Empirical Likelihood Method

Maciej Ostapiuk and Maciej Beręsewicz

Empirical Likelihood Method

To consider the empirical likelihood under non-ignorable missing data, one has to discuss the distribution function of multidimensional random variable $(\boldsymbol{X},Y)$ which is determined by its distribution function $F(x,y)$. There are no assumptions on $F(x,y)$ (except the fact that it has to fit CDF assumptions) but there is a setting in Kim and Shao (2013), that $$\begin{equation}\label{eq: setting on (X,Y)} \mathop{\mathrm{\mathbb{E}}}\{U(\boldsymbol{\theta}_0; X,Y)\} = 0 \end{equation}$$ which is a $m$-dimensional, linearly independent vector, $U(\cdot) \in \mathcal{C}^2$ and $\boldsymbol{\theta}_0 \in \Omega \subseteq \mathbb{R}^p$. if $m = p$, then a consistent estimator of $\boldsymbol{\theta}_0$ is a solution of: $$\begin{equation}\label{eq: consistent estimator of btheta under m = p} \sum_{i=1}^{n} U(\boldsymbol{\theta}, \boldsymbol{x}_i, y). \end{equation}$$ When $m > p$ the model is called , thus one has to adjust the methodology which results in different optimization problem, since might not provide solution at all. This adjustment is also proposed in Kim and Shao (2013) and the EL approach is concentrated around finding a solution $\boldsymbol{\theta}$ that maximizes the empirical likelihood function of $\boldsymbol{\theta}$: $$\begin{equation}\label{eq: EL solution for m>p} L(\boldsymbol{\theta}) = \mathop{\mathrm{arg\,max}}_{\boldsymbol{\theta}}\left\{\prod_{i=1}^n p_i: p_i>0, \sum_{i=1}^np_i \wedge \sum_{i=1}^n p_i U(\boldsymbol{\theta}; \boldsymbol{x}_i, y_i) = 0\right\}. \end{equation}$$ Using the Lagrange multiplier method, lets denote $$\begin{equation} \hat{p}_i(\boldsymbol{\theta}) =\frac{1}{n} \frac{1}{1+\hat{\lambda}_{\boldsymbol{\theta}}'U(\theta; \boldsymbol{x}_i, y_i)}. \end{equation}$$ Thus, the MEL estimator is being obtained by maximizing $$\begin{equation} \label{eq: empirical likelihood method estimator} l_e(\boldsymbol{\theta}) = \sum_{i=1}^n\log\{\hat{p}_i(\boldsymbol{\theta})\}. \end{equation}$$ When dealing with any missingness in data, (recall, that we do have $\boldsymbol{x}_i$ for any individual and $y_i$ is only observed for respondents) the scoring of propensity is applied in such fashion:

• let $R$ be a response indicator ($R_i=1$ if $y_i$ is observed and $R_i=0$ if $y_i$ is unobserved),
• denote response propensity as $\pi(\boldsymbol{x};\phi) = P(R =1|\boldsymbol{x})$ (MNAR mechanism provides lack of $y_i$ in conditioning),
• let $S(\phi; R, \boldsymbol{x})$ be the score function of $\phi$ s.t. $\mathop{\mathrm{\mathbb{E}}}\{S(\phi; R, \boldsymbol{x})\} = 0$ holds.

Thus, $\mathop{\mathrm{\mathbb{E}}}\{S(\phi; R, \boldsymbol{x})\} = 0$ is one of the moment conditions, along with second one of form: $$\begin{equation}\label{eq: second moment condition for EL by PS} \mathop{\mathrm{\mathbb{E}}}\left\{\frac{R}{\pi(\boldsymbol{X};\phi)}U(\boldsymbol{\theta},\boldsymbol{X}, Y)\right\} = 0 \end{equation}$$ Those two moment conditions allows us to perform propensity-score-based MELE by maximizing: $$\begin{equation}\label{eq: MELE by PS} L(\boldsymbol{\theta}, \phi) = \mathop{\mathrm{arg\,max}}\left\{\prod_{i=1}^n p_i: \boldsymbol{p}\in B(\boldsymbol{\theta}, \phi) \right\}, \end{equation}$$ where $$\begin{equation} B(\boldsymbol{\theta}, \phi) = \left\{\boldsymbol{p}: p_i>0, \sum_{i=1}^np_i, \sum_{i=1}^n p_i\frac{R_i}{\pi(\boldsymbol{x}_i;\phi)} U(\boldsymbol{\theta}; \boldsymbol{x}_i, y_i) = 0\, \sum_{i=1}^n p_i S(\phi; R_i, \boldsymbol{x}_i) = 0 \right\}. \end{equation}$$

References

Kim, J. K., and J. Shao. 2013. Statistical Methods for Handling Incomplete Data. Chapman & Hall Book. Taylor & Francis. https://books.google.pl/books?id=MY4AAAAAQBAJ.