5. Properties of the estimators

Andrés Gutiérrez, Leonardo Trujillo and Pedro Luis do Nascimento Silva

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Following Cassel, Särndal and Wretman (1976), the aim of considering a survey sampling approach is to gather information from just a subset (sample) of units in the finite population that enable us to obtain conclusion for the whole population. During this process, the statistician must face the randomness sources defining the complex stochastic behavior of the inferential process. Although this paper considers the sampling design as the probability measure determining the inference for the parameters and the model, it is necessary to understand that the proposed Markovian model provides another correctly defined measure of probability. Now we obtain some properties of the estimators proposed in the last section.

The aim of this paper is to incorporate the sampling weights in the proposed model and then it is important to get approximately unbiased estimators with respect to the probability measure related to the sampling design for $\theta$ and $\mu$ . The following results show some properties of the proposed estimators considered under the complex survey design. In terms of notation, the probability measure induced for the sampling design will be denoted with the subindex $p.$ The following results provide the maximum likelihood estimators for the parameters of interest when instead of getting a sample, the measurement is obtained through a census or complete enumeration of the individuals in the population.

Result 5.1 Suppose there is complete access to the whole population and the log-likelihood function of the model is given by (4.1), then the maximum likelihood estimators, under the model assumptions are

$$\begin{array}{cc}{\psi }_U& =\frac{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{N}_{ij}}}+{\displaystyle {\sum }_i{R}_i}}{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{N}_{ij}}}+{\displaystyle {\sum }_i{R}_i}+{\displaystyle {\sum }_j{C}_j}+M}\hfill \\ \\ {\rho }_{RR\mathrm{,}U}& =\frac{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{N}_{ij}}}}{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{N}_{ij}}}+{\displaystyle {\sum }_i{R}_i}}\hfill \\ \\ {\rho }_{MM\mathrm{,}U}& =\frac{M}{{\displaystyle {\sum }_j{C}_j}+M}\hfill \\ \\ {\eta }_{i\mathrm{,}U}^{(v+1)}& =\frac{{\displaystyle {\sum }_j{N}_{ij}}+R_i+{\displaystyle {\sum }_j\left(C_j{\widehat{\eta }}_i^{(v)}{\widehat{p}}_{ij}^{(v)}/{\displaystyle {\sum }_i{\widehat{\eta }}_i^{(v)}{\widehat{p}}_{ij}^{(v)}}\right)}}{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{N}_{ij}}}+{\displaystyle {\sum }_i{R}_i}+{\displaystyle {\sum }_j{C}_j}}(5.1)\hfill \end{array}$$

$$p_{ij\mathrm{,}U}^{(v+1)}=\frac{N_{ij}+\left(C_j{\widehat{\eta }}_i^{(v)}{p}_{ij}^{(v)}/{\displaystyle {\sum }_i{\widehat{\eta }}_i^{(v)}{\widehat{p}}_{ij}^{(v)}}\right)}{{\displaystyle {\sum }_j{N}_{ij}}+{\displaystyle {\sum }_j\left(C_j{\widehat{\eta }}_i^{(v)}{\widehat{p}}_{ij}^{(v)}/{\displaystyle {\sum }_i{\widehat{\eta }}_i^{(v)}{\widehat{p}}_{ij}^{(v)}}\right)}}(5.2)$$

where (5.1) and (5.2) must be jointly iterated to convergence.

Result 5.2 Under the model assumptions, a maximum likelihood estimator of ${\mu }_{ij}$ is

$${\mu }_{ij\mathrm{,}U}=N\times {\eta }_{i\mathrm{,}U}\times p_{ij\mathrm{,}U}$$

where $N$ corresponds to the population size and ${\eta }_{i\mathrm{,}U}$ and $p_{ij\mathrm{,}U}$ are defined by the last result, respectively.

