Assessing the coverage of confidence intervals under nonresponse. A case study on income mean and quantiles in some municipalities from the 2015 Mexican Intercensal Survey
Section 2. Three methods for estimating confidence intervals

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2.1 Estimation using two-phase sampling

We consider a finite population $U\mathrm{<mo>=</mo>}\left\{\mathrm{1,}\mathrm{2,}\dots \mathrm{,}N\right\}$ and a probability sample $s\subset U$ of fixed size $n,$ with first and second-order inclusion probabilities ${\pi }_k$ and ${\pi }_{kl},$ $k\mathrm{,}l\in U,$ $k\ne l.$ Let $y_k$ be the $k^{\text{th}}$ value of the variable of interest $y,$ and let ${\theta }_0$ be a population parameter and ${\widehat{\theta }}_0$ an estimator of ${\theta }_0.$

We assume that the value $y_k$ is available for a subset $r\subset s$ only. Let ${\varphi }_k$ denote the response probability for unit $k.$ Let $I_k$ be a response indicator variable such that $I_k\mathrm{<mo>=</mo>}1$ for $k\in r$ and $I_k\mathrm{<mo>=</mo>}0$ for $k\in s\backslash r.$ We also assume that there is a vector of auxiliary variables $x$ observed for all $k\in s.$ We make the missing at random (MAR) assumption:

$$P\mathrm{<mo\; stretchy="false">(</mo>}{I}_k\mathrm{<mo>=</mo>}1\mathrm{<mo>|</mo>}{y}_k\mathrm{,}{x}_k\mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}P\mathrm{<mo\; stretchy="false">(</mo>}{I}_k\mathrm{<mo>=</mo>}1\mathrm{<mo>|</mo>}{x}_k\mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}{\varphi }_k\forall k\in U\mathrm{.}$$

The response probabilities ${\varphi }_k$ are used for adjusting the design weights. We assume that the sampling design and the response mechanism are independent as in Berger (2020). Borrowing from two-phase sampling theory (Särndal et al., 1992, Section 9.3) the weights adjusted for nonresponse are defined as ${\pi }_k^{*-1}\mathrm{<mo>=</mo>}1\mathrm{<mo>/</mo>}\mathrm{<mo\; stretchy="false">(</mo>}{\pi }_k{\varphi }_k\mathrm{<mo\; stretchy="false">)</mo>}$ and the second-order inclusion probabilities as ${\pi }_{kl}^{*}\mathrm{<mo>=</mo>}{\pi }_{kl}{\varphi }_k{\varphi }_l\mathrm{,}k\mathrm{,}l\in U,$ $k\ne l.$

The interest lies in estimating the population mean $\overline{Y}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in U}{y}_k/N}$ and the population quantile given by

where $y_{\mathrm{<mo\; stretchy="false">(</mo>}i\mathrm{<mo\; stretchy="false">)</mo>}}$  is the value for the $i^{\text{th}}$ unit arranged in increasing order, $d\mathrm{<mo>=</mo>}\min \left\{l\mathrm{<mo>:</mo>}qN\mathrm{<mo><</mo>}{N}_{\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{,}l\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,N\right\}$ and $N_{\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\in U}I\left(y_j\le y_{\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}}\right)}\mathrm{,}l\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,N.$ Formula (2.1) is obtained by considering a piecewise linear interpolation of the step distribution function $F\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in U}I\mathrm{<mo\; stretchy="false">(</mo>}{y}_k\le y\mathrm{<mo\; stretchy="false">)</mo>}/N},$ where $I\left(y_k\le y\right)\mathrm{<mo>=</mo>}1$ when $y_k\le y.$

