Va}}{{\text{r}}_{\text{URS}}}\left( {{{\widehat{{{\bf N}}}}_t}} \right)$. Moreover, denote by ${\text{Co}}{{\text{v}}_{\text{URS}}}\left( {{{\widehat{{{\bf N}}}}_1},{{\widehat{{{\bf N}}}}_2}} \right)$ the covariance matrix between the two estimators. As R sites are randomly and independently thrown onto ${\cal G}$, the replication procedure gives rise to R pairs of estimates $\left( {{{\widehat{{{\bf N}}}}_{{1,1}}},{{\widehat{{{\bf N}}}}_{{1,2}}}} \right), \ldots, \left( {{{\widehat{{{\bf N}}}}_{{R,1}}},{{\widehat{{{\bf N}}}}_{{R,2}}}} \right)$ which constitute R independent realizations of the pair $\left( {{{\widehat{{{\bf N}}}}_1},{{\widehat{{{\bf N}}}}_2}} \right)$. From the above-mentioned results on independently and identically distributed random vectors, the arithmetic mean vector of the N _i,t s, say ${\widehat{{\overline {{\bf N}} }}_t}$, is an unbiased, consistent and asymptotically normal estimator of N _t with variance–covariance matrix which is unbiasedly and consistently estimated by ${{{\bf V}}_{{N,t}}} = {{{{{{\bf S}}_t}}} \left/ {R} \right.}$, where S _t is the empirical variance–covariance matrix of the ${\widehat{{{\bf N}}}_{{i,t}}}$s, while the covariance matrix is unbiasedly and consistently estimated by ${{{\bf C}}_N} = {{{{{{\bf S}}_{{1,2}}}}} \left/ {R} \right.}$, where

$$ {{{\bf S}}_{{1,2}}} = \frac{1}{{R - 1}}{\sum\limits_{{i = 1}}^R {\left( {{{\widehat{{{\bf N}}}}_{{i,1}}} - {{\widehat{{\overline {{\bf N}} }}}_1}} \right)\left( {{{\widehat{{{\bf N}}}}_{{i,2}}} - {{\widehat{{\overline {{\bf N}} }}}_2}} \right)}^{\text{T}}} $$

is the empirical covariance matrix of the ${\widehat{{{\bf N}}}_{{i,1}}}{\text{s}}$ and ${\widehat{{{\bf N}}}_{{i,2}}}{\text{s}}$. From the Delta Method, ${\widehat{{\overline {{\bf P}} }}_t} = {{{{{\widehat{{\overline {{\bf N}} }}}_t}}} \left/ {{\left( {{{{\bf 1}}^{\text{T}}}{{\widehat{{\overline {{\bf N}} }}}_t}} \right)}} \right.}$ is a consistent and asymptotically normal estimator of P _t with variance–covariance matrix which is consistently estimated by ${{{\bf V}}_{{P,t}}} = \left( {{{\bf I}} - {{\widehat{{\overline {{\bf P}} }}}_t}{{{\bf 1}}^{\text{T}}}} \right){{{\bf V}}_{{N,t}}}\left( {{{\bf I}} - {{\bf 1}}\widehat{{\overline {{\bf P}} }}_t^{\text{T}}} \right)$ and covariance matrix which is consistently estimated by ${{{\bf C}}_P} = \left( {{{\bf I}} - {{\widehat{{\overline {{\bf P}} }}}_1}{{{\bf 1}}^{\text{T}}}} \right){{{\bf C}}_N}\left( {{{\bf I}} - {{\bf 1}}\widehat{{\overline {{\bf P}} }}_2^{\text{T}}} \right)$. From these last results, the difference ${\widehat{{\overline {{\bf P}} }}_2} - {\widehat{{\overline {{\bf P}} }}_1}$ turns out to be a consistent and asymptotically normal estimator of ${{{\bf P}}_2} - {{{\bf P}}_1}$, with variance–covariance matrix which is consistently estimated by ${{{\bf V}}_{{P,1}}} + {{{\bf V}}_{{P,2}}} - {{{\bf C}}_P} - {{\bf C}}_P^{\text{T}}$. Hence, the difference estimator $\widehat{{\overline D }} = {\widehat{{\overline C }}_2} - {\widehat{{\overline C }}_1} = {{{\bf c}}^{\text{T}}}\left( {{{\widehat{{\overline {{\bf P}} }}}_2} - {{\widehat{{\overline {{\bf P}} }}}_1}} \right)$ is a consistent and asymptotically normal estimator of D with variance which is consistently estimated by $V_D^2 = {{{\bf c}}^{\text{T}}}\left( {{{{\bf V}}_{{P,1}}} + {{{\bf V}}_{{P,2}}} - {{{\bf C}}_P} - {{\bf C}}_P^{\text{T}}} \right){{\bf c}}$.