Note that both $\theta$ and $\mu$ can be defined as descriptive population quantities. Based on the inference approach induced by the maximum likelihood method, there exist estimators ${\theta }_U={\left({{\psi }^{\prime }}_U\mathrm{,}{{\rho }^{\prime }}_{RR,U},{{\rho }^{\prime }}_{MM,U}\mathrm{,}{{\eta }^{\prime }}_U,{p^{\prime }}_U\right)}^{\prime }$ and ${\mu }_U={\left({\mu }_{\mathrm{11,}U}\mathrm{,}\dots \mathrm{,}{\mu }_{ij\mathrm{,}U}\mathrm{,}\dots \mathrm{,}{\mu }_{GG\mathrm{,}U}\right)}^{\prime }$ defined as the corresponding descriptive population quantities making ${\theta }_{mpv}$ and ${\mu }_{mpv}$ consistent with regard to the sampling design in the sense of definition 2 in Pfeffermann (1993). Note also that ${\theta }_U$ and ${\mu }_U$ can be calculated only if there is access to the whole finite population.

Following Pessoa and Silva (1998, p. 79), it is possible to assess that under some regularity conditions, it follows that ${\theta }_U-\theta =o_p\left(1\right)$ and ${\mu }_U-\mu =o_p\left(1\right)$ . Also, as in many sampling surveys, both the population and the sample size are generally large, then an appropriate estimator of ${\theta }_U$ is also an appropriate estimator for $\theta ,$ and an appropriate estimator for ${\mu }_U$ will be an appropriate estimator for $\mu .$

In the next section, we explore the properties of the estimators proposed above and we discuss about their suitability for our research problem.

5.1 Properties of the count estimators

Result 5.3 The estimators ${\widehat{N}}_{ij},{\widehat{R}}_i,{\widehat{C}}_j,\widehat{M}$ , and $\widehat{N}$ defined in Section 4 are unbiased with regard to the sampling design.

The proof is quite immediate. The weighting factor $w_k$ corresponds to the inverse of ${\pi }_k,$ the inclusion probability associated to the $k$ -th element. All the estimators are of the Horvitz-Thompson class and therefore are unbiased.

Result 5.4 Making $w_k=1/{\pi }_k$ , the corresponding variances for ${\widehat{N}}_{ij},{\widehat{R}}_i,{\widehat{C}}_j,\widehat{M}$ and $\widehat{N},$ are given by

$$\begin{array}{lll}Va{r}_p\left({\widehat{N}}_{ij}\right)\hfill & =\hfill & {\displaystyle \sum _U}{\displaystyle \sum _U}{\Delta }_{kl}\frac{y_{1ik}{y}_{2jk}}{{\pi }_k}\frac{y_{1il}{y}_{2jl}}{{\pi }_l}\hfill \\ Va{r}_p\left({\widehat{R}}_i\right)\hfill & =\hfill & {\displaystyle \sum _U}{\displaystyle \sum _U}{\Delta }_{kl}\frac{y_{1ik}\left(1-z_{2k}\right)}{{\pi }_k}\frac{y_{1il}\left(1-z_{2l}\right)}{{\pi }_l}\hfill \\ Va{r}_p\left({\widehat{C}}_j\right)\hfill & =\hfill & {\displaystyle \sum _U}{\displaystyle \sum _U}{\Delta }_{kl}\frac{y_{2jk}\left(1-z_{1k}\right)}{{\pi }_k}\frac{y_{2jl}\left(1-z_{1l}\right)}{{\pi }_l}\hfill \\ Va{r}_p\left(\widehat{M}\right)\hfill & =\hfill & {\displaystyle \sum _U}{\displaystyle \sum _U}{\Delta }_{kl}\frac{\left(1-z_{1k}\right)}{{\pi }_k}\frac{\left(1-z_{1l}\right)}{{\pi }_l}\hfill \\ Va{r}_p\left(\widehat{N}\right)\hfill & =\hfill & {\displaystyle \sum _U}{\displaystyle \sum _U}{\Delta }_{kl}\frac{v_k}{{\pi }_k}\frac{v_l}{{\pi }_l}.\hfill \end{array}$$

