## Bernoulli

• Bernoulli
• Volume 16, Number 4 (2010), 1208-1223.

### Second order ancillary: A differential view from continuity

#### Abstract

Second order approximate ancillaries have evolved as the primary ingredient for recent likelihood development in statistical inference. This uses quantile functions rather than the equivalent distribution functions, and the intrinsic ancillary contour is given explicitly as the plug-in estimate of the vector quantile function. The derivation uses a Taylor expansion of the full quantile function, and the linear term gives a tangent to the observed ancillary contour. For the scalar parameter case, there is a vector field that integrates to give the ancillary contours, but for the vector case, there are multiple vector fields and the Frobenius conditions for mutual consistency may not hold. We demonstrate, however, that the conditions hold in a restricted way and that this verifies the second order ancillary contours in moderate deviations. The methodology can generate an appropriate exact ancillary when such exists or an approximate ancillary for the numerical or Monte Carlo calculation of $p$-values and confidence quantiles. Examples are given, including nonlinear regression and several enigmatic examples from the literature.

#### Article information

Source
Bernoulli, Volume 16, Number 4 (2010), 1208-1223.

Dates
First available in Project Euclid: 18 November 2010

https://projecteuclid.org/euclid.bj/1290092903

Digital Object Identifier
doi:10.3150/10-BEJ248

Mathematical Reviews number (MathSciNet)
MR2759176

Zentralblatt MATH identifier
1207.62041

#### Citation

Fraser, Ailana M.; Fraser, D.A.S.; Staicu, Ana-Maria. Second order ancillary: A differential view from continuity. Bernoulli 16 (2010), no. 4, 1208--1223. doi:10.3150/10-BEJ248. https://projecteuclid.org/euclid.bj/1290092903

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