Influence Function for the Quantile Ratio Index

Computes the influence function of the quantile ratio index (QRI) in the context of finite population for all observations, as defined in Scarpa et al. (2025) , under simple and complex sampling. See Deville (1999) for an introduction to the definition of influence function in finite population theory.

Usage

if_qri(y, weights = NULL, type = 6)

Arguments

y: A numeric vector of data values
weights: A numeric vector of sampling weights (optional). If NULL, equal weights are assumed.
type: Quantile estimation type: integer 4–9 or HD for Harrell–Davis (default: 6) See csquantile documentation for a complete description.

Value

A numeric vector of influence function values (one for each observation)

Details

The influence function for the QRI is computed on each observation as $$ \widehat{z}_i = - \int_0^1 \frac{ \left( \frac{\frac{p}{2} - \mathbf{1}(y_i \leq \widehat{Q}(p/2))} {\widehat{f}(\widehat{Q}(p/2)) \, \widehat{N}} \right) \widehat{Q}(1 - p/2) - \left( \frac{(1 - \frac{p}{2}) - \mathbf{1}(y_i \leq \widehat{Q}(1 - p/2))} {\widehat{f}(\widehat{Q}(1 - p/2)) \, \widehat{N}} \right) \widehat{Q}(p/2) }{ \widehat{Q}(1 - p/2)^2 } \, dp $$

where:

$\widehat{Q}(p)$ is the weighted sample quantile of order $p$, computed using the internal function csquantile(),
$\widehat{f}(\cdot)$ denotes the estimated income density function,
$\widehat{N} = \sum_i w_i$ is the estimated population size.

The density function $\widehat{f}(y)$ is estimated via a Gaussian kernel smoother: $$ \widehat{f}(y) = \frac{1}{\widehat{N}} \sum_{j \in s} w_j K\!\left(\frac{y - y_j}{h}\right) = \frac{1}{\widehat{N}\, h \sqrt{2\pi}} \sum_{j \in s} w_j \exp\!\left\{ -\frac{(y - y_j)^2}{2h^2} \right\}, $$ where $K(\cdot)$ is the Gaussian kernel.

The bandwidth is chosen as: $$ h = 0.79 \cdot \mathrm{IQR} \cdot \widehat{N}^{-1/5}, $$ where $\mathrm{IQR}$ is the interquartile range of the weighted sample.

References

Deville J (1999). “Variance estimation for complex statistics and estimators: linearization and residual techniques.” Survey methodology, 25, 193–204.

Scarpa S, Ferrante MR, Sperlich S (2025). “Inference for the quantile ratio inequality index in the context of survey data.” Journal of Survey Statistics and Methodology. doi:10.1093/jssam/smaf024 .

Examples


# On synthetic data
eq_synth <- rlnorm(30, 9, 0.7)
IF_synth <- if_qri(y = eq_synth)

# On real data
eq <- synthouse$eq_income[1:30] ## Take some observations (as example)
w <- synthouse$weight[1:30]
IF_qri <- if_qri(y = eq, weights = w, type = 6)