These population parameters are respectively estimated by

$$\widehat{\overline{Y}}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in r}{y}_k/{\pi }_k^{*}}}{{\displaystyle {\sum }_{k\in r}1/{\pi }_k^{*}}}\mathrm{,}(2.2)$$

and

$${\widehat{Y}}_q\mathrm{<mo>=</mo>}{y}_{\mathrm{<mo\; stretchy="false">(</mo>}d-\mathrm{1\; <mo\; stretchy="false">)</mo>}}+\frac{\left(y_{\mathrm{<mo\; stretchy="false">(</mo>}d\mathrm{<mo\; stretchy="false">)</mo>}}-y_{\mathrm{<mo\; stretchy="false">(</mo>}d-\mathrm{1\; <mo\; stretchy="false">)</mo>}}\right)\left(q\widehat{N}-{\widehat{N}}_{\mathrm{<mo\; stretchy="false">(</mo>}d-\mathrm{1\; <mo\; stretchy="false">)</mo>}}\right)}{{\widehat{N}}_{\mathrm{<mo\; stretchy="false">(</mo>}d\mathrm{<mo\; stretchy="false">)</mo>}}-{\widehat{N}}_{\mathrm{<mo\; stretchy="false">(</mo>}d-\mathrm{1\; <mo\; stretchy="false">)</mo>}}}\mathrm{,}(2.3)$$

where $d\mathrm{<mo>=</mo>}\min \left\{l\mathrm{<mo>:</mo>}q\widehat{N}\mathrm{<mo><</mo>}{\widehat{N}}_{\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{,}l\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,n_r\right\},$ $\widehat{N}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in r}1/{\pi }_k^{*}},$ ${\widehat{N}}_{\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in r}I\mathrm{<mo\; stretchy="false">(</mo>}{y}_k\le y_{\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mtext>\hspace{0.17em}</mtext>\; <mtext>\hspace{0.17em}</mtext>\; <mo\; stretchy="false">)</mo>}/{\pi }_k^{*}}$ and $n_r\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in s}{I}_k}.$

These estimators and the CIs described in the following subsection are based on the assumption that the response probabilities ${\varphi }_k$ are known, unlike Berger (2020) and Kim and Kim (2007). However, we use ${\widehat{\varphi }}_k$ instead of ${\varphi }_k$ in the simulation studies, where

$${\widehat{\varphi }}_k\mathrm{<mo>=</mo>}\frac{\exp \left(x_k^T\widehat{\beta }\right)}{1+\exp \left(x_k^T\widehat{\beta }\right)}\forall k\in s\mathrm{,}$$

with $\widehat{\beta }$ obtained by fitting a logistic regression using $s.$ This leads to the estimators known as the empirical double expansion estimators (Haziza and Beaumont, 2017).

2.2 Methods for estimating confidence intervals

2.2.1 Linearization

The linearization method relies on the assumption that the distribution of ${\widehat{\theta }}_0$ is approximately normal. A CI for ${\theta }_0$ is

$$\left[{\widehat{\theta }}_0-z_{1-\alpha /2}{\left[V\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{\theta }}_0\mathrm{<mo\; stretchy="false">)</mo>}\right]}^{1/2}\text{}\mathrm{,}{\widehat{\theta }}_0+z_{1-\alpha /2}{\left[V\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{\theta }}_0\mathrm{<mo\; stretchy="false">)</mo>}\right]}^{1/2}\right]\mathrm{,}(2.4)$$

where $1-\alpha$ is the confidence level, also known as nominal coverage; see Särndal et al. (1992, expression 5.2.3). In practice $V\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{\theta }}_0\mathrm{<mo\; stretchy="false">)</mo>}$ is estimated. For the estimators given by (2.2) and (2.3), a variance estimator is given by

$$\widehat{V}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{\theta }}_0\mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in r}{\displaystyle \sum _{l\in r}\frac{\mathrm{<mo\; stretchy="false">(</mo>}{\pi }_{kl}^{*}-{\pi }_k^{*}{\pi }_l^{*}\mathrm{<mo\; stretchy="false">)</mo>}}{{\pi }_{kl}^{*}}\frac{{\widehat{z}}_k}{{\pi }_k^{*}}\frac{{\widehat{z}}_l}{{\pi }_l^{*}}}}\mathrm{,}(2.5)$$

