rif_function: This function calculate the recentered influence function of...

In wenjingwang/drifr: Decomposit Recentered Influence Function Regression with R

Description

This function calculate the recentered influence function of quantile statistic

Usage

rif_function(formula, data, method, tau = NULL, kernel = NULL)

Arguments

formula	formula of RIF regression
data	data with varibales
method	the distrubiton statistic to calculate recentered influence funciton, which can be choose from "mean", "quantle", "variance","gini".
tau	quantile used in method "quantile".
kernel	kernel used for kernel estimating of the dependent variable, selecting from "gaussian", "epanechnikov", "rectangular", "triangular", "biweight", "cosine", "optcosine", used in method "quantile".

Details

The rif_quantile is a function to calculate the recenterd influence function value of different distribution statistics, such as mean, quantile, variance and gini index. The case of mean: The influence function is:

IF(y;μ)=lim_{ε \rightarrow} \frac{[(1-ε)μ+ε y -μ]}{ε} =y-μ

and the recentered influence function is:

RIF(y;μ)=IF(y;μ)+μ=y

The case of quantile: The influence function is: The τ-th quantile of the distribution F is defined as the functional, Q(F, τ) = inf{y|F(y)≥qslant}τ, or as q_{τ} for short, and its influential function is:

IF{y;q_{τ}}=\frac{τ-I{y ≤qslant q_{τ}}}{f_{Y}(q_{τ})}

The recentered influential function of the τ^{th} quantile is

RIF(y;q_{τ}) = q_{τ}+IF(y;q_{τ})= q_{τ}+(τ-I{y ≤qslant q_{τ}})/f_{Y}(q_{τ)})

The case of variance: The influence function of the variance is well-known to be

IF(y;σ^{2})=(y-\int{z \cdot dF_{Y}(z)})^2-σ^2

And the recenterd influence function is:

RIF(y; σ^2)=(y-\int{z \cdot dF_{Y}(z)})^2=(Y-μ)^2

The case of Gini: The Gini coefficient is defined as:

ν^{GC}(F_{Y})=1-2μ^{-1}R(F_{Y})

where R(F_{Y})=\int_{0}^{1} GL(p;F_{Y})dp with p(y)=F_{Y}(y) and where GL(p;F_{Y}) the generalized Lorenz ordinate of F_{Y} is given by GL(p;F_{Y})=\int_{-Inf}{F^{-1}(p)zdF_{Y}(z)}The generalized Lorenz curve tracks the cumulateive total of y divided by total population size against the cumulative distribution function and the generalized Lorenz ordinate can be interpreted as the proportion of earnings going to the 100p The influence function of the Gini coefficient is:

IF(y;ν^{GC})=A_{2}(F_{Y})+B_{2}(F_{Y})y+C_{2}(y;F_{Y})

where,A_{2}(F_{Y})=2μ^{-1}R(F_Y), B_2(F_Y)=2μ^{-2}R(F_Y), C_2(y;F_Y)=-2μ^{-1}[y[1-p(y)]+GL(p(y);F_Y)], thus the recentered influence function of Gini is:

RIF(y;ν^GC)=1+B_2(F_Y)y+C_2(y;F_Y)

In estimation, the GL coordinates are computed using a series of discrete data points y_1, y_2,...y_N, where observations have been ordered so that y_1 ≤q y_2 ≤q y_3 ...≤q y_N

Author(s)

Wenjing Wang [e-mail]

References

Firpo S, Fortin N M, Lemieux T(2009). Unconditional quantile regressions. Econometrica, 77(3): 953-973.

wenjingwang/drifr documentation built on May 4, 2019, 5:20 a.m.