Unbiased estimators for these variances, respectively, are given by

$$\begin{array}{l}\begin{array}{lll}{\widehat{Var}}_p\left({\widehat{N}}_{ij}\right)\hfill & =\hfill & {\displaystyle \sum _s}{\displaystyle \sum _s}\frac{{\Delta }_{kl}}{{\pi }_{kl}}\frac{y_{1ik}{y}_{2jk}}{{\pi }_k}\frac{y_{1il}{y}_{2jl}}{{\pi }_l}\hfill \\ {\widehat{Var}}_p\left({\widehat{R}}_i\right)\hfill & =\hfill & {\displaystyle \sum _s}{\displaystyle \sum _s}\frac{{\Delta }_{kl}}{{\pi }_{kl}}\frac{y_{1ik}\left(1-z_{2k}\right)}{{\pi }_k}\frac{y_{1il}\left(1-z_{2l}\right)}{{\pi }_l}\hfill \\ {\widehat{Var}}_p\left({\widehat{C}}_j\right)\hfill & =\hfill & {\displaystyle \sum _s}{\displaystyle \sum _s}\frac{{\Delta }_{kl}}{{\pi }_{kl}}\frac{y_{2jk}\left(1-z_{1k}\right)}{{\pi }_k}\frac{y_{2jl}\left(1-z_{1l}\right)}{{\pi }_l}\hfill \end{array}\\ \begin{array}{lll}{\widehat{Var}}_p\left(\widehat{M}\right)\hfill & =\hfill & {\displaystyle \sum _s}{\displaystyle \sum _s}\frac{{\Delta }_{kl}}{{\pi }_{kl}}\frac{\left(1-z_{1k}\right)}{{\pi }_k}\frac{\left(1-z_{1l}\right)}{{\pi }_l}\hfill \\ {\widehat{Var}}_p\left(\widehat{N}\right)\hfill & =\hfill & {\displaystyle \sum _s}{\displaystyle \sum _s}\frac{{\Delta }_{kl}}{{\pi }_{kl}}\frac{v_k}{{\pi }_k}\frac{v_l}{{\pi }_l}.\hfill \end{array}\end{array}$$

On the other hand, if the $w_k$ correspond to calibration weights, then all the estimators considered are asymptotically unbiased and proofs are given in Deville and Särndal (1992). Their corresponding variances are given by Kim and Park (2010).

5.2 Properties of the model probabilities estimators

Result 5.5 The first-order Taylor approximation for the estimator ${\psi }_{mpv}$ , defined at the result 4.2 above, around the point $\left(N_{ij}\mathrm{,}{R}_i\mathrm{,}{C}_j\mathrm{,}M\right)$ and $i\mathrm{,}j=1,\dots \mathrm{,}G$ , is given by the expression

$$\begin{array}{ll}{\widehat{\psi }}_{mpv}\hfill & \cong {\widehat{\psi }}_0\hfill \\ \hfill & ={\psi }_U+a_1{\displaystyle \sum _i}{\displaystyle \sum _j}\left({\widehat{N}}_{ij}-N_{ij}\right)+a_1{\displaystyle \sum _i}\left({\widehat{R}}_i-R_i\right)\hfill \\ \hfill & +a_2{\displaystyle \sum _j}\left({\widehat{C}}_j-C_j\right)+a_2\left(\widehat{M}-M\right)\hfill \end{array}$$

with

$$\begin{array}{l}{a}_1=\frac{{\displaystyle {\sum }_j{C}_j}+M}{{\left({\displaystyle {\sum }_i{\displaystyle {\sum }_j{N}_{ij}}}+{\displaystyle {\sum }_i{R}_i}+{\displaystyle {\sum }_j{C}_j}+M\right)}^2}\\ a_2=-\frac{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{N}_{ij}}}+{\displaystyle {\sum }_i{R}_i}}{{\left({\displaystyle {\sum }_i{\displaystyle {\sum }_j{N}_{ij}}}+{\displaystyle {\sum }_i{R}_i}+{\displaystyle {\sum }_j{C}_j}+M\right)}^2}.\end{array}$$

