2. Framework

Alina Matei and M. Giovanna Ranalli

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Let $U$ be a finite population of size $N,$ indexed by $k$ from $1$ to $N\mathrm{.}$ Let $s$ denote the set of sample labels, so that $s\subset U\mathrm{,}$ drawn from the population using a probabilistic sampling design $p\left(s\right).$ The sample size is denoted by $n\mathrm{.}$ Let ${\pi }_k={\displaystyle {\sum }_{s;s∍k}p\left(s\right)}$ be the probability of including unit $k$ in the sample. It is assumed that ${\pi }_k>0,k=1,\dots \mathrm{,}N\mathrm{.}$ Not all units selected in $s$ respond to the survey. Denote by $r\subseteq s$ the set of respondents, and by $\overline{r}=s\backslash r$ the set of nonrespondents. The response mechanism is given by the distribution $q\left(r|s\right)$ such that for every fixed $s$ we have

$$q\left(r|s\right)\ge \mathrm{0,}\text{ for all }r\in {ℛ}_s\text{ and }{\displaystyle \sum _{s\in {ℛ}_s}}q\left(r|s\right)=\mathrm{1,}\text{ where }{ℛ}_s=\left\{r|r\subseteq s\right\}\mathrm{.}$$

Under unit nonresponse we define the response indicator $R_k=1$ if unit $k\in r$ and $0$ if $k\in \overline{r}.$ Thus $r=\left\{k\in s|R_k=1\right\}\mathrm{.}$ We assume that these random variables are independent of one another and of the sample selection mechanism (Oh and Scheuren 1983). Since only the units in $r$ are observed, a response model is used to estimate the probability of responding to the survey of a unit $k\in U\mathrm{,}$ $p_k=P\left(k\in r|k\in s\right)=P\left(R_k=1|k\in s\right),$ which is a function of the sample and must be positive.

Suppose that in the survey there are $m$ variables of particular interest. Each respondent is exposed to these $m$ questionnaire variables, labelled $\ell =1,\dots \mathrm{,}m\mathrm{.}$ Suppose that the goal is to estimate the population total of some variables of interest and, in particular, of the variable of interest $y_j,$ i.e., $Y_j={\displaystyle {\sum }_{k=1}^N{y}_{kj}},$ with $y_{kj}$ being the value taken by $y_j$ on unit $k.$ In the ideal case, if the response distribution $q\left(r|s\right)$ is known, then the $p_k’\text{s}$ would be known and available to estimate $Y_j$ using a reweigthing approach. Suppose also that item nonresponse is present for variable $y_j\mathrm{.}$ Let $r_j=\left\{k\text{ answers }{y}_j|k\in r\right\}$ be the set of respondents for variable $y_j.$ As in the case of unit nonresponse we assume that the units in $r_j$ respond independently of each other. Let $q_{kj}=P\left(k\text{ answers }{y}_j|k\in r\right)\mathrm{.}$ The final set of weights to be used into a fully reweighting approach to handle unit and item nonresponse is given by $1/\left({\pi }_k{p}_k{q}_{kj}\right)\mathrm{,}$ for all $k\in r_j\mathrm{,}$ assuming $q_{kj}>0.$ These weights can be for example used in a three-phase fashion in the following Horvitz-Thompson (HT) estimator

$${\widehat{Y}}_{j,pq,\text{true}}={\displaystyle \sum _{k\in r_j}}\frac{y_{kj}}{{\pi }_k{p}_k{q}_{kj}},(2.1)$$

(see Legg and Fuller 2009, for the properties of estimators under three-phase sampling).

Usually, $p_k$ and $q_{kj}$ are unknown and should be estimated. A nonresponse adjusted estimator is then constructed by replacing $p_k$ and $q_{kj}$ with estimates ${\widehat{p}}_k$ and ${\widehat{q}}_{kj}$ in (2.1). The following sections provide details with this regard.

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