CondCopulas (version 0.2.0)

estimateCondQuantiles: 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)

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.

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.

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