Statistical matching using fractional imputation 3. Fractional imputationStatistical matching using fractional imputation 3. Fractional imputation

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We now describe the fractional imputation methods for statistical matching without using the CI assumption. The use of fractional imputation for statistical matching was originally presented in Chapter 9 of Kim and Shao (2013) under the IV assumption. In this paper, we present the methodology without requiring the IV assumption. We only assume that the specified model is fully identified. The identifiability of the specified model can be easily checked in the computation of the proposed procedure.

To explain the idea, note that $y_1$ is missing in Sample B and our goal is to generate $y_1$ from the conditional distribution of $y_1$ given the observations. That is, we wish to generate $y_1$ from

$$f\left(y_1|x\mathrm{,}{y}_2\right)\propto f\left(y_2|x\mathrm{,}{y}_1\right)f\left(y_1|x\right)\mathrm{.}(3.1)$$

To generate $y_1$ from (3.1), we can consider the following two-step imputation:

1. Generate $y_1^{*}$ from ${\widehat{f}}_a\left(y_1|x\right).$
2. Accept $y_1^{*}$ if $f\left(y_2|x\mathrm{,}{y}_1^{*}\right)$ is sufficiently large.

Note that the first step is the usual method under the CI assumption. The second step incorporates the information in $y_2.$ The determination of whether $f\left(y_2|x\mathrm{,}{y}_1^{*}\right)$ is sufficiently large required for Step 2 is often made by applying a Markov Chain Monte Carlo (MCMC) method such as the Metropolis-Hastings algorithm (Chib and Greenberg 1995). That is, let $y_1^{\left(t-1\right)}$ be the current value of $y_1$ in the Markov Chain. Then, we accept $y_1^{*}$ with probability

$$R\left(y_1^{*}\mathrm{,}{y}_1^{\left(t-1\right)}\right)\mathrm{<mo>=</mo>}\min \left\{\mathrm{1,}\frac{f\left(y_2|x\mathrm{,}{y}_1^{*}\right)}{f\left(y_2|x\mathrm{,}{y}_1^{\left(t-1\right)}\right)}\right\}\mathrm{.}$$

Such algorithms can be computationally cumbersome because of slow convergence of the MCMC algorithm.

Parametric fractional imputation of Kim (2011) enables generating imputed values in (3.1) without requiring MCMC. The following EM algorithm by fractional imputation can be used:

1. For each $i\in B,$ generate $m$ imputed values of $y_{1i},$ denoted by $y_{1i}^{\mathrm{<mo>*</mo>}\left(1\right)}\mathrm{,}\dots \mathrm{,}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(m\right)},$ from ${\widehat{f}}_a\left(y_1|x_i\right),$ where ${\widehat{f}}_a\left(y_1|x\right)$ denotes the estimated density for the conditional distribution of $y_1$ given $x$ obtained from Sample A.
2. Let ${\widehat{\theta }}_t$ be the current parameter value of $\theta$ in $f\left(y_2|x\mathrm{,}{y}_1\right).$ For the $j^{\text{th}}$ imputed value $y_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)},$ assign the fractional weight
3. $$w_{ij\left(t\right)}^{*}\propto f\left(y_{2i}|x_i\mathrm{,}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{<mo>;</mo>}{\widehat{\theta }}_t\right)$$
4. such that ${\displaystyle {\sum }_{j\mathrm{<mo>=</mo>1}}^m}{w}_{ij}^{*}\mathrm{<mo>=</mo>1.}$

3. Solve the fractionally imputed score equation for $\theta$
4. $${\displaystyle \sum _{i\in B}}{w}_{ib}{\displaystyle \sum _{j\mathrm{<mo>=</mo>1}}^m}{w}_{ij\left(t\right)}^{*}S\left(\theta \mathrm{<mo>;</mo>}{x}_i\mathrm{,}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{,}{y}_{2i}\right)\mathrm{<mo>=</mo>0}(3.2)$$
5. to obtain ${\widehat{\theta }}_{t+1},$ where $S\left(\theta \mathrm{<mo>;</mo>}x\mathrm{,}{y}_1\mathrm{,}{y}_2\right)\mathrm{<mo>=</mo>}\partial \log f\left(y_2|x\mathrm{,}{y}_1\mathrm{<mo>;</mo>}\theta \right)/\partial \theta ,$ and $w_{ib}$ is the sampling weight of unit $i$ in Sample B.