Appendix 2. Theoretical results on estimation under systematic and stratified sampling

If the R sites/plots are thrown onto the same reference region ${\cal G}$, TSS invariably outperforms URS, in the sense that under TSS, $\widehat{{\overline {{\bf N}} }}$ is unbiased with variance–covariance matrix such that ${\text{Va}}{{\text{r}}_{\text{URS}}}\left( {\widehat{{\overline {{\bf N}} }}} \right) \geqslant {\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline {{\bf N}} }}} \right)$ (e.g. Barabesi and Franceschi 2011), where E_TSS and V_TSS denote expectations, variances and covariances arising from TSS scheme. Interestingly, TSS displays variances and covariances decreasing with R ^–3/2 (Barabesi and Franceschi 2011) while URS displays variances and covariances decreasing with R ^–1. Accordingly, for large R, TSS gives rise to relevant gains in precision with respect to the URS. Moreover, under weak assumptions, the asymptotic normality of $\widehat{{\overline {{\bf N}} }}$ is preserved in the case of TSS (Barabesi and Franceschi 2011). Hence, from an enlarged version of the Delta Method (e.g. Shao and Tu 1995, p.448), under TSS, the estimators $\widehat{{\overline {{\bf P}} }}$, $\widehat{{\overline C }}$ and $\widehat{{\overline D }}$ considered in the “Estimation under uniform random sampling” section turn out to be consistent and asymptotically normal with variances and covariances decreasing with R ^–3/2. Finally, under TSS, V _N constitutes a conservative estimator for ${\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline {{\bf N}} }}} \right)$ in the sense that ${{\text{E}}_{\text{TSS}}}\left( {{{{\bf V}}_N}} \right) = {\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline {{\bf N}} }}} \right) + {{\bf H}}$ where H is a positive definite matrix (the proof of this result is simply based on the independence of ${\widehat{{{\bf N}}}_i}$s and is not reported for brevity), while V _P , $V_C^2$ and $V_D^2$ are asymptotically conservative, in the sense that they are asymptotically equivalent to conservative estimators for ${\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline {{\bf P}} }}} \right)$, ${\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline C }}} \right)$ and ${\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline D }}} \right)$.

To prove this result, consider that under TSS, $\widehat{{\overline {{\bf P}} }}$ (being a consistent estimator of P) converges in probability to P as R increases, i.e. $\widehat{{\overline {{\bf P}} }}\mathop{ \to }\limits^p {{\bf P}}$. Hence, $\widehat{{\overline {{\bf P}} }}$ can be rewritten as $\widehat{{\overline {{\bf P}} }} = {{\bf P}} + {{{\bf D}}_R}$ where ${{{\bf D}}_R}\mathop{ \to }\limits^p {{\bf 0}}$ and 0 is a vector of zeros. In accordance with this notation, V _P can be rewritten as

$$ {{{\bf V}}_P} = \left[ {{{\bf I}} - \left( {{{\bf P}} + {{{\bf D}}_R}} \right){{{\bf 1}}^{\text{T}}}} \right]{{{\bf V}}_N}\left[ {{{\bf I}} - {{\bf 1}}{{\left( {{{\bf P}} + {{{\bf D}}_R}} \right)}^{\text{T}}}} \right] = \left( {{{\bf I}} - {{\bf P}}{{{\bf 1}}^{\text{T}}} - {{{\bf D}}_R}{{{\bf 1}}^{\text{T}}}} \right){{{\bf V}}_N}\left( {{{\bf I}} - {{\bf 1}}{{{\bf P}}^{\text{T}}} + {{\bf 1D}}_R^{\text{T}}} \right) = \left( {{{\bf I}} - {{\bf P}}{{{\bf 1}}^{\text{T}}}} \right){{{\bf V}}_N}\left( {{{\bf I}} - {{\bf 1}}{{{\bf P}}^{\text{T}}}} \right) - \left( {{{\bf I}} - {{\bf P}}{{{\bf 1}}^{\text{T}}}} \right){{{\bf V}}_N}{{\bf 1D}}_R^{\text{T}} - {{{\bf D}}_R}{{{\bf 1}}^{\text{T}}}{{{\bf V}}_N}\left( {{{\bf I}} - {{\bf 1}}{{{\bf P}}^{\text{T}}}} \right) + {{{\bf D}}_R}{{{\bf 1}}^{\text{T}}}{{{\bf V}}_N}{{\bf 1D}}_R^{\text{T}} = \left( {{{\bf I}} - {{\bf P}}{{{\bf 1}}^{\text{T}}}} \right){{{\bf V}}_N}\left( {{{\bf I}} - {{\bf 1}}{{{\bf P}}^{\text{T}}}} \right) + {{{\bf G}}_R} $$

