LorenzRegression (version 2.2.0)

Lorenz.Reg: Fits a Lorenz regression

Description

Lorenz.Reg fits the Lorenz regression of a response with respect to several covariates.

Usage

Lorenz.Reg(
  formula,
  data,
  weights,
  na.action,
  penalty = c("none", "SCAD", "LASSO"),
  grid.arg = c("h", "SCAD.nfwd", "eps", "kernel", "a", "gamma"),
  grid.value = NULL,
  ...
)

Value

An object of class "LR" for the non-penalized Lorenz regression or of class "PLR" for a penalized Lorenz regression.

Several methods are available for both classes to facilitate model analysis. Use summary.LR or summary.PLR to summarize the model fits. Extract the coefficients of the single-index model using coef.LR or coef.PLR. Measures of explained inequality (Gini coefficient and Lorenz-\(R^2\)) are retrieved using ineqExplained.LR or ineqExplained.PLR. Obtain predictions with predict.LR or predict.PLR, and fitted values with fitted.LR or fitted.PLR. For visual representations of explained inequality, use autoplot.LR and plot.LR, or autoplot.PLR and plot.PLR.

The object of class "LR" is a list containing the following components:

theta

The estimated vector of parameters.

Gi.expl

The estimated explained Gini coefficient.

LR2

The Lorenz-\(R^2\) of the regression.

The object of class "PLR" is a list containing the following components:

path

A list where the different elements correspond to the values of the grid parameter. Each element is a matrix where the first line displays the vector of lambda values. The second and third lines display the evolution of the Lorenz-\(R^2\) and explained Gini coefficient along that vector. The next lines display the evolution of the BIC score. The remaining lines display the evolution of the estimated coefficients of the single-index model.

lambda.idx

the index of the optimal lambda obtained by the BIC method

grid.idx

the index of the optimal grid parameter obtained by the BIC method.

In both cases, the list also provides technical information, such as the specified formula, weights and call, as well as the design matrix x and the response vector y.

Arguments

formula

An object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted.

data

An optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which Lorenz.Reg is called.

weights

An optional vector of sample weights to be used in the fitting process. Should be NULL or a numeric vector.

na.action

A function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The 'factory-fresh' default is na.omit. Another possible value is NULL, no action. Value na.exclude can be useful.

penalty

A character string specifying the type of penalty on the size of the estimated coefficients of the single-index model. The default value is "none", in which case a non-penalized Lorenz regression is fitted using Lorenz.GA. Other possible values are "LASSO" and "SCAD", in which case a penalized Lorenz regression is fitted using Lorenz.FABS or Lorenz.SCADFABS respectively.

grid.arg

A character string specifying the tuning parameter for which a grid is to be constructed, see Details.

grid.value

A numeric vector specifying the grid values, see Details.

...

Additional parameters corresponding to arguments passed in Lorenz.GA, Lorenz.FABS or Lorenz.SCADFABS, depending on the argument chosen in penalty.

Details

In the penalized case, the model is fitted for a grid of values of two parameters : the penalty parameter (lambda) and one tuning parameter specified by the arguments grid.arg and grid.value. The possibles values for grid.arg are tuning parameters of the functions Lorenz.FABS and Lorenz.SCADFABS : ''h'' (the default), ''SCAD.nfwd'',''eps'', ''kernel'', ''a'' and ''gamma''. The values for the grid are specified with grid.value. The default is NULL, in which case no grid is constructed

References

Heuchenne, C. and A. Jacquemain (2022). Inference for monotone single-index conditional means: A Lorenz regression approach. Computational Statistics & Data Analysis 167(C).

Jacquemain, A., C. Heuchenne, and E. Pircalabelu (2024). A penalised bootstrap estimation procedure for the explained Gini coefficient. Electronic Journal of Statistics 18(1) 247-300.

See Also

Lorenz.GA, Lorenz.SCADFABS, Lorenz.FABS, Lorenz.boot

Examples

data(Data.Incomes)
set.seed(123)
data <- Data.Incomes[sample(1:200,40),]
# 1. Non-penalized regression
NPLR <- Lorenz.Reg(Income ~ ., data = Data.Incomes, penalty = "none", popSize = 15)
# 2. Penalized regression
PLR <- Lorenz.Reg(Income ~ ., data = Data.Incomes, penalty = "SCAD",
                  eps = 0.06, grid.arg = "h",
                  grid.value=c(0.5,1,2)*nrow(Data.Incomes)^(-1/5.5))
# Print method
print(NPLR)
print(PLR)
# Summary method
summary(NPLR)
summary(PLR)
# Coef method
coef(NPLR)
coef(PLR)
# ineqExplained method
ineqExplained(NPLR)
ineqExplained(PLR)
# Predict method
## One can predict either the index or the response
predict(NPLR,type="response")
predict(PLR,type="response")
# Plot method
## The default displays the explained and observed Lorenz curve.
plot(NPLR)
plot(PLR)
## It is also possible to display a residuals plot.
plot(PLR,type="residuals")
## For PLR only, one can obtain a traceplot of the penalized coefficients
plot(PLR,type="traceplot")