A few remarks on a small example by Jean-Claude Deville regarding non-ignorable non-response Section 4. Estimation using the maximum likelihood methodA few remarks on a small example by Jean-Claude Deville regarding non-ignorable non-response Section 4. Estimation using the maximum likelihood method

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4.1 MAR

The probability distribution is multinomial. For MAR, the following likelihood function applies:

$$\begin{array}{ll}\mathcal{L}\left(n_{HD}\mathrm{,}{n}_{FD}\mathrm{,}{p}_H\mathrm{,}{p}_F\right)\hfill & \mathrm{<mo>=</mo>}\frac{n_{H\mathrm{.}}\mathrm{<mo>!</mo>}}{r_{HD}\mathrm{<mo>!</mo>}{r}_{HS}\mathrm{<mo>!</mo>}{m}_H\mathrm{<mo>!</mo>}}{\left(\frac{n_{HD}{p}_H}{n_{H\mathrm{.}}}\right)}^{r_{HD}}{\left(\frac{\left(n_{H\mathrm{.}}-n_{HD}\right)p_H}{n_{H\mathrm{.}}}\right)}^{r_{HS}}{\left(\frac{n_{H\mathrm{.}}\left(1-p_H\right)}{n_{H\mathrm{.}}}\right)}^{m_H}\hfill \\ \hfill & \times \frac{n_{F\mathrm{.}}\mathrm{<mo>!</mo>}}{r_{FD}\mathrm{<mo>!</mo>}{r}_{FS}\mathrm{<mo>!</mo>}{m}_F\mathrm{<mo>!</mo>}}{\left(\frac{n_{FD}{p}_F}{n_{F\mathrm{.}}}\right)}^{r_{FD}}{\left(\frac{\left(n_{F\mathrm{.}}-n_{FD}\right)p_F}{n_{F\mathrm{.}}}\right)}^{r_{FS}}{\left(\frac{n_{F\mathrm{.}}\left(1-p_F\right)}{n_{F\mathrm{.}}}\right)}^{m_F}\mathrm{.}\hfill \end{array}$$

By setting to zero the partial derivatives of the log-likelihood with respect to parameters $p_H$ and $p_F,$ we obtain two equations with two unknowns. The solution yields the estimators

$$\begin{array}{ll}{\widehat{p}}_H\hfill & \mathrm{<mo>=</mo>\; 1}-\frac{m_H}{n_{H\mathrm{.}}}\mathrm{,}\hfill \\ {\widehat{p}}_F\hfill & \mathrm{<mo>=</mo>\; 1}-\frac{m_F}{n_{F\mathrm{.}}}\mathrm{.}\hfill \end{array}$$

By setting to zero the derivatives with respect to $n_{HD}$ and $n_{FD},$ we obtain the estimators

$${\widehat{n}}_{HD}\mathrm{<mo>=</mo>}\frac{r_{HD}}{{\widehat{p}}_H}\text{and}{\widehat{n}}_{FD}\mathrm{<mo>=</mo>}\frac{r_{FD}}{{\widehat{p}}_F}\mathrm{.}$$

Therefore,

$${\widehat{n}}_{\mathrm{.}D}\mathrm{<mo>=</mo>}{\widehat{n}}_{HD}+{\widehat{n}}_{FD}\mathrm{<mo>=</mo>}\frac{r_{HD}}{{\widehat{p}}_H}+\frac{r_{FD}}{{\widehat{p}}_F}\mathrm{.}$$

These estimators are exactly the same as those obtained using the method of moments.

4.2 NMAR

For NMAR, the following likelihood function applies:

$$\begin{array}{ll}\mathcal{L}\left(n_{HD}\mathrm{,}{n}_{FD}\mathrm{,}{q}_D\mathrm{,}{p}_S\right)\hfill & \mathrm{<mo>=</mo>}\frac{n_{H\mathrm{.}}\mathrm{<mo>!</mo>}}{r_{HD}\mathrm{<mo>!</mo>}{r}_{HS}\mathrm{<mo>!</mo>}{m}_H\mathrm{<mo>!</mo>}}{\left(\frac{n_{HD}{q}_D}{n_{H\mathrm{.}}}\right)}^{r_{HD}}{\left(\frac{\left(n_{H\mathrm{.}}-n_{HD}\right)q_S}{n_{H\mathrm{.}}}\right)}^{r_{HS}}{\left(\frac{n_{HD}\left(1-q_D\right)+\left(n_{H\mathrm{.}}-n_{HD}\right)\left(1-q_S\right)}{n_{H\mathrm{.}}}\right)}^{m_H}\hfill \\ \hfill & \times \frac{n_{F\mathrm{.}}\mathrm{<mo>!</mo>}}{r_{FD}\mathrm{<mo>!</mo>}{r}_{FS}\mathrm{<mo>!</mo>}{m}_F\mathrm{<mo>!</mo>}}{\left(\frac{n_{FD}{q}_D}{n_{F\mathrm{.}}}\right)}^{r_{FD}}{\left(\frac{\left(n_{F\mathrm{.}}-n_{FD}\right)q_S}{n_{F\mathrm{.}}}\right)}^{r_{FS}}{\left(\frac{n_{FD}\left(1-q_D\right)+\left(n_{F\mathrm{.}}-n_{FD}\right)\left(1-q_S\right)}{n_{F\mathrm{.}}}\right)}^{m_F}\mathrm{.}\hfill \end{array}$$

By setting to zero the partial derivatives of the log-likelihood with respect to the four parameters $q_D,$ $q_S,$ $n_{HD}$ and $n_{FD},$ we obtain a system of four rather complicated second-order equations with four unknowns. We used a symbolic computation software program to verify that the solution given by the method of moments is a solution to this system of equations. Obviously, since the system is second-order, there is a second solution. However, for Deville’s example, the second solution yields negative values, which are not valid for estimating probabilities and numbers of people.

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