where ${{{\bf G}}_R}\mathop{ \to }\limits^p {{\bf 0}}$. Accordingly V _P is an estimator of ${\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline {{\bf P}} }}} \right)$ which is asymptotically equivalent to ${{\bf V}}_P^{*} = \left( {{{\bf I}} - {{\bf P}}{{{\bf 1}}^{\text{T}}}} \right){{{\bf V}}_N}\left( {{{\bf I}} - {{\bf 1}}{{{\bf P}}^{\text{T}}}} \right)$. Then, since ${{\text{E}}_{{TSS}}}\left( {{{{\bf V}}_N}} \right) = {\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline {{\bf N}} }}} \right) + {{\bf H}}$, where H is a positive definite matrix, it follows that

$$ {{\text{E}}_{{TSS}}}\left( {{{\bf V}}_P^{*}} \right) = {{\text{E}}_{{TSS}}}\left\{ {\left( {{{\bf I}} - {{\bf P}}{{{\bf 1}}^{\text{T}}}} \right){{{\bf V}}_N}\left( {{{\bf I}} - {{\bf 1}}{{{\bf P}}^{\text{T}}}} \right)} \right\} = \left( {{{\bf I}} - {{\bf P}}{{{\bf 1}}^{\text{T}}}} \right){{\text{E}}_{{TSS}}}\left( {{{{\bf V}}_N}} \right)\left( {{{\bf I}} - {{\bf 1}}{{{\bf P}}^{\text{T}}}} \right) = \left( {{{\bf I}} - {{\bf P}}{{{\bf 1}}^{\text{T}}}} \right)\left\{ {{\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline {{\bf N}} }}} \right) + {{\bf H}}} \right\}\left( {{{\bf I}} - {{\bf 1}}{{{\bf P}}^{\text{T}}}} \right) = \left( {{{\bf I}} - {{\bf P}}{{{\bf 1}}^{\text{T}}}} \right){\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline {{\bf N}} }}} \right)\left( {{{\bf I}} - {{\bf 1}}{{{\bf P}}^{\text{T}}}} \right) + \left( {{{\bf I}} - {{\bf P}}{{{\bf 1}}^{\text{T}}}} \right){{\bf H}}\left( {{{\bf I}} - {{\bf 1}}{{{\bf P}}^{\text{T}}}} \right) = {\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline {{\bf P}} }}} \right) + {{\bf Q}} $$

with Q positive definite. Hence, ${{\bf V}}_P^{*}$ is a conservative estimator of ${\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline {{\bf P}} }}} \right)$, so that $V_C^{{*2}} = {{{\bf c}}^{\text{T}}}{{\bf V}}_P^{*}{{\bf c}}$ and $V_D^{{*2}} = {{{\bf c}}^{\text{T}}}\left( {{{\bf V}}_{{P,1}}^{*} + {{\bf V}}_{{P,2}}^{*} - {{\bf C}}_P^{*} - {{\bf C}}_P^{{{\text{*T}}}}} \right){{\bf c}}$ are conservative estimators of ${\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline C }}} \right)$ and ${\text{Va}}{{\text{r}}_{\text{TSS}}}\left( {\widehat{{\overline D }}} \right)$, where ${{\bf V}}_{{P,t}}^{*} = \left( {{{\bf I}} - {{{\bf P}}_t}{{{\bf 1}}^{\text{T}}}} \right){{{\bf V}}_{{N,t}}}\left( {{{\bf I}} - {{\bf 1P}}_t^{\text{T}}} \right)$ for $t = 1,2$ and ${{{\bf C}}_P} = \left( {{{\bf I}} - {{{\bf P}}_1}{{{\bf 1}}^{\text{T}}}} \right){{{\bf C}}_N}\left( {{{\bf I}} - {{\bf 1P}}_2^{\text{T}}} \right)$.

