Solves the quadratic inequality \(a\beta^2 + b\beta + c \leq 0\) derived from the score test inversion to determine the topology and boundaries of the confidence set for \(\beta\).
GetCItypebd(CIcoef)Numeric vector of length 3. The coefficients \((a, b, c)\) obtained
from GetCIcoef.
Numeric vector of length 3. Format: c(CItype, LowerBound, UpperBound).
The confidence set is defined as \(\{ \beta : a\beta^2 + b\beta + c \leq 0 \}\). Depending on the sign of \(a\) and the discriminant \(\Delta = b^2 - 4ac\), this set can take one of four forms:
Type 1: Bounded Interval (\(a \ge 0, \Delta \ge 0\)). The parabola opens upward with real roots. The CI is the closed interval \([\beta_{min}, \beta_{max}]\).
Type 2: Disjoint Union (Donut) (\(a < 0, \Delta \ge 0\)). The parabola opens downward with real roots. The CI is the union of two infinite rays: \((-\infty, \beta_{min}] \cup [\beta_{max}, \infty)\).
Type 3: Real Line (\(a < 0, \Delta < 0\)).
The parabola is always negative. The CI includes the entire real line.
Bounds are returned as placeholders c(-100, 100).
Type 4: Empty Set (\(a \ge 0, \Delta < 0\)). The parabola is always positive. The confidence set is empty, implying the model is rejected at the specified significance level for all \(\beta\).