4. Go to Step 2 and continue until convergence.

When the model is identified, the EM sequence obtained from the above PFI method will converge. If the specified model is not identifiable then there is no unique solution to maximizing the observed likelihood and the above EM sequence does not converge. In (3.2), note that, for sufficiently large $m,$

$$\begin{array}{ll}{\displaystyle \sum _{j\mathrm{<mo>=</mo>1}}^m}{w}_{ij\left(t\right)}^{*}S\left(\theta \mathrm{<mo>;</mo>}{x}_i\mathrm{,}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{,}{y}_{2i}\right)\hfill & \cong \frac{{\displaystyle \int }S\left(\theta \mathrm{<mo>;</mo>}{x}_i\mathrm{,}{y}_1\mathrm{,}{y}_{2i}\right)f\left(y_{2i}|x_i\mathrm{,}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{<mo>;</mo>}{\widehat{\theta }}_t\right){\widehat{f}}_a\left(y_1|x_i\right)d{y}_1}{{\displaystyle \int }f\left(y_{2i}|x_i\mathrm{,}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{<mo>;</mo>}{\widehat{\theta }}_t\right){\widehat{f}}_a\left(y_1|x_i\right)d{y}_1}\hfill \\ \hfill & \mathrm{<mo>=</mo>}E\left\{S\left(\theta \mathrm{<mo>;</mo>}{x}_i\mathrm{,}{Y}_1\mathrm{,}{y}_{2i}\right)|x_i\mathrm{,}{y}_{2i}\mathrm{<mo>;</mo>}{\widehat{\theta }}_t\right\}\mathrm{.}\hfill \end{array}$$

If $y_{i1}$ is categorical, then the fractional weight can be constructed by the conditional probability corresponding to the realized imputed value (Ibrahim 1990). Step 2 is used to incorporate observed information of $y_{i2}$ in Sample B. Note that Step 1 is not repeated for each iteration. Only Step 2 and Step 3 are iterated until convergence. Because Step 1 is not iterated, convergence is guaranteed and the observed likelihood increases, as long as the model is identifiable. See Theorem 2 of Kim (2011).

Remark 3.1 In Section 2, we introduce IV only because this is what it is typically done in the literature to ensure identifiability. The proposed method itself does not rely on this assumption. To illustrate a situation where we can identify the model without introducing the IV assumption, suppose that the model is

$$\begin{array}{ll}{y}_2\hfill & \mathrm{<mo>=</mo>}{\beta }_0+{\beta }_{1}x+{\beta }_2{y}_1+e_2\hfill \\ y_1\hfill & \mathrm{<mo>=</mo>}{\alpha }_0+{\alpha }_{1}x+e_1\hfill \end{array}$$

with $e_1\sim N\left(\mathrm{0,}{x}^2{\sigma }_1^2\right)$  and $e_2|e_1\sim N\left(\mathrm{0,}{\sigma }_2^2\right).$  Then

$$f\left(y_2|x\right)\mathrm{<mo>=</mo>}{\displaystyle \int }f\left(y_2|x\mathrm{,}{y}_1\right)f\left(y_1|x\right)d{y}_1$$

is also a normal distribution with mean $\left({\beta }_0+{\beta }_2{\alpha }_0\right)+\left({\beta }_1+{\beta }_2{\alpha }_1\right)x$  and variance ${\sigma }_2^2+{\beta }_2^2{\sigma }_1^2{x}^2.$  Under the data structure in Table 1.1, such a model is identified without assuming the IV assumption. The assumption of no interaction between $y_1$  and $x$  in the model for $y_2$  is key to ensuring the model is identifiable.

