Build tau-by-K weights for a 2-component Normal mixture — make_weights_normal

Constructs a weight matrix $W$ that maps RIF tau-slopes $b(\tau)$ into component-parameter effects under a two-Normal mixture baseline.

Usage

make_weights_normal_mixture(
  taus,
  q_tau,
  p1,
  mu1,
  sd1,
  mu2,
  sd2,
  include_membership = FALSE,
  tiny = 1e-12
)

Arguments

taus: Numeric vector of quantile levels.
q_tau: Numeric vector of baseline quantiles at taus (type=8 recommended).
p1: Proportion of class 1 in (0,1); class 2 is 1 - p1.
mu1, sd1: Mean and standard deviation of component 1.
mu2, sd2: Mean and standard deviation of component 2.
include_membership: Logical; include a first column "gamma" if TRUE.
tiny: Small positive floor for stabilizing divisions (default: mixture pdf clamped to tiny).

Value

A numeric matrix (T x K) with column names:

If include_membership = FALSE: c("beta_1","beta_2")
Else: c("gamma","beta_1","beta_2")

Details

Let $Y \sim p_1 N(\mu_1, \sigma_1^2) + (1-p_1) N(\mu_2, \sigma_2^2)$. For baseline unconditional quantiles $q_\tau$ and mixture pdf $f(q_\tau)$, the default columns of $W$ correspond to component means: $$W_1(\tau) = \frac{p_1 f_1(q_\tau)}{f(q_\tau)}, \quad W_2(\tau) = \frac{(1-p_1) f_2(q_\tau)}{f(q_\tau)}$$

If include_membership = TRUE, a first column is added for membership (log-odds) perturbation: $$W_\gamma(\tau) = - \frac{p_1 (1-p_1)}{f(q_\tau)} \left\{F_1(q_\tau) - F_2(q_\tau)\right\}$$

Examples

taus <- seq(0.10, 0.90, by = 0.05)
y <- rnorm(2000, 2, 1)
q_tau <- as.numeric(quantile(y, taus, type = 8))
W <- make_weights_normal_mixture(
  taus, q_tau,
  p1 = 0.5, mu1 = 1.2, sd1 = 0.45,
  mu2 = 3.0, sd2 = 0.7,
  include_membership = TRUE
)
dim(W); colnames(W)
#> [1] 17  3
#> [1] "gamma"  "beta_1" "beta_2"