where ${\widehat{z}}_k\mathrm{<mo>=</mo>}\left(y_k-\widehat{\overline{Y}}\right)/\widehat{N}$ for $\widehat{\overline{Y}}$ (Särndal et al., 1992, Result 5.7.1) and ${\widehat{z}}_k\mathrm{<mo>=</mo>}-\left(I(y_k\le {\widehat{Y}}_q)-q\right)/\left(f\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{Y}}_q\mathrm{<mo\; stretchy="false">)</mo>}\widehat{N}\right)$ for ${\widehat{Y}}_q$ (Deville, 1999). The density function $f$ was obtained in two ways: a) using a Gaussian kernel as in Osier (2009) and b) using the nearest neighbour technique as in Graf and Tillé (2014). We present the results pertaining to a), since the technique in b) led to similar results.

We note that using (2.5) with ${\widehat{\varphi }}_k$ instead of ${\varphi }_k$ might lead to overestimation of the variance of the empirical double expansion estimator and to wider CIs; see the expression (17) in Kim and Kim (2007) associated with the estimators of $\overline{Y}.$

2.2.2 Empirical likelihood method

The empirical likelihood approach assumes that ${\theta }_0$ is the unique solution of the estimating equation $G\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in U}{g}_k\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}0}$ for a given function $g_k.$ In particular, we use:

1. $g_k\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}{y}_k-\theta$ for ${\theta }_0\mathrm{<mo>=</mo>}\overline{Y}.$
2. $g_k\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}\rho \mathrm{<mo\; stretchy="false">(</mo>}{y}_k\mathrm{,}\theta \mathrm{<mo\; stretchy="false">)</mo>}-q$ for ${\theta }_0\mathrm{<mo>=</mo>}{Y}_q,$ where $\rho \mathrm{<mo\; stretchy="false">(</mo>}{y}_k\mathrm{,}\theta \mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}I\mathrm{<mo\; stretchy="false">(</mo>}{y}_k\mathrm{<mo><</mo>}{y}_{\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mtext>\hspace{0.17em}</mtext>\; <mtext>\hspace{0.17em}</mtext>\; <mo\; stretchy="false">)</mo>}+I\mathrm{<mo\; stretchy="false">(</mo>}{y}_k\mathrm{<mo>=</mo>}{y}_{\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mtext>\hspace{0.17em}</mtext>\; <mtext>\hspace{0.17em}</mtext>\; <mo\; stretchy="false">)</mo>}\mathrm{<mo\; stretchy="false">(</mo>}\theta -y_{\mathrm{<mo\; stretchy="false">(</mo>}l-\mathrm{1\; <mo\; stretchy="false">)</mo>}}\mathrm{<mtext>\hspace{0.17em}</mtext>\; <mtext>\hspace{0.17em}</mtext>\; <mo\; stretchy="false">)</mo>}/\mathrm{<mo\; stretchy="false">(</mo>}{y}_{\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}}-y_{\mathrm{<mo\; stretchy="false">(</mo>}l-\mathrm{1\; <mo\; stretchy="false">)</mo>}}\mathrm{<mtext>\hspace{0.17em}</mtext>\; <mtext>\hspace{0.17em}</mtext>\; <mo\; stretchy="false">)</mo>}$ and $l\mathrm{<mo>=</mo>}\min \left\{j\mathrm{<mo>:</mo>}{y}_{\mathrm{<mo\; stretchy="false">(</mo>}j\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>></mo>}\theta \right\}.$

The empirical log-likelihood function in Berger and De La Riva Torres (2016) for a one-stage sampling design without stratification or auxiliary information is

$${\text{ℓ}}_{\max }\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}\underset{m_k\mathrm{<mo>:</mo>}k\in s}{{\displaystyle \max }}\left\{{\displaystyle \sum _{k\in s}\log \mathrm{<mo\; stretchy="false">(</mo>}{m}_k\mathrm{<mo\; stretchy="false">)</mo>\; <mo>:</mo>}{m}_k\mathrm{<mo>></mo>}0}\mathrm{,}{\displaystyle \sum _{k\in s}{m}_k{g}_k\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}0}\mathrm{,}{\displaystyle \sum _{k\in s}{m}_k{\pi }_k\mathrm{<mo>=</mo>}n}\right\}\mathrm{,}(2.6)$$

where $\left\{m_k\mathrm{<mo>:</mo>}k\in s\right\}$ satisfies the design and the parameter constraints ${\displaystyle {\sum }_{k\in s}{m}_k{\pi }_k\mathrm{<mo>=</mo>}n}$ and ${\displaystyle {\sum }_{k\in s}{m}_k{g}_k\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}0}.$

