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Quantile Ratio Index Estimator

Source: R/qri.R
qri.Rd

Computes the quantile ratio index (QRI) estimator for measuring inequality on simple and complex sampling data

Usage

qri(y, weights = NULL, M = 100, type = 6, na.rm = TRUE)

Arguments

y

A numeric vector of strictly positive values (e.g. income, wealth, expenditure).

weights

A numeric vector of sampling weights (optional)

M

Integer, number of quantile ratios to average (default: 100)

type

Quantile estimation type: integer 4-9 or HD for Harrell-Davis (default: 6). See csquantile documentation for a complete description.

na.rm

Logical, should missing values be removed? (default: TRUE)

Value

A scalar numeric value representing the estimated inequality by the quantile ratio index (QRI).

Details

Consider a random sample \(s\) of size, where \(w_j\), \(j \in s\), defines the sampling weight associated to the \(j\)-th individual. The QRI estimator is defined as:

$$\widehat{QRI} = \frac{1}{M}\sum_{m=1}^M\left(1- \frac{\widehat{Q}(p_{m/2})}{\widehat{Q}(1 - p_{m/2})}\right)$$

where \(p_m = p_m = (m-0.5)/M\). The estimated quantiles \(\widehat{Q}(p)\) are computed via the function csquantile(), which accounts for sampling weights and the specified quantile type. This allows \(\widehat{QRI}\) to be used both for simple random samples and for complex survey data with design weights.

This index was proposed by Prendergast and Staudte (2018) , and extended to survey data by Scarpa et al. (2025) .

References

Prendergast LA, Staudte RG (2018). “A simple and effective inequality measure.” The American Statistician, 72, 328–343.

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 .

See also

qri_grouped for QRI estimation from pre-aggregated (grouped/binned) data, superpop_qri for the theoretical QRI of a parametric distribution, if_qri for the influence function used in variance estimation.

Other inequality indicators based on quantiles: inequantiles(), palma_ratio(), plot_inequality_curve(), qsr(), ratio_quantiles()

Examples


data(synthouse)

eq <- synthouse$eq_income ### Income data

# Compute unweighted QRI with default type 6 quantile estimator
qri(y = eq)
#> [1] 0.5611575

# Consider the sampling weights and change quantile estimation type
w <- synthouse$weight
qri(y = eq, weights = w, type = 5)
#> [1] 0.5690882

# Compare QRI across macro-regions (NUTS1)
tapply(1:nrow(synthouse), synthouse$NUTS1, function(area) {
  qri(y = synthouse$eq_income[area],
      weights = synthouse$weight[area],
      type = 6)
})
#>         C         N        NE        NO         S 
#> 0.5847404 0.5639082 0.5691184 0.5796789 0.5601799