Instead of generating $y_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}$ from ${\widehat{f}}_a\left(y_1|x_i\right),$ we can consider a hot-deck fractional imputation (HDFI) method, where all the observed values of $y_{1i}$ in Sample A are used as imputed values. In this case, the fractional weights in Step 2 are given by

$$w_{ij}^{*}\left({\widehat{\theta }}_t\right)\propto w_{ij0}^{*}f\left(y_{2i}|x_i\mathrm{,}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{<mo>;</mo>}{\widehat{\theta }}_t\right)\mathrm{,}$$

where

$$w_{ij0}^{*}\mathrm{<mo>=</mo>}\frac{{\widehat{f}}_a\left(y_{1j}|x_i\right)}{{\displaystyle \sum _{k\in A}}{w}_{ka}{\widehat{f}}_a\left(y_{1j}|x_k\right)}\mathrm{.}(3.3)$$

The initial fractional weight $w_{ij0}^{*}$ in (3.3) is computed by applying importance weighting with

$${\widehat{f}}_a\left(y_{1j}\right)\mathrm{<mo>=</mo>}{\displaystyle \int }{\widehat{f}}_a\left(y_{1j}|x\right){\widehat{f}}_a\left(x\right)dx\propto {\displaystyle \sum _{i\in A}}{w}_{ia}{\widehat{f}}_a\left(y_{1j}|x_i\right)$$

as the proposal density for $y_{1j}.$ The M-step is the same as for parametric fractional imputation. See Kim and Yang (2014) for more details on HDFI. In practice, we may use a single imputed value for each unit. In this case, the fractional weights can be used as the selection probability in Probability-Proportional-to-Size (PPS) sampling of size $m\mathrm{<mo>=</mo>\; 1.}$

For variance estimation, we can either use a linearization method or a resampling method. We first consider variance estimation for the maximum likelihood estimator (MLE) of $\theta .$ If we use a parametric model $f\left(y_1|x\right)\mathrm{<mo>=</mo>}f\left(y_1|x\mathrm{<mo>;</mo>}{\theta }_1\right)$ and $f\left(y_2|x\mathrm{,}{y}_1\mathrm{<mo>;</mo>}{\theta }_2\right),$ the MLE of $\theta \mathrm{<mo>=</mo>}\left({\theta }_1\mathrm{,}{\theta }_2\right)$ is obtained by solving

$$\left[S_1\left({\theta }_1\right)\mathrm{,}{\overline{S}}_2\left({\theta }_1\mathrm{,}{\theta }_2\right)\right]\mathrm{<mo>=</mo>}\left(\mathrm{0,0}\right)\mathrm{,}(3.4)$$

where $S_1\left({\theta }_1\right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in A}}{w}_{ia}{S}_{i1}\left({\theta }_1\right),$ $S_{i1}\left({\theta }_1\right)\mathrm{<mo>=</mo>}\partial \log f\left(y_{1i}|x_i\mathrm{<mo>;</mo>}{\theta }_1\right)/\partial {\theta }_1$ is the score function of ${\theta }_1,$

$${\overline{S}}_2\left({\theta }_1\mathrm{,}{\theta }_2\right)\mathrm{<mo>=</mo>}E\left\{S_2\left({\theta }_2\right)|X\mathrm{,}{Y}_2\mathrm{<mo>;</mo>}{\theta }_1\mathrm{,}{\theta }_2\right\}\mathrm{,}$$