Appendix 3. Theoretical results on estimation at European level

Consider L countries participating in the forest monitoring network and denote by ${\widehat{{\overline {{\bf N}} }}_1}, \ldots, {\widehat{{\overline {{\bf N}} }}_L}$ the estimates of their abundance vectors achieved by means of separate, independent surveys performed in each country by means of plot sampling with sites selected in accordance with TSS or PSS. Since each ${\widehat{{\overline {{\bf N}} }}_l}$ constitutes an unbiased, consistent and asymptotically normal estimator of N _l with variance–covariance matrix which can be conservatively estimated by ${{{\bf V}}_{{N,l}}} = {{{{{{\bf S}}_l}}} \left/ {{{R_l}}} \right.}$, where S _l is the empirical variance–covariance matrix of the R _l estimates for the lth country and R _l is the number of sites adopted in the country, then, the sum ${\widehat{{\overline {{\bf N}} }}_E} = {\widehat{{\overline {{\bf N}} }}_1} + \ldots + {\widehat{{\overline {{\bf N}} }}_L}$ is an unbiased and consistent (${R_1}, \ldots, {R_L} \to \infty$) estimator of N _E with a variance–covariance matrix which (owing to the independence of the L estimates) is conservatively estimated by ${{{\bf V}}_{{N,E}}} = {{{\bf V}}_{{N,1}}} + \ldots + {{{\bf V}}_{{N,L}}}$. Moreover, if the R _l s are supposed to increase with constant ratios R _l /R _h , then as ${R_1}, \ldots, {R_L} \to \infty$, ${\widehat{{{\bf N}}}_E}$ is an asymptotically normal estimator of N _E .

The properties of ${\widehat{{\overline {{\bf N}} }}_E}$ (unbiasedness, consistency and asymptotic normality) once again allow for the application of the enlarged version of the Delta Method (Shao and Tu 1995, p.448). Thus, ${\widehat{{\overline {{\bf P}} }}_E} = {{{{{\widehat{{\overline {{\bf N}} }}}_E}}} \left/ {{\left( {{{{\bf 1}}^{\text{T}}}{{\widehat{{\overline {{\bf N}} }}}_E}} \right)}} \right.}$ constitutes a consistent and asymptotically normal estimator for P _E , while ${{{\bf V}}_{{P,E}}} = \left( {{{\bf I}} - {{\widehat{{\overline {{\bf P}} }}}_E}{{{\bf 1}}^{\text{T}}}} \right){{{\bf V}}_{{N,E}}}\left( {{{\bf I}} - {{\bf 1}}\widehat{{\overline {{\bf P}} }}_E^{\text{T}}} \right)$ constitutes an asymptotically conservative estimator of the variance–covariance matrix. Finally, as to C parameters at European level, ${\widehat{{\overline C }}_E} = {{{\bf c}}^{\text{T}}}{\widehat{{\overline {{\bf P}} }}_E}$ constitutes a consistent and asymptotically normal estimator for C _E , while $V_{{C,E}}^2 = {{{\bf c}}^{\text{T}}}{{{\bf V}}_{{P,E}}}{{\bf c}}$ constitutes an asymptotically conservative estimator for the variance.

