Conformal Prediction Intervals for Semi-Functional Partial Linear Regression Under β-Mixing Dependence

Abstract

We study prediction intervals for the semi-functional partial linear model (SFPLM) under stationary, geometrically β-mixing dependence. We analyze a split conformal procedure based on a three way data partition with buffer gaps, a functional principal component projection semi-metric on the functional covariate, profiled least squares estimation of the parametric component, kernel estimation of the nonparametric component and of the conditional standard deviation, and a studentized absolute residual score. Marginal validity of split conformal prediction with a trained score under β-mixing is available from generic results of Oliveira et al. and of Barber and Pananjady, without any buffer and without accuracy requirements on the fitted estimators. Our main result is complementary to those guarantees: a finite sample marginal lower coverage bound whose theoretical finite-sample coverage penalty decomposes additively into seven interpretable components expressed in the structural primitives of the SFPLM, quantifying the price of replacing the ideal SFPLM score by the estimated score inside the proof. The penalty plays no role in the computation of the interval, involves unknown structural constants, and is not an operational correction. The bound requires no parametric error model, but it is not assumption free; it holds under explicit structural conditions, including geometric β-mixing, conditionally centered sub-Gaussian errors, a fractal small ball regime for the projected functional covariate, and local regularity of the score distribution. Simulations under a protocol frozen before outcome computation, spanning mild and strong dependence, a misspecification stress test, and a dependent-score design with exactly quantified score autocorrelation, show near nominal coverage for all methods, with the gapped and contiguous variants statistically indistinguishable in coverage. Studentization showed no systematic coverage advantage, while interval-length differences were systematic. The value of the analysis lies in the explicit model-specific estimation layer of the coverage decomposition, not in a numerical gain over naive split conformal.