Bootstrap Confidence Sets for the Number of Components in Finite Mixture Regression Models

Source abstract

Finite mixture regression analyses usually treat the selected number of components as fixed, although neighboring orders may be difficult to distinguish. This paper develops a set-valued procedure for the order of a Gaussian finite mixture regression. A Bayesian information criterion (BIC) fit supplies a reference order, and the remaining candidate orders are compared with it through likelihood-ratio statistics calibrated by a conditional parametric bootstrap that keeps the observed regressors fixed; the calibration therefore reflects the separation between the component regression functions at the realized design. For a sample-split version, explicit conditions on the design, the parameter space and the separation of the components yield consistent bootstrap tests and asymptotic coverage of at least 1−α for Monte Carlo tests with a fixed number of draws and of at least 1−α−ϵ for an untruncated sequential implementation with resampling risk ϵ. The size result additionally requires a locally uniform limit for the nonregular likelihood-ratio statistic. For the full-sample version, we show why smaller orders are rarely retained so that the set is typically the BIC order extended upwards; its coverage tends to one at fixed well-separated parameters, and its finite-sample coverage is determined by the probability that BIC overfits and by the power of the upward tests. In Monte Carlo experiments with sequential bootstrap decisions of at most 499 draws, the empirical coverage is close to the nominal 95% for well-separated components, equals 0.753 when a mixing proportion is 0.05, and does not exceed 0.15 when two regression functions nearly coincide. Applications to Fisher’s iris data and to tone perception data illustrate the set-valued summary of order uncertainty and its dependence on the variance constraint.