Result 5.6 The first-order Taylor approximation for the estimator ${\widehat{\rho }}_{RR\mathrm{,}mpv}$ , defined at the result 4.2 above, around the point $\left(N_{ij}\mathrm{,}{R}_i\right)$ and $i\mathrm{,}j=1,\dots \mathrm{,}G$ , is given by the expression

$$\begin{array}{ll}{\widehat{\rho }}_{RR\mathrm{,}mpv}& \cong {\widehat{\rho }}_{RR\mathrm{,0}}\hfill \\ & ={\rho }_{RR\mathrm{,}U}+a_3{\displaystyle \sum _i}{\displaystyle \sum _j}\left({\widehat{N}}_{ij}-N_{ij}\right)+a_4{\displaystyle \sum _i}\left({\widehat{R}}_i-R_i\right)\hfill \end{array}$$

with

$$\begin{array}{l}{a}_3=\frac{{\displaystyle {\sum }_i{R}_i}}{{\left({\displaystyle {\sum }_i{\displaystyle {\sum }_j{N}_{ij}}}+{\displaystyle {\sum }_i{R}_i}\right)}^2}\\ a_4=-\frac{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{N}_{ij}}}}{{\left({\displaystyle {\sum }_i{\displaystyle {\sum }_j{N}_{ij}}}+{\displaystyle {\sum }_i{R}_i}\right)}^2}.\end{array}$$

Result 5.7 The first-order Taylor approximation for the estimator ${\widehat{\rho }}_{MM\mathrm{,}mpv}$ , defined at the result 4.2 above, around the point $\left(C_j\mathrm{,}M\right)$ and $j=1,\dots \mathrm{,}G,$ is given by the expression

$$\begin{array}{ll}{\widehat{\rho }}_{MM\mathrm{,}mpv}& \cong {\widehat{\rho }}_{MM\mathrm{,0}}\hfill \\ & ={\rho }_{MM\mathrm{,}U}+a_5{\displaystyle \sum _j}\left({\widehat{C}}_j-C_j\right)+a_6\left(\widehat{M}-M\right)\hfill \end{array}$$

with

$$\begin{array}{l}{a}_5=-\frac{M}{{\left({\displaystyle {\sum }_j{C}_j}+M\right)}^2}\\ a_6=-\frac{{\displaystyle {\sum }_j{C}_j}}{{\left({\displaystyle {\sum }_j{C}_j}+M\right)}^2}.\end{array}$$

Result 5.8 The estimators ${\widehat{\psi }}_{mpv},{\widehat{\rho }}_{MM\mathrm{,}mpv}$ and ${\widehat{\rho }}_{RR\mathrm{,}mpv},$ are approximately unbiased for ${\psi }_U,{\rho }_{MM\mathrm{,}U},{\rho }_{RR\mathrm{,}U}.$

Result 5.9 The estimators ${\widehat{\eta }}_{i\mathrm{,}mpv}$ and ${\widehat{p}}_{ij\mathrm{,}mpv}$ , are approximately unbiased for ${\eta }_{i\mathrm{,}U}$ and $p_{ij\mathrm{,}U}$ .

Result 5.10 The approximate variances for the estimators ${\widehat{\psi }}_{mpv},{\widehat{\rho }}_{MM\mathrm{,}mpv}$ and ${\widehat{\rho }}_{RR\mathrm{,}mpv},$ are given by

