Mixtures of Linear Regressions with Measurement Error in the Response, with an Application to Gamma-Ray Burst Data

Gamma-ray bursts are intense, energetic explosions of gamma rays that are usually accompanied by an afterglow, which is a longer-lived emission that is detected at longer wavelengths, like X-ray, infrared, and radio. Classic gamma-ray burst data is often analyzed using some sort of regression model (e.g., linear, piecewise linear, or a broken-power law model) to relate the flux of the burst to the time since the event. While these models may provide good fits, there is also often a “flaring” phenomena that tends to noticeably deviate from the fitted model. One way we can characterize such a phenomena relative to the underlying general trend is through a mixture-of-regressions model. Some applications in astronomy, like color-luminosity relations for field galaxies, are known to have the variables in the models prone to both intrinsic scatter and measurement error. This assumption is also tenable for gamma-ray burst data where the variance of heteroscedastic measurement errors can be reasonably known. Thus, we introduce a mixture-of-linear-regressions model where the variance of the measurement error is roughly known. Estimation is accomplished using an expectation-maximization (EM) algorithm framework with a weighted least squares estimator that was developed for the non-mixture setting. The finite-sampling behavior of our proposed model’s estimates is examined by a simulation study. We also demonstrate the efficacy of this approach on a dataset involving the flux measurements of gamma-ray bursts, where the variance of the measurement error for the flux measurements (the response) are known. Our results for this data problem are compared with estimates obtained using other traditional models, including the linear regression model and the mixture-of-linear-regressions model.