$S_2\left({\theta }_2\right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in B}}{w}_{ib}{S}_{i2}\left({\theta }_2\right),$ and $S_{i2}\left({\theta }_2\right)\mathrm{<mo>=</mo>}\partial \log f\left(y_{2i}|x_i\mathrm{,}{y}_{1i}\mathrm{<mo>;</mo>}{\theta }_2\right)/\partial {\theta }_2$ is the score function of ${\theta }_2.$ Note that we can write ${\overline{S}}_2\left({\theta }_1\mathrm{,}{\theta }_2\right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in B}}{w}_{ib}E\left\{S_{i2}\left({\theta }_2\right)|x_i\mathrm{,}{y}_{2i}\mathrm{<mo>;</mo>}\theta \right\}.$ Thus,

$$\begin{array}{ll}\frac{\partial }{\partial {{\theta }^{ \prime }}_{}}{\overline{S}}_2\left(\theta \right)\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in B}}{w}_{ib}\frac{\partial }{\partial {{\theta }^{ \prime }}_{}}\left[\frac{{\displaystyle \int }{S}_{i2}\left({\theta }_2\right)f\left(y_1|x_i\mathrm{<mo>;</mo>}{\theta }_1\right)f\left(y_{2i}|x_i\mathrm{,}{y}_1\mathrm{<mo>;</mo>}{\theta }_2\right)d{y}_1}{{\displaystyle \int }f\left(y_1|x_i\mathrm{<mo>;</mo>}{\theta }_1\right)f\left(y_{2i}|x_i\mathrm{,}{y}_1\mathrm{<mo>;</mo>}{\theta }_2\right)d{y}_1}\right]\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in B}}{w}_{ib}E\left\{S_{i2}\left({\theta }_2\right)S_{i1}\left({\theta }_1\right)\prime |x_i\mathrm{,}{y}_{2i}\mathrm{<mo>;</mo>}\theta \right\}\hfill \\ \hfill & -{\displaystyle \sum _{i\in B}}{w}_{ib}E\left\{S_{i2}\left({\theta }_2\right)|x_i\mathrm{,}{y}_{2i}\mathrm{<mo>;</mo>}\theta \right\}E\left\{S_{i1}\left({\theta }_1\right)\prime |x_i\mathrm{,}{y}_{2i}\mathrm{<mo>;</mo>}\theta \right\}\hfill \end{array}$$

and

$$\begin{array}{ll}\frac{\partial }{\partial {{\theta }^{ \prime }}_{}}{\overline{S}}_2\left(\theta \right)\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in B}}{w}_{ib}\frac{\partial }{\partial {{\theta }^{ \prime }}_{}}\left[\frac{{\displaystyle \int }{S}_{i2}\left({\theta }_2\right)f\left(y_1|x_i\mathrm{<mo>;</mo>}{\theta }_1\right)f\left(y_{2i}|x_i\mathrm{,}{y}_1\mathrm{<mo>;</mo>}{\theta }_2\right)d{y}_1}{{\displaystyle \int }f\left(y_1|x_i\mathrm{<mo>;</mo>}{\theta }_1\right)f\left(y_{2i}|x_i\mathrm{,}{y}_1\mathrm{<mo>;</mo>}{\theta }_2\right)d{y}_1}\right]\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in B}}{w}_{ib}E\left\{\frac{\partial }{\partial {{\theta }^{ \prime }}_{}}{S}_{i2}\left({\theta }_2\right)|x_i\mathrm{,}{y}_{2i}\mathrm{<mo>;</mo>}\theta \right\}\hfill \\ \hfill & +{\displaystyle \sum _{i\in B}}{w}_{ib}E\left\{S_{i2}\left({\theta }_2\right)S_{i2}\left({\theta }_2\right)\prime |x_i\mathrm{,}{y}_{2i}\mathrm{<mo>;</mo>}\theta \right\}\hfill \\ \hfill & -{\displaystyle \sum _{i\in B}}{w}_{ib}E\left\{S_{i2}\left({\theta }_2\right)|x_i\mathrm{,}{y}_{2i}\mathrm{<mo>;</mo>}\theta \right\}E\left\{S_{2i}\left({\theta }_2\right)\prime |x_i\mathrm{,}{y}_{2i}\mathrm{<mo>;</mo>}\theta \right\}\mathrm{.}\hfill \end{array}$$

