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We consider the estimation of a common period for a set of functions sampledat irregular intervals. The problem arises in astronomy, where the functionsrepresent a star's brightness observed over time through different photometricfilters. While current methods can estimate periods accurately provided thatthe brightness is well--sampled in at least one filter, there are no existingmethods that can provide accurate estimates when no brightness function iswell--sampled. In this paper we introduce two new methods for period estimationwhen brightnesses are poorly--sampled in all filters. The first, multibandgeneralized Lomb-Scargle (MGLS), extends the frequently used Lomb-Scarglemethod in a way that na\"{i}vely combines information across filters. Thesecond, penalized generalized Lomb-Scargle (PGLS), builds on the first by moreintelligently borrowing strength across filters. Specifically, we incorporateconstraints on the phases and amplitudes across the different functions using anon--convex penalized likelihood function. We develop a fast algorithm tooptimize the penalized likelihood by combining block coordinate descent withthe majorization-minimization (MM) principle. We illustrate our methods onsynthetic and real astronomy data. Both advance the state-of-the-art in periodestimation; however, PGLS significantly outperforms MGLS when all functions areextremely poorly--sampled.

Estimating a Common Period for a Set of Irregularly Sampled Functions with Applications to Periodic Variable Star Data