$$\begin{array}{cc}A{V}_p\left({\widehat{\psi }}_{mpv}\right)& =V_p\left({\displaystyle \sum _s}\frac{E_k^{\psi }}{{\pi }_k}\right)={\displaystyle \sum _U}{\displaystyle \sum _U}{\Delta }_{kl}\frac{E_k^{\psi }}{{\pi }_k}\frac{E_l^{\psi }}{{\pi }_l}\hfill \\ A{V}_p\left({\widehat{\rho }}_{RR\mathrm{,}mpv}\right)& =V_p\left({\displaystyle \sum _s}\frac{E_k^{RR}}{{\pi }_k}\right)={\displaystyle \sum _U}{\displaystyle \sum _U}{\Delta }_{kl}\frac{E_k^{RR}}{{\pi }_k}\frac{E_l^{RR}}{{\pi }_l}\hfill \\ A{V}_p\left({\widehat{\rho }}_{MM\mathrm{,}mpv}\right)& =V_p\left({\displaystyle \sum _s}\frac{E_k^{MM}}{{\pi }_k}\right)={\displaystyle \sum _U}{\displaystyle \sum _U}{\Delta }_{kl}\frac{E_k^{MM}}{{\pi }_k}\frac{E_l^{MM}}{{\pi }_l}\hfill \end{array}$$

where

$$\begin{array}{cc}{E}_k^{\psi }& =a_1\left(2-z_{2k}\right)+a_2\left(1-z_{1k}\right)\left(2-z_{2k}\right)\hfill \\ E_k^{RR}& =a_3+a_4\left(1-z_{2k}\right)\hfill \\ E_k^{MM}& =a_5\left(1-z_{1k}\right)+a_6\left(1-z_{1k}\right)\left(1-z_{2k}\right).\hfill \end{array}$$

Result 5.11 Unbiased estimators for the approximate variances of the estimators ${\widehat{\psi }}_{mpv},{\widehat{\rho }}_{MM\mathrm{,}mpv}$ and ${\widehat{\rho }}_{RR\mathrm{,}mpv},$ are given by

$$\begin{array}{cc}\widehat{V}\left({\widehat{\psi }}_{mpv}\right)& ={\displaystyle \sum _s}{\displaystyle \sum _s}\frac{{\Delta }_{kl}}{{\pi }_{kl}}\frac{e_k^{\psi }}{{\pi }_k}\frac{e_l^{\psi }}{{\pi }_l}\hfill \\ \widehat{V}\left({\widehat{\rho }}_{RR\mathrm{,}mpv}\right)& ={\displaystyle \sum _s}{\displaystyle \sum _s}\frac{{\Delta }_{kl}}{{\pi }_{kl}}\frac{e_k^{RR}}{{\pi }_k}\frac{e_l^{RR}}{{\pi }_l}\hfill \\ \widehat{V}\left({\widehat{\rho }}_{MM\mathrm{,}mpv}\right)& ={\displaystyle \sum _s}{\displaystyle \sum _s}\frac{{\Delta }_{kl}}{{\pi }_{kl}}\frac{e_k^{MM}}{{\pi }_k}\frac{e_l^{MM}}{{\pi }_l}\hfill \end{array}$$

respectively, where

$$\begin{array}{cc}{e}_k^{\psi }& ={\widehat{a}}_1\left(2-z_{2k}\right)+{\widehat{a}}_2\left(1-z_{1k}\right)\left(2-z_{2k}\right)\hfill \\ e_k^{RR}& ={\widehat{a}}_3+{\widehat{a}}_4\left(1-z_{2k}\right)\hfill \\ e_k^{MM}& ={\widehat{a}}_5\left(1-z_{1k}\right)+{\widehat{a}}_6\left(1-z_{1k}\right)\left(1-z_{2k}\right)\hfill \end{array}$$

