estimateCondQuantiles: Compute kernel-based conditional quantiles

In CondCopulas: Estimation and Inference for Conditional Copula Models

Compute kernel-based conditional quantiles

Description

This function is supposed to be used with computeKernelMatrix. Assume that we observe a sample (X_{i,1}, X_{i,3}), i=1, \dots, n. We want to estimate the conditional quantiles of X_1 given X_3 = x_3 at point u_1 using the following kernel-based estimator

\hat Q(u_1 | X_3 = x_3) := \hat P^{(-1)}(u_1 \leq x_1 | X_3 = x_3),

where

\hat P(X_1 \leq x_1 | X_3 = x_3) := \frac{\sum_{l=1}^n 1 \{X_(l,1) \leq x_1 \} K_h(X_(l,3) - x_3)} {\sum_{l=1}^n K_h(X_(l,3) - x_3)},

for every u_1 in probsX1 and every x_3 in newX3. The matrixK3 should be a matrix of the values K_h(X_(l,3) - x_3) such as the one produced by computeKernelMatrix(observedX3, newX3, kernel, h).

Usage

estimateCondQuantiles(observedX1, probsX1, matrixK3)

Arguments

observedX1	a sample of observations of X1 of size n
probsX1	a sample of probabilities at which we want to compute the quantiles for the variable X1, of size p1
matrixK3	a matrix of kernel values of dimension (p2 , n) \big(K_h(X3[i] - U3[j])\big)_{i,j} such as given by computeKernelMatrix.

Value

A matrix of dimensions (p1,p2) whose (i,j) entry is \hat Q(u_1 | X_3 = x_3) with u_1 = probsX1[i] and x_3 = newX3[j], where newX3[j] is the vector that was used to construct matrixK3.

Examples

Y = MASS::mvrnorm(n = 100, mu = c(0,0), Sigma = cbind(c(1, 0.9), c(0.9, 1)))
matrixK = computeKernelMatrix(observedX = Y[,2] , newX = c(0, 1, 2.5),
  kernel = "Gaussian", h = 0.8)
matrixnp = estimateCondQuantiles(observedX1 = Y[,2],
  probsX1 = c(0.3, 0.5) , matrixK3 = matrixK)
matrixnp

CondCopulas documentation built on Sept. 11, 2024, 9:10 p.m.