Mathematical Classification of Switching Types and Hierarchical Structures  <div> Three Switching Types, Six Hierarchical Structures, Coefficient of Variation Lower Bound, and Discretization Plateau Effect in the k-Regime Linear Switching Model </div>

Source abstract

<div> In the k-regime linear switching model, the switching between adjacent regimes is fully characterized by the parameter difference Δθ_j = θ_{j+1} − θ_j. This paper restricts the comparison to the union of adjacent regime pairs (R_j, R_{j+1}) and establishes a mathematical classification of switching types under the covariate-centering convention (E[X]=0). </div> <div> <br> </div> <div> Theorem 1 (Switching Type Classification): under the switching existence assumption (θ_j ≠ θ_{j+1}), the parameter difference between adjacent regimes must be one of three types—pure slope switching (Δα_j=0, Δβ_j≠0), full-parameter switching (Δα_j≠0, Δβ_j≠0), or pure intercept switching (Δα_j≠0, Δβ_j=0); the fourth combination is excluded by switching existence. The centering convention is necessary: only the zero/nonzero status of Δβ_j is translation-invariant, and the distinction between Type I and Type II depends on centering. </div> <div> <br> </div> <div> Theorem 2 (Exhaustiveness of Six Hierarchical Structures): under the standard model (A0–A8), the hierarchical structure of the system belongs to one and only one of four types—single-layer system (including pure intercept switching), single-switching system, multi-switching system, dual-switching system—which are mutually exclusive and exhaustive; after introducing additional moderating variables, hierarchical switching systems and cyclic switching systems may arise, and the six types jointly exhaust all hierarchical structures. </div> <div> <br> </div> <div> Theorem 3 (Completeness of the Switching Type Sequence): the switching type sequence S is a complete invariant of the hierarchical partial order P(S), and the mapping P is a bijection from the sequence space to the ordered partition space. Corollary 4 (Hierarchical Structure Counting) provides a combinatorial counting formula for the total number of hierarchical structures. </div> <div> <br> </div> <div> Theorem 4 (Boundary Precision Lower Bound): in the single-boundary model, the coefficient of variation lower bound of the boundary estimator is inversely proportional to the square of the global effect size S_eff. Corollary 2 (Multi-Boundary Per-Boundary Lower Bound): in the multi-boundary model, the coefficient of variation lower bound of the per-boundary spacing estimator is jointly determined by the local effect sizes on both sides of the boundary, with both conservative and exact bounds provided. Corollary 3 (Switching Type–Effect Size Joint Corollary): the lower bounds of the three switching types admit explicit decomposition forms. </div> <div> <br> </div> <div> Section 6.3 provides a switching type inference framework based on the joint Wald test, including explicit construction of the covariance matrix, classification rules (including an “unclassifiable” case), and multiple comparison corrections. Definitions (proxy variable, orthogonal variables, semi-continuous variable, processual variable, cascade system, purity spectrum) provide testable definitions of six extended configurations, distinguishing complete formalizations from working definitions. </div> <div> <br> </div> <div> Numerical verification based on two sets of real simulations (n=1000 and n=2000, each with MC=500) confirms classification identifiability, the necessity of centering, the directional correctness of the lower bound, classification accuracy, and feasibility under multi-dimensional covariates, and yields two original numerical findings. </div> <div> <br> </div> <div> Numerical Finding 1 (Discretization Plateau Effect): the coefficient of variation of the Chow argmax estimator approaches a plateau after S_eff ≥ 5, and the plateau value approximately halves as the sample size doubles—from 2.6×10^−3 at n=1000 to 1.1×10^−3 at n=2000—confirming that the plateau originates from the discretization of observations. </div> <div> <br> </div> <div> Numerical Finding 2 (Sample Size Sensitivity of Weak-Effect-Size Classification): at S_eff = 0.2, doubling the sample size reduces the “unclassifiable” proportion from about 12% to about 0.5%–0.8%, and raises the Type II correct rate from 28% to 61%. The two findings are unified under the mechanism of “insufficient information”: the former is limited by the discretization of boundary location information, while the latter is overwhelmed by noise in parameter difference information; sample size is the common remedy for both. The proofs in this paper directly cite the Information Lower Bound Theorem (Tang, 2026aj, Theorem 3.1) and introduce no additional assumptions. </div>