and

$$\begin{array}{l}{\widehat{a}}_1=\frac{{\displaystyle {\sum }_j{\widehat{C}}_j}+\widehat{M}}{{\left({\displaystyle {\sum }_i{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}+{\displaystyle {\sum }_i{\widehat{R}}_i}+{\displaystyle {\sum }_j{\widehat{C}}_j}+\widehat{M}\right)}^2}\\ {\widehat{a}}_2=-\frac{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}+{\displaystyle {\sum }_i{\widehat{R}}_i}}{{\left({\displaystyle {\sum }_i{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}+{\displaystyle {\sum }_i{\widehat{R}}_i}+{\displaystyle {\sum }_j{\widehat{C}}_j}+\widehat{M}\right)}^2}\\ {\widehat{a}}_3=\frac{{\displaystyle {\sum }_i{\widehat{R}}_i}}{{\left({\displaystyle {\sum }_i{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}+{\displaystyle {\sum }_i{\widehat{R}}_i}\right)}^2}\\ {\widehat{a}}_4=-\frac{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}}{{\left({\displaystyle {\sum }_i{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}+{\displaystyle {\sum }_i{\widehat{R}}_i}\right)}^2}\\ {\widehat{a}}_5=-\frac{\widehat{M}}{{\left({\displaystyle {\sum }_j{\widehat{C}}_j}+\widehat{M}\right)}^2}\\ {\widehat{a}}_6=-\frac{{\displaystyle {\sum }_j{\widehat{C}}_j}}{{\left({\displaystyle {\sum }_j{\widehat{C}}_j}+\widehat{M}\right)}^2}.\end{array}$$

Result 5.12 The approximate variances for the estimators ${\widehat{\eta }}_{i\mathrm{,}mpv}$ and ${\widehat{p}}_{ij\mathrm{,}mpv}$ are given by

$$\begin{array}{ll}A{V}_p\left({\widehat{\eta }}_{i\mathrm{,}mpv}\right)& =\frac{1}{{\left(J_{{\eta }_i}\right)}^2}{\displaystyle \sum _U}{\displaystyle \sum _U}{\Delta }_{kl}\frac{u_k\left({\eta }_i\right)}{{\pi }_k}\frac{u_l\left({\eta }_i\right)}{{\pi }_l}\hfill \\ A{V}_p\left({\widehat{p}}_{ij\mathrm{,}mpv}\right)& =\frac{1}{{\left(J_{p_{ij}}\right)}^2}{\displaystyle \sum _U}{\displaystyle \sum _U}{\Delta }_{kl}\frac{u_k\left(p_{ij}\right)}{{\pi }_k}\frac{u_l\left(p_{ij}\right)}{{\pi }_l}\hfill \end{array}$$

where

$$\begin{array}{l}{u}_k\left({\eta }_i\right)=\frac{{\displaystyle {\sum }_j{y}_{1ik}{y}_{2jk}}+y_{1ik}\left(1-z_{2k}\right)}{{\eta }_i}+{\displaystyle \sum _j}{y}_{2jk}\left(1-z_{1k}\right)\frac{p_{ij}}{{\displaystyle {\sum }_i{\eta }_i{p}_{ij}}}+\left(1-z_{1k}\right)\left(1-z_{2k}\right)-1\\ \\ u_k\left(p_{ij}\right)=\frac{y_{1ik}{y}_{2jk}}{p_{ij}}+y_{1ik}\left(1-z_{2k}\right)+y_{2jk}\left(1-z_{1k}\right)\frac{{\eta }_i}{{\displaystyle {\sum }_i{\eta }_i{p}_{ij}}}+\left(1-z_{1k}\right)\left(1-z_{2k}\right){\eta }_i\\ -\frac{1}{\widehat{N}}\left({\displaystyle \sum _j}{\widehat{N}}_{ij}+{\widehat{R}}_i+\widehat{M}{\eta }_i+{\displaystyle \sum _j}{\widehat{C}}_j\left(\frac{{\eta }_i{p}_{ij}}{{\displaystyle {\sum }_i{\eta }_i{p}_{ij}}}\right)\right)\\ \\ J_{{\eta }_i}=-\frac{2}{{\eta }_i^2}{\displaystyle \sum _U}{y}_{1ik}+\frac{1}{{\eta }_i^2}{\displaystyle \sum _U}{y}_{1ik}{z}_{2k}-{\displaystyle \sum _U}\left(1-z_{1k}\right){\displaystyle \sum _j}\frac{y_{2jk}{p}_{ij}^2}{{\left({\displaystyle {\sum }_i{\eta }_i{p}_{ij}}\right)}^2}\\ J_{p_{ij}}=-\frac{1}{p_{ij}^2}{\displaystyle \sum _U}{y}_{1ik}{y}_{2jk}-\frac{{\eta }_i^2}{{\left({\displaystyle {\sum }_i{\eta }_i{p}_{ij}}\right)}^2}{\displaystyle \sum _U}{y}_{2jk}\left(1-z_{1k}\right).\end{array}$$