In the presence of nonresponse, we use (2.6) replacing ${\displaystyle {\sum }_{k\in s}{m}_k{g}_k\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}0}$ with ${\displaystyle {\sum }_{k\in s}{m}_k{I}_k{g}_k\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}}/{\varphi }_k\mathrm{<mo>=</mo>}0.$ A CI for ${\theta }_0$ is given by

$$\left[\min \left\{\theta \mathrm{<mo>:</mo>}\widehat{R}\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}\le {\chi }_1^2\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}\right\}\mathrm{,}\max \left\{\theta \mathrm{<mo>:</mo>}\widehat{R}\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}\le {\chi }_1^2\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}\right\}\right]\mathrm{,}(2.7)$$

where $\widehat{R}\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}2\left\{{\text{ℓ}}_{\max }\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{\theta }}_0\mathrm{<mo\; stretchy="false">)</mo>}-{\text{ℓ}}_{\max }\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}\right\}$ and ${\chi }_1^2\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ is the $\left(1-\alpha \right)$ -quantile of the ${\chi }_1^2$ distribution. The estimator ${\widehat{\theta }}_0\mathrm{<mo>:=</mo>}{\text{argmax}}_{\left\{\theta \right\}}{\text{ℓ}}_{\max }\left(\theta \right)$ corresponds to (2.2) and (2.3), respectively.

We computed (2.7) using a root search method, calculating $\widehat{R}\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}$ for several values of $\theta ,$ where ${\text{ℓ}}_{\max }\mathrm{<mo\; stretchy="false">(</mo>}\theta \mathrm{<mo\; stretchy="false">)</mo>}$ for a given value $\theta$ was obtained by a modified Newton-Raphson algorithm as in Wu (2004).

2.2.3 Woodruff method for quantiles

The method of Woodruff (1952) is based on the estimated distribution function $\widehat{F}\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{<mo\; stretchy="false">)</mo>\; .}$ For a quantile $Y_q,$ the variance of $\widehat{F}\mathrm{<mo\; stretchy="false">(</mo>}{Y}_q\mathrm{<mo\; stretchy="false">)</mo>}$ can be approximated using the Taylor linearization method with linearized variable $z_k\mathrm{<mo>=</mo>}\left(I\mathrm{<mo\; stretchy="false">(</mo>}{y}_k\le Y_q\mathrm{<mo\; stretchy="false">)</mo>}-q\right)/N,$ while the variance is estimated using (2.5) with ${\widehat{z}}_k\mathrm{<mo>=</mo>}\left(I\mathrm{<mo\; stretchy="false">(</mo>}{y}_k\le {\widehat{Y}}_q\mathrm{<mo\; stretchy="false">)</mo>}-q\right)/\widehat{N}.$ Assuming normality of $\widehat{F}\mathrm{<mo\; stretchy="false">(</mo>}{Y}_q\mathrm{<mo\; stretchy="false">)</mo>}$ and using (2.4), it is possible to find a CI $\left[c_1\mathrm{,}{c}_2\right]$ for $F\mathrm{<mo\; stretchy="false">(</mo>}{Y}_q\mathrm{<mo\; stretchy="false">)</mo>},$ which leads to $\left[{\widehat{F}}^{-1}\mathrm{<mo\; stretchy="false">(</mo>}{c}_1\mathrm{<mo\; stretchy="false">)</mo>\; ,}{\widehat{F}}^{-1}\mathrm{<mo\; stretchy="false">(</mo>}{c}_2\mathrm{<mo\; stretchy="false">)</mo>}\right]$ for $Y_q.$

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