On conformally covariant powers of the Laplacian

We propose and discuss recursive formulas for conformally covariant powers $P_{2N}$ of the Laplacian (GJMS-operators). For locally conformally flat metrics, these describe the non-constant part of any GJMS-operator as the sum of a certain linear combination of compositions of lower order GJMS-operators (primary part) and a second-order operator which is defined by the Schouten tensor (secondary part). We complete the description of GJMS-operators by proposing and discussing recursive formulas for their constant terms, i.e., for Branson's $Q$-curvatures, along similar lines. We confirm the picture in a number of cases. Full proofs are given for spheres of any dimension and arbitrary signature. Moreover, we prove formulas of the respective critical third power $P_6$ in terms of the Yamabe operator $P_2$ and the Paneitz operator $P_4$, and of a fourth power in terms of $P_2$, $P_4$ and $P_6$. For general metrics, the latter involves the first two of Graham's extended obstruction tensors. In full generality, the recursive formulas remain conjectural. We describe their relation to the theory of residue families and the associated $Q$-curvature polynomials.Comment: We extend the previous description of GJMS-operators to general metrics (Conjecture 11.1

A nonparametric adjustment for tests of changing mean

When testing for a change in mean of a time series, the null hypothesis is no change in mean. However, a change in mean causes a bias in the estimation of serial correlation parameters. This bias can cause nonmonotonic power to the point that if the change is big enough, power can go to zero. In this paper, we show that a nonparametric correction can restore power. The procedure is illustrated with a small Monte Carlo experiment.

Summation formulas for GJMS-operators and Q-curvatures on the M\"obius sphere

For the M\"obius spheres $S^{q,p}$, we give alternative elementary proofs of the recursive formulas for GJMS-operators and $Q$-curvatures due to the first author [Geom. Funct. Anal. 23, (2013), 1278-1370; arXiv:1108.0273]. These proofs make essential use of the theory of hypergeometric series.Comment: 21 pages; final version; updated presentation in view of more recent work of the first author and of Fefferman and Graha