Result 5.13 Unbiased estimators for the approximate variances of the estimators ${\widehat{\eta }}_{i\mathrm{,}mpv}$ and ${\widehat{p}}_{ij\mathrm{,}mpv}$ are given by

$$\begin{array}{cc}{\widehat{V}}_p\left({\widehat{\eta }}_{i\mathrm{,}mpv}\right)& =\frac{1}{{\left({\widehat{J}}_{{\widehat{\eta }}_i}\right)}^2}{\displaystyle \sum _s}{\displaystyle \sum _s}\frac{{\Delta }_{kl}}{{\pi }_{kl}}\frac{{\widehat{u}}_k\left({\widehat{\eta }}_i\right)}{{\pi }_k}\frac{{\widehat{u}}_l\left({\widehat{\eta }}_i\right)}{{\pi }_l}\hfill \\ {\widehat{V}}_p\left({\widehat{p}}_{ij\mathrm{,}mpv}\right)& =\frac{1}{{\left({\widehat{J}}_{{\widehat{p}}_{ij}}\right)}^2}{\displaystyle \sum _s}{\displaystyle \sum _s}\frac{{\Delta }_{kl}}{{\pi }_{kl}}\frac{{\widehat{u}}_k\left({\widehat{p}}_{ij}\right)}{{\pi }_k}\frac{{\widehat{u}}_l\left({\widehat{p}}_{ij}\right)}{{\pi }_l}\hfill \end{array}$$

where

$$\begin{array}{cc}{\widehat{u}}_k\left({\widehat{\eta }}_i\right)& =\frac{{\displaystyle {\sum }_j{y}_{1ik}{y}_{2jk}}+y_{1ik}\left(1-z_{2k}\right)}{{\widehat{\eta }}_i}+{\displaystyle \sum _j}{y}_{2jk}\left(1-z_{1k}\right)\frac{{\widehat{p}}_{ij\mathrm{,}mpv}}{{\displaystyle {\sum }_i{\widehat{\eta }}_{i\mathrm{,}mpv}{\widehat{p}}_{ij\mathrm{,}mpv}}}+\left(1-z_{1k}\right)\left(1-z_{2k}\right)\hfill \\ {\widehat{u}}_k\left({\widehat{p}}_{ij}\right)& =\frac{y_{1ik}{y}_{2jk}}{{\widehat{p}}_{ij\mathrm{,}mpv}}+y_{1ik}\left(1-z_{2k}\right)+y_{2jk}\left(1-z_{1k}\right)\frac{{\widehat{\eta }}_{i\mathrm{,}mpv}}{{\displaystyle {\sum }_i{\widehat{\eta }}_{i\mathrm{,}mpv}{p}_{ij\mathrm{,,}mpv}}}+\left(1-z_{1k}\right)\left(1-z_{2k}\right){\widehat{\eta }}_{i\mathrm{,}mpv}\hfill \end{array}$$

