Who's Afraid of Reduced-Rank Parameterizations of Multivariate Models? Theory and Example

Scott Gilbert and Petr Zemcik

Reduced-rank restrictions can add useful parsimony to coefficient matrices of multivariate models, but their use is limited by the daunting complexity of the methods and their theory. The present work takes the easy road, focusing on unifying themes and simplified methods. For Gaussian and non-Gaussian (GLM, GAM, etc.) multivariate models, the present work gives a unified, explicit theory for the general asymptotic (normal) distribution of maximum likelihood estimators (MLE). MLE can be complex and computationally difficult, but we show a strong asymptotic equivalence between MLE and a relatively simple minimum (Mahalanobis) distance estimator. The latter method yields particularly simple tests of rank, and we describe its asymptotic behavior in detail. We also examine the method's performance in simulation and via analytical and empirical examples.

Omezeni hodnosti matice mohou podstatne zjednodusit matici koeficientu v modelech s vice promennymi, ale jejich pouziti limituje slozitost metod a jejich teorie. Nas clanek se vydava jednodussi cestou se zamerenim na metodologicke zobecneni a zaroven zjednoduseni. Pro gaussovske a negaussovske modely vice promennych (v anglicke literatu e oznacovane GLM, GAM, atd.) poskytujeme jednotnou, explicitni teorii pro obecne asymptoticke (normalni) rozdeleni estimatoru metody maximalni verohodnosti (EMMV). EMMV muze mit slozitou formu a nemusi byt snadne jej spocitat, nicmene tuto prekazku resime pomoci dukazu asymptoticke ekvivalence mezi EMMV a relativne jednoduchym (Mahalanobis) estimatorem nejmensi vzdalenosti. Tato metoda je vhodna obzvlast pro testy omezeni hodnosti matice a my popiseme detailne jeji asymptoticke vlastnosti v tomto kontextu. Navic zahrneme studii metody v simulacich a analytickych i empirickych prikladech.