As to the inference on change, denote by ${\widehat{{\overline {{\bf N}} }}_{{l,t}}}$ the plot sampling estimators of ${{{\bf N}}_{{l,t}}}\left( {t = 1,2} \right)$. Hence, ${\widehat{{\overline {{\bf N}} }}_{{l,t}}}$ is unbiased, consistent and asymptotically normal, while ${{{\bf V}}_{{N,l,t}}} = {{{{{{\bf S}}_{{l,t}}}}} \left/ {{{R_l}}} \right.}$ is a conservative estimator of the variance–covariance matrix of N _l , _t and ${{{\bf C}}_{{N,l}}} = {{{{{{\bf S}}_{{l,1,2}}}}} \left/ {{{R_l}}} \right.}$ is the estimator of the covariance matrix of N _l ,₁ and N _l ,₂, where S _l , _t is the empirical variance–covariance matrix at period t and S _{l, 1,2} is the empirical covariance matrix between period 1 and 2. Accordingly, from the previous results of Appendix 3, ${\widehat{{\overline {{\bf N}} }}_{{E,t}}} = {\widehat{{\overline {{\bf N}} }}_{{1,t}}} + \ldots + {\widehat{{\overline {{\bf N}} }}_{{L,t}}}$ is an unbiased, consistent and asymptotically normal estimator of N _E,t with variance–covariance matrix which can be conservatively estimated by ${{{\bf V}}_{{N,E,t}}} = {{{\bf V}}_{{N,1,t}}} + \ldots + {{{\bf V}}_{{N,L,t}}}$. Moreover, since correlation exists only among estimators achieved in the same country at different times, the covariance matrix of ${\widehat{{\overline {{\bf N}} }}_{{E,1}}}$ and ${\widehat{{\overline {{\bf N}} }}_{{E,2}}}$ can be estimated by ${{{\bf C}}_{{N,E}}} = {{{\bf C}}_{{N,1}}} + \ldots + {{{\bf C}}_{{N,L}}}$. From the enlarged version of the Delta Method, the relative abundance vector estimator ${\widehat{{\overline {{\bf P}} }}_{{E,t}}} = {{{{{\widehat{{\overline {{\bf N}} }}}_{{E,t}}}}} \left/ {{\left( {{{{\bf 1}}^{\text{T}}}{{\widehat{{\overline {{\bf N}} }}}_{{E,t}}}} \right)}} \right.}$ is a consistent and asymptotically normal estimator of P _E,t . Moreover, ${{{\bf V}}_{{P,E,t}}} = \left( {{{\bf I}} - {{\widehat{{\overline {{\bf P}} }}}_{{E,t}}}{{{\bf 1}}^{\text{T}}}} \right){{{\bf V}}_{{N,E,t}}}\left( {{{\bf I}} - {{\bf 1}}\widehat{{\overline {{\bf P}} }}_{{\left. {E,t} \right)}}^{\text{T}}} \right)$ is an asymptotically conservative estimator of the variance–covariance matrix of ${\widehat{{\overline {{\bf P}} }}_{{E,t}}}$, while ${{{\bf C}}_{{P,E}}} = \left( {{{\bf I}} - {{\widehat{{\overline {{\bf P}} }}}_{{E,1}}}{{{\bf 1}}^{\text{T}}}} \right){{{\bf C}}_{{N,E}}}\left( {{{\bf I}} - {{\bf 1}}\widehat{{\overline {{\bf P}} }}_{{E,2}}^{\text{T}}} \right)$ is an estimator for the covariance matrix of ${\widehat{{\overline {{\bf P}} }}_{{E,1}}}$ and ${\widehat{{\overline {{\bf P}} }}_{{E,2}}}$. From these last results, the difference ${\widehat{{\overline {{\bf P}} }}_{{E,2}}} - {\widehat{{\overline {{\bf P}} }}_{{E,1}}}$ turns out to be a consistent and asymptotically normal estimator of ${{{\bf P}}_{{E,2}}} - {{{\bf P}}_{{E,1}}}$, while ${{{\bf V}}_{{P,E,1}}} + {{{\bf V}}_{{P,E,2}}} - {{{\bf C}}_{{P,E}}} - {{\bf C}}_{{P,E}}^{\text{T}}$ is an asymptotically conservative estimator of the variance–covariance matrix. Hence, the difference estimator ${\widehat{{\overline D }}_E} = {\widehat{{\overline C }}_{{E,2}}} - {\widehat{{\overline C }}_{{E,1}}} = {{{\bf c}}^{\text{T}}}\left( {{{\widehat{{\overline {{\bf P}} }}}_{{E,2}}} - {{\widehat{{\overline {{\bf P}} }}}_{{E,1}}}} \right)$ is a consistent and asymptotically normal estimator of D _E with variance which can be conservatively estimated by $V_{{D,E}}^2 = {{{\bf c}}^{\text{T}}}\left( {{{{\bf V}}_{{P,E,1}}} + {{{\bf V}}_{{P,E,2}}} - {{{\bf C}}_{{P,E}}} - {{\bf C}}_{{\left. {P,E} \right)}}^{\text{T}}} \right){{\bf c}}$.