and

$$\begin{array}{ll}{\widehat{J}}_{{\widehat{\eta }}_i}& =-\frac{2}{{\widehat{\eta }}_{i\mathrm{,}mpv}^2}{\displaystyle \sum _U}{y}_{1ik}+\frac{1}{{\widehat{\eta }}_{i\mathrm{,}mpv}^2}{\displaystyle \sum _U}{y}_{1ik}{z}_{2k}-{\displaystyle \sum _U}(1-z_{1k}){\displaystyle \sum _j}\frac{y_{2jk}{\widehat{p}}_{ij\mathrm{,}mpv}^2}{{\left({\displaystyle {\sum }_i{\widehat{\eta }}_{i\mathrm{,}mpv}{\widehat{p}}_{ij\mathrm{,}mpv}}\right)}^2}\hfill \\ {\widehat{J}}_{{\widehat{p}}_{ij}}& =-\frac{1}{{\widehat{p}}_{ij\mathrm{,}mpv}^2}{\displaystyle \sum _U}{y}_{1ik}{y}_{2jk}-\frac{{\widehat{\eta }}_{i\mathrm{,}mpv}^2}{{\left({\displaystyle {\sum }_i{\widehat{\eta }}_{i\mathrm{,}mpv}{\widehat{p}}_{ij\mathrm{,}mpv}}\right)}^2}{\displaystyle \sum _U}{y}_{2jk}\left(1-z_{1k}\right).\hfill \end{array}$$

5.3 Properties of gross flows estimators

Result 5.14 Under the model assumptions, the first-order Taylor approximation of the gross flows estimator given by ${\widehat{\mu }}_{ij}$ and defined in result 4.4, around the point $(N\mathrm{,}{\eta }_{i\mathrm{,}U}\mathrm{,}{p}_{ij\mathrm{,}U})$ and $i\mathrm{,}j=1,\dots \mathrm{,}G$ , is given by

$$\begin{array}{ll}{\widehat{\mu }}_{ij\mathrm{,}mpv}& \cong {\widehat{\mu }}_{ij\mathrm{,0}}\hfill \\ & ={\mu }_{ij\mathrm{,}U}+a_7\left({\widehat{N}}_{ij}-N_{ij}\right)+a_8\left({\widehat{\eta }}_{i\mathrm{,}mpv}-{\eta }_{i\mathrm{,}U}\right)+a_9\left({\widehat{p}}_{ij\mathrm{,}mpv}-p_{ij\mathrm{,}U}\right)\hfill \end{array}$$

with

$$\begin{array}{c}{a}_7={\eta }_{i\mathrm{,}U}{p}_{ij\mathrm{,}U}\\ a_8=N_{ij}{p}_{ij\mathrm{,}U}\\ a_9=N_{ij}{\eta }_{i,U}.\end{array}$$

Result 5.15 The gross flows estimator ${\widehat{\mu }}_{ij\mathrm{,}mpv}$ is approximately unbiased for ${\mu }_{ij\mathrm{,}U}.$

Result 5.16 The following expression approximate the variance for ${\widehat{\mu }}_{ij\mathrm{,}mpv}$

$$A{V}_p\left({\widehat{\mu }}_{ij\mathrm{,}mpv}\right)\cong a_7^{2}Va{r}_p\left({\widehat{N}}_{ij}\right)+a_8^{2}A{V}_p\left({\widehat{\eta }}_{i\mathrm{,}mpv}\right)+a_9^{2}A{V}_p\left({\widehat{p}}_{ij}\right).(5.3)$$

Result 5.17 An approximately unbiased estimator for the asymptotic variance in (5.3) is given by

$${\widehat{V}}_p\left({\widehat{\mu }}_{ij\mathrm{,}mpv}\right)={\widehat{a}}_7^2{\widehat{V}}_p\left({\widehat{N}}_{ij}\right)+a_8^2{\widehat{V}}_p\left({\widehat{\eta }}_{i\mathrm{,}mpv}\right)+a_9^2{\widehat{V}}_p\left({\widehat{p}}_{ij}\right)$$

with

$$\begin{array}{l}{\widehat{a}}_7={\widehat{\eta }}_{i\mathrm{,}U}{\widehat{p}}_{ij\mathrm{,}U}\\ {\widehat{a}}_8={\widehat{N}}_{ij}{\widehat{p}}_{ij\mathrm{,}U}\\ {\widehat{a}}_9={\widehat{N}}_{ij}{\widehat{\eta }}_{i\mathrm{,}U}.\end{array}$$

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