Now, $\partial {\overline{S}}_2\left(\theta \right)/\partial {{\theta }^{ \prime }}_{}$ can be consistently estimated by

$${\widehat{B}}_{21}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in B}}{w}_{ib}{\displaystyle \sum _{j\mathrm{<mo>=</mo>1}}^m}{w}_{ij}^{*}{S}_{2ij}^{*}\left({\widehat{\theta }}_2\right){\left\{S_{1ij}^{*}\left({\widehat{\theta }}_1\right)-{\overline{S}}_{1i}^{*}\left({\widehat{\theta }}_1\right)\right\}}^{\prime }\mathrm{,}(3.5)$$

where $S_{1ij}^{*}\left({\widehat{\theta }}_1\right)\mathrm{<mo>=</mo>}{S}_1\left({\widehat{\theta }}_1\mathrm{<mo>;</mo>}{x}_i\mathrm{,}{y}_{1i}^{*\left(j\right)}\right),$ $S_{2ij}^{*}\left({\widehat{\theta }}_2\right)\mathrm{<mo>=</mo>}{S}_2\left({\widehat{\theta }}_2\mathrm{<mo>;</mo>}{x}_i\mathrm{,}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{,}{y}_{2i}\right),$ and ${\overline{S}}_{1i}^{*}\left({\widehat{\theta }}_1\right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^m}{w}_{ij}^{*}{S}_1\left({\widehat{\theta }}_1\mathrm{<mo>;</mo>}{x}_i\mathrm{,}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\right).$ Also, $\partial {\overline{S}}_2\left(\theta \right)/\partial {{\theta }^{ \prime }}_{}$ can be consistently estimated by

$$-{\widehat{I}}_{22}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in B}}{w}_{ib}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^m}{w}_{ij}^{*}{\dot{S}}_{2ij}^{*}\left({\widehat{\theta }}_2\right)-{\widehat{B}}_{22}(3.6)$$

where

$${\widehat{B}}_{22}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in B}}{w}_{ib}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^m}{w}_{ij}^{*}{S}_{2ij}^{*}\left({\widehat{\theta }}_2\right){\left\{S_{2ij}^{*}\left({\widehat{\theta }}_2\right)-{\overline{S}}_{2i}^{*}\left({\widehat{\theta }}_2\right)\right\}}^{\prime }\mathrm{,}$$

${\dot{S}}_{2ij}^{*}\left({\theta }_2\right)\mathrm{<mo>=</mo>}\partial S_2\left({\theta }_2\mathrm{<mo>;</mo>}{x}_i\mathrm{,}{y}_{1i}^{*\left(j\right)}\mathrm{,}{y}_{2i}\right)/\partial {{\theta }^{ \prime }}_{}$ and ${\overline{S}}_{2i}^{*}\left({\theta }_2\right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^m}{w}_{ij}^{*}{S}_{2ij}^{*}\left({\theta }_2\right).$

Using a Taylor expansion with respect to ${\theta }_1,$

$$\begin{array}{ll}{\overline{S}}_2\left({\widehat{\theta }}_1\mathrm{,}{\theta }_2\right)\hfill & \cong {\overline{S}}_2\left({\theta }_1\mathrm{,}{\theta }_2\right)-E\left\{\frac{\partial }{\partial {{\theta }^{ \prime }}_{}}{\overline{S}}_2\left(\theta \right)\right\}{\left[E\left\{\frac{\partial }{\partial {{\theta }^{ \prime }}_{}}{S}_1\left({\theta }_1\right)\right\}\right]}^{-1}{S}_1\left({\theta }_1\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\overline{S}}_2\left(\theta \right)+K{S}_1\left({\theta }_1\right)\mathrm{,}\hfill \end{array}$$

