A cautionary note on Clark Winsorization Section 3. MethodA cautionary note on Clark Winsorization Section 3. Method

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We first introduce notation needed to describe the Clark Winsorization, which follows Mulry et al. (2014). For the $i^{\text{th}}$ business in a survey sample of size $n$ for the month of observation $t,Y_{ti}$ is the collected characteristic (e.g., sales), $w_{ti}$ is its survey weight (which may or may not be equivalent to the inverse probability of selection), and $X_{ti}$ is a variable highly correlated with $Y_{ti},$ such as previous month’s revenue. The monthly total $Y_t$ is estimated by ${\widehat{Y}}_t$ defined by ${\widehat{Y}}_t={\displaystyle {\sum }_{i=1}^n{w}_{ti}{Y}_{ti}}.$

For ease of notation, we suppress the index for the month of observation $t$ in the remainder of this section. In MRTS, the survey weight $w_i$ is the (possibly modified) sampling weight since the missing data treatment is imputation.

The general form of the one-sided Winsorized estimator of the total is designed for large values and is written as ${\widehat{Y}}^{*}={\displaystyle {\sum }_{i=1}^n{w}_i{Z}_i}$ where $Z_i=\min \left\{Y_i,K_i+\left(Y_i-K_i\right)/w_i\right\}.$

Detection of observation $i$ as an influential value by Clark Winsorization occurs when $Z_i\ne Y_i.$ More than one observation may be identified. Note that using $Z_i=\min \left\{Y_i,K_i\right\}$ would ensure bounded influence and a robust estimator. However, this may lead to a large bias in ${\widehat{Y}}^{*}.$

To implement the method, Clark assumes a general model where the $Y_i$ are characterized as independent realizations of random variables with $E\left(Y_i\right)={\mu }_i$ and $\mathrm{var}\left(Y_i\right)={\sigma }_i^2.$ Then the approach approximates the $K_i$ that minimizes the MSE under the model by setting $K_i={\mu }_i+L{\left(w_i-1\right)}^{-1},$ which requires estimating ${\mu }_i$ and $L.$ Clark’s approach builds on a method developed by Kokic and Bell (1994) that derived a $K$ for each stratum rather than for each individual unit.

For an estimate of ${\mu }_i,$ Chambers et al. (2000) suggest using the results of a robust regression. In our application, we used the SAS Procedure ROBUSTREG (SAS 2014) to implement the weighted least median of squares (LMS) robust regression method. The LMS robust regression uses weights to compensate for the heteroscedasticity visible in Figure 2.1. Other considered methods appeared too sensitive with our data, designating some observations as influential when they were not large enough to have an excessive effect on the estimated total in our empirical data sets. In different applications, different robust regression methods could exhibit superior performance and should be considered. Our prediction model estimates ${\mu }_i$ with $b{X}_i$ where $b$ is the regression coefficient and $X_i$ is the previous month’s observation, chosen because $X_i$ and $Y_i$ tend to be highly correlated and no administrative data are available on a monthly basis. To estimate $L,$ the Clark Winsorization procedure uses the estimate of ${\mu }_i$ to estimate weighted residuals

$D_i=\left(Y_i-{\mu }_i\right)\left(w_i-1\right)$  by  ${\widehat{D}}_i=\left(Y_i-b{X}_i\right)\left(w_i-1\right).$

Certainty units have weighted residual values of zero, assuming that no other weight adjustments are performed (e.g., for unit nonresponse, for post-stratification). Next, the method sorts the estimates of the residuals in decreasing order ${\widehat{D}}_{\left(1\right)},{\widehat{D}}_{\left(2\right)},\dots ,{\widehat{D}}_{\left(n\right)}.$ Then the Clark method finds the largest value of $k,$ called $k^{*},$ such that $\left(k+1\right){\widehat{D}}_{\left(k\right)}-{\displaystyle {\sum }_{j=1}^k{\widehat{D}}_{\left(j\right)}}$ is positive, then estimates $L$ by $\widehat{L}={\left(k^{*}+1\right)}^{-1}{\displaystyle {\sum }_{j=1}^{k^{*}}{\widehat{D}}_{\left(j\right)}}.$ Finally, the estimate of $K_i$ is formed by ${\widehat{K}}_i=b{X}_i+\widehat{L}{\left(w_i-1\right)}^{-1},$ which is used to determine the values of $Z_i$ for the estimate of the total ${\widehat{Y}}^{*}.$ Chambers et al. (2000) recommend forming the estimate of $L$ for the procedure by using an average of estimates of $L$ from several previous months of data. However, our examples in Section 4 use only the previous month because we use data from a simulated stationary series constructed to reflect the different means and variances in the sampling strata for an industry in the MRTS. The stationary series was created by constructing a simulated population from MRTS data and applying an ARMA model to generate the time series. Thus, additions and deletions to the MRTS sample over time (i.e., births and deaths) are not incorporated in the simulation design. Consequently, averaging over several previous months offers no advantage over the point estimate from the previous month. In addition, we used the Winzorized values as auxiliary values $\left(X_i\right)$ in the application of the procedure to the subsequent month in order to study the propagation of the effects of the adjustment in the production setting. Although influential values were induced by adding a large amount to an observation selected at random from a stratum with one of the largest weights, the calculation of the value of $L$ used all the sample observations with weights greater than one. More details on the construction of the series may be found in Mulry et al. (2014). We have not explored using an average of estimates of $L$ from several previous months with simulated MRTS data that incorporated seasonality, volatility, and changes in economic conditions or with empirical MRTS data. Such an average of estimates of $L$ may be useful in other designs and surveys that exhibit more stable behavior, such as annual rather than monthly implementations.

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