fm_var: Fuzzy monetary poverty estimation

Description

This function estimates the variance of the fuzzy monetary poverty index

Usage

fm_var(
  predicate,
  weight,
  fm,
  ID = NULL,
  type = "bootstrap_naive",
  R = 100,
  M = NULL,
  stratum,
  psu,
  f = 0.01,
  verbose = FALSE,
  HCR,
  interval = c(1, 10),
  alpha = NULL,
  hh.size,
  z_min,
  z_max,
  z1,
  z2,
  b,
  z,
  Xs,
  total,
  breakdown = NULL,
  fixed = FALSE,
  data = NULL
)

Value

An object of class FuzzyMonetary containing the estimate of variance with the method selected. if breakdown is not NULL, the variance is estimated for each sub-domain.

Arguments

predicate: A numeric vector representing the poverty predicate (i.e. income or expenditure)
weight: A numeric vector of sampling weights of the same length of predicate. if NULL weights will set equal to n (n = sample size)
fm: The membership function (default is "verma". Other options are "ZBM", "belhadj2015", "belhadj2011", "chakravarty", "cerioli", "verma1999" and "TFR". See Betti et. al., 2023)
ID: A numeric or character vector of IDs. if NULL (the default) it is set as the row sequence
type: The variance estimation method chosen. One between bootstrap_naive (default), bootstrap_calibrated or jackknife
R: The number of bootstrap replicates. Default is 500
M: The size of bootstrap samples. Default is nrow(data)
stratum: The vector identifying the stratum (if 'jackknife' is chosen as variance estimation technique)
psu: The vector identifying the psu (if 'jackknife' is chosen as variance estimation technique)
f: The finite population correction fraction (if 'jackknife' is chosen as variance estimation technique)
verbose: Logical. whether to print the proceeding of the variance estimation procedure
HCR: If fm="verma" or fm="verma1999" or fm="TFR" . The value of the head count ratio used to compute alpha so that the membership function equals the HCR
interval: If fm="verma" or fm="verma1999" or fm="TFR". A numeric vector of length two to look for the value of alpha (if not supplied)
alpha: The value of the exponent in equations of "verma", "verma1999" and "TFR". If NULL it is calculated so that it equates the expectation of the membership function to HCR.
hh.size: If fm="ZBM". A numeric vector of household size
z_min: A parameter of the membership function if fm="belhadj2011", i.e. the z_min: $mu=1 for 0 <y_i<z_min$ (see: See Betti et. al, 2023)
z_max: A parameter of the membership function if fm="belhadj2011", i.e. the z_max: $mu=0 for y_i>z_max$ (see: See Betti et. al, 2023)
z1: A parameter of the membership function if fm="belhadj2015" or fm="cerioli". For "belhadj2015" z1: $mu=1 for y_i<z1$ while for "cerioli" $mu=1 for 0 <y_i<z1$ (see: See Betti et. al, 2023)
z2: A parameter of the membership function if fm="belhadj2015" or fm="cerioli". For "belhadj2015" z2: $mu=0 for y_i>z2$ while for "cerioli" the z1: $mu=0 for y_i>z2$ (see: See Betti et. al, 2023)
b: A parameter of the membership function if fm="belhadj2015". The shape parameter (if b=1 the mf is linear between z1 and z2)
z: A parameter of the membership function if fm="chakravarty", i.e. $mu=0 for y_i>=z$ (see: See Betti et. al, 2023)
Xs: A matrix (i x j) of calibration variables. i number of units, j number of variables
total: A Vector of population totals of dimension 1 x j
breakdown: A factor of sub-domains to calculate estimates for (using the same alpha). If numeric will be coerced to a factor
fixed: Whether the membership function needs to be re-calculated at each bootstrap or jackknife replicate (default is FALSE)
data: An optional data frame containing the variables to be used

References

Belhadj, B. (2011). A new fuzzy unidimensional poverty index from an information theory perspective. Empirical Economics, 40(1):687–704.

Belhadj, B. (2015). Employment measure in developing countries via minimum wage and poverty new fuzzy approach. Opsearch, 52(1):329–339.

Betti, G., Cheli, B., Lemmi, A., and Verma, V. (2006). Multidimensional and longitudinal poverty: an integrated fuzzy approach. In Betti, G. and Lemmi, A., editors, Fuzzy set approach to multidimensional poverty measurement, pages 115–137. Springer, Boston, USA.

Betti, G., D’Agostino, A., Lemmi, A., & Neri, L. (2023). The fuzzy approach to poverty measurement. In Research Handbook on Measuring Poverty and Deprivation Edited by Silber, J. (pp. 489-500). Edward Elgar Publishing.

Betti, G. and Verma, V. (1999). Measuring the degree of poverty in a dynamic and comparative context: a multi-dimensional approach using fuzzy set theory. In Proceedings, iccs-vi, volume 11, pages 289–300.

Cerioli, A. and Zani, S. (1990). A fuzzy approach to the measurement of poverty. In Income and Wealth Distribution, Inequality and Poverty: Proceedings of the Second International Conference on Income Distribution by Size: Generation, Distribution, Measurement and Applications., 272–284. Springer, Boston, USA.

Chakravarty, S. R. (2006). An Axiomatic Approach to Multidimensional Poverty Measurement via Fuzzy Sets. Fuzzy Set Approach to Multidimensional Poverty Measurement, 49-72.

Cheli, B. and Lemmi, A. (1995). A ’totally’ fuzzy and relative approach to the multidimensional analysis of poverty. 24(1):115–134.

Zedini, A. and Belhadj, B. (2015). A new approach to unidimensional poverty analysis: Application to the Tunisian case. Review of Income and Wealth, 61(3):465–476.

Betti, G., Gagliardi, F., & Verma, V. (2018). Simplified Jackknife variance estimates for fuzzy measures of multidimensional poverty. International Statistical Review, 86(1), 68-86.

Examples

Run this code

#The following examples are based on the dataset eusilc
#included in the package.

#Example 1 using bootstrap and breakdown

#fm = "verma"

fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090,
       fm = "verma", breakdown = NULL, type = "bootstrap_calibrated",
       alpha = 4, Xs = eusilc[,4:6], total = c(20, 30, 40))

#fm = "belhadj2015"

fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090,
       fm = "belhadj2015", breakdown = eusilc$db040, type = "bootstrap_naive",
       z1 = 100, z2 = 15000, b = 2)


#Example 2 using jackknife without breakdown

#fm = "verma1999"

fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090,
       fm = "verma1999",  type = "jackknife",
       stratum = eusilc$stratum , psu = eusilc$psu,
       alpha = 4)

#fm = "cerioli"

fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090,
       fm = "cerioli",  type = "jackknife",
       stratum = eusilc$stratum , psu = eusilc$psu,
       z1 = 1000, z2 = 12000)

Run the code above in your browser using DataLab