and we can write

$$V\left({\widehat{\theta }}_2\right)\doteq {\left\{E\left(\frac{\partial }{\partial {{\theta }^{ \prime }}_{}}{\overline{S}}_2\right)\right\}}^{-1}V\left\{{\overline{S}}_2\left(\theta \right)+K{S}_1\left({\theta }_1\right)\right\}{\left\{E\left(\frac{\partial }{\partial {{\theta }^{ \prime }}_{}}{\overline{S}}_2\right)\right\}}^{-1^{\prime }}\mathrm{.}$$

Writing

$${\overline{S}}_2\left(\theta \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in B}}{w}_{ib}{\overline{s}}_{2i}\left(\theta \right)\mathrm{,}$$

with ${\overline{s}}_{2i}\left(\theta \right)\mathrm{<mo>=</mo>}E\left\{S_{i2}\left({\theta }_2\right)|x_i\mathrm{,}{y}_{2i}\mathrm{<mo>;</mo>}\theta \right\},$ a consistent estimator of $V\left\{{\overline{S}}_2\left(\theta \right)\right\}$ can be obtained by applying a design-consistent variance estimator to ${\displaystyle {\sum }_{i\in B}}{w}_{ib}{\widehat{s}}_{2i}$ with ${\widehat{s}}_{2i}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^m}{w}_{ij}^{*}{S}_{2ij}^{*}\left({\widehat{\theta }}_2\right).$ Under simple random sampling for Sample B, we have

$$\widehat{V}\left\{{\overline{S}}_2\left(\theta \right)\right\}\mathrm{<mo>=</mo>}{n}_B^{-2}{\displaystyle \sum _{i\in B}}{\widehat{s}}_{2i}{{\widehat{s}}^{ \prime }}_{}\mathrm{.}$$

Also, $V\left\{K{S}_1\left({\theta }_1\right)\right\}$ is consistently estimated by

$${\widehat{V}}_2\mathrm{<mo>=</mo>}\widehat{K}\widehat{V}\left(S_1\right){\widehat{K}}^{\prime }\mathrm{,}$$

where $\widehat{K}\mathrm{<mo>=</mo>}{\widehat{B}}_{21}{\widehat{I}}_{11}^{-1},$ ${\widehat{B}}_{21}$ is defined in (3.5), and ${\widehat{I}}_{11}\mathrm{<mo>=</mo>}-\partial S_1\left({\theta }_1\right)/\partial {{\theta }^{ \prime }}_{}$ evaluated at ${\theta }_1\mathrm{<mo>=</mo>}{\widehat{\theta }}_1.$ Since the two terms ${\overline{S}}_2\left(\theta \right)$ and $S_1\left({\theta }_1\right)$ are independent, the variance can be estimated by

$$\widehat{V}\left(\widehat{\theta }\right)\doteq {\widehat{I}}_{22}^{-1}\left[\widehat{V}\left\{{\overline{S}}_2\left(\theta \right)\right\}+{\widehat{V}}_2\right]{\widehat{I}}_{22}^{-1^{\prime }}\mathrm{,}$$

where ${\widehat{I}}_{22}$ is defined in (3.6).

More generally, one may consider estimation of a parameter $\eta$ defined as a root of the census estimating equation ${\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N}U\left(\eta \mathrm{<mo>;</mo>}{x}_i\mathrm{,}{y}_{1i}\mathrm{,}{y}_{2i}\right)\mathrm{<mo>=</mo>\; 0.}$ Variance estimation of the FI estimator of $\eta$ computed from ${\displaystyle {\sum }_{i\in B}}{w}_{ib}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^m}{w}_{ij}^{*}U\left(\eta \mathrm{<mo>;</mo>}{x}_i\mathrm{,}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{,}{y}_{2i}\right)\mathrm{<mo>=</mo>\; 0}$ is discussed in Appendix B.

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