Foundations of multivariate inference using modern computers Original Research Article

Fisher suggested in 1930s algebraically structured pivot functions (PFs) whose distribution does not depend on unknown parameters. These pivots provided a foundation for (asymptotic) statistical inference. T.W. Anderson [Introduction to Multivariate Statistical Analysis, Wiley, New York, 1958, p. 116] introduced the concept of a critical function of observables, which finds the rejection probability of a test for Fisherʹs pivot. H.D. Vinod [J. Econometrics 86 (1998) 387] shows that V.P. Godambeʹs [Biometrika 78 (1985) 419] pivot function (GPF) based on Godambe–Durbin ‘estimating funtionsʹ (EFs) from [Ann. Math. Statist. 31 (1960) 1208] are particularly robust compared to pivots by B. Efron and D.V. Hinkley [Biometrica 65 (1978) 457] and R.M. Royall [Internat. Statist. Rev. 54 (2) (1986) 221]. Vinod argues that numerically computed algebraic roots of GPFs based on algebraically scaled score functions can fill a long-standing need of the bootstrap literature for robust pivots. This paper considers D.R. Coxʹs [Biometrica 62 (1975) 269] example in detail and reports on a simulation for it. This paper also discusses new pivots for Poisson mean, binomial probability and normal standard deviation. We propose inference methods for a modified standard deviation designed to represent financial risk. In the context of regression problems, we propose and discuss Godambe-type multivariate pivots (denoted by GPF2) which are asymptotically χ2.