“Optimal” calibration weights under unit nonresponse in survey sampling
Section 3. A simulation study

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Properties of the estimators were studied by means of a Monte Carlo simulation. We used an authentic population called KYBOK, which consists of $N\mathrm{<mo>=</mo>\; 832}$ clerical municipalities in Sweden in 1992. (This population was also used for simulation purposes in Särndal and Lundström (2005) and Andersson and Särndal (2016).)

The study variable $y_k$ is “Expenditure on administration and maintenance” $(t_y\mathrm{<mo>=</mo>}\text{1,023,983}).$ The population is divided into four groups with respect to size, from the smallest to the largest. The group sizes are $N_1\mathrm{<mo>=</mo>\; 218},$ $N_2\mathrm{<mo>=</mo>\; 272},$ $N_3\mathrm{<mo>=</mo>\; 290}$ and $N_4\mathrm{<mo>=</mo>\; 52.}$ The moon vector is $x_k^o={\left(x_{1k}^o\mathrm{,}\dots \mathrm{,}{x}_{4k}^o\right)}^{\prime },$ where $x_{ik}^o\mathrm{<mo>=</mo>\; 1}$ if the unit $k$ belongs to population group $i$ and otherwise 0, $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}4.$ The quantitative star variable $x_k^{*}$ is the square root of “Revenue advances”, which is highly positively correlated with $y_k.$

The sample size/expected sample size was 300 and we used the exponential response probability

$${\theta }_k\mathrm{<mo>=</mo>\; 1}-\text{exp}\left(-c\cdot x_k^{*}\right)\mathrm{,}k\in U\mathrm{,}(3.1)$$

where $c$ is chosen according to the desired average response probability; in this study varying between 0.60 and 0.86 (the latter value being the chosen response probability in e.g., Särndal and Lundström (2005)). Two sampling designs have been considered separately: simple random sampling and Poisson sampling. In the latter case ${\pi }_k\propto x_k^{*}.$ For each combination of design, sample size/expected sample size and average response probability, 10,000 samples were generated. For each such sample $s,$ a response set $r$ was created by performing independent Bernoulli trials with probability ${\theta }_k$ of success, $k\in s.$

The estimators of main interest are ${\widehat{t}}_{y\text{cal}},$ ${\widehat{t}}_{y\text{opt}}$ and ${\widehat{t}}_{y\text{optm}},$ but in this simulation study we will also include an example of parametric propensity modelling based on ${\widehat{t}}_{y\text{cal}}.$ A simple choice is the logistic (logit) model

$${\theta }_k\mathrm{<mo>=</mo>}\frac{\text{exp}\left(x^{\prime }B\right)}{1+\text{exp}\left(x^{\prime }B\right)}\mathrm{,}(3.2)$$

where we let $x^{\prime }\mathrm{<mo>=</mo>}\left(1x^{*}\right).$ For each sample with its observed nonresponse maximum likelihood estimation was used to obtain $\widehat{B},$ yielding estimates ${\left(\widehat{\theta }\right)}_{k\in r}\text{}.$ To obtain ${\widehat{t}}_{y\text{cal}\text{logit}}$ the design weights $d_k$ are then replaced by $d_k\left(1/{\widehat{\theta }}_k\right)$ before calibration. The logit model (3.2) is misspecified since the true response probability is determined by (3.1).

An arbitrary estimator ${\widehat{t}}_y$ is assessed by the empirical (simulation estimated) bias $\left(\widehat{B}\right),$ variance $\left(\widehat{V}\right)$ and mean squared error $(\widehat{\text{MSE}}):$

$$\widehat{B}\mathrm{<mo>=</mo>}\widehat{E}\left({\widehat{t}}_y\right)-t_y\mathrm{<mo>=</mo>}\frac{1}{K}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^K}{\widehat{t}}_{yj}-t_y$$

$$\widehat{V}\mathrm{<mo>=</mo>}\frac{1}{K}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^K}{\left({\widehat{t}}_{yj}-\widehat{E}\left({\widehat{t}}_y\right)\right)}^2$$

$$\widehat{\text{MSE}}\mathrm{<mo>=</mo>}{\widehat{B}}^2+\widehat{V}\mathrm{,}$$

where $K\mathrm{<mo>=</mo>}\text{10,000}.$

Observe that expressions such as “the bias has increased” should be interpreted in the following as an increase of the bias in absolute value.

3.1  Results

As a benchmark for the study where auxiliary information is not used at the design stage, let us first consider the results for simple random sampling in Table 3.1. This is a case where ${\widehat{t}}_{y\text{cal}}\mathrm{<mo>=</mo>}{\widehat{t}}_{y\text{opt}}.$ (Actually, to get equality the “star” information is $x_k^{*}\mathrm{<mo>=</mo>}{\left(\mathrm{1,}{x}_k^{*}\right)}^{\prime }$ for ${\widehat{t}}_{y\text{cal}}.)$ As will hold throughout this study the bias of ${\widehat{t}}_{y\text{cal}\text{logit}}$ is considerably larger than the bias of ${\widehat{t}}_{y\text{cal}},$ which is a natural effect from the construction of ${\widehat{t}}_{y\text{cal}\text{logit}}$ based on a misspecified nonresponse model. Furthermore, of these two estimators ${\widehat{t}}_{y\text{cal}\text{logit}}$ has always the largest variance.

Looking instead at the results in Table 3.2 for Poisson sampling, we can first observe that for ${\widehat{t}}_{y\text{cal}}$ both the bias and the variance are larger than under simple random sampling. ${\widehat{t}}_{y\text{opt}},$ on the other hand, has highly reduced bias under Poisson sampling compared with simple random sampling, whereas there is a slight increase in the variance. Then, turning to the proposed modified estimator ${\widehat{t}}_{y\text{optm}}$ we observe a further reduction in bias, except for ${\overline{\theta }}_U\mathrm{<mo>=</mo>\; 0.86.}$ Actually, the bias has a monotonic behaviour and changes sign from positive to negative for ${\overline{\theta }}_U\approx \mathrm{0.64.}$ However, compared with ${\widehat{t}}_{y\text{opt}}$ the variance of ${\widehat{t}}_{y\text{optm}}$ is increased due to the inclusion of ${\widehat{\overline{\theta }}}_U$ in (2.4), thus leading to a trade-off between the bias and the variance. We also note that of these two estimators ${\widehat{t}}_{y\text{optm}}$ displays the largest MSE values, since the dominating part of the MSE is the variance for these low levels of bias.

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