Calculus of Taylor Deviations

• Activity Number: 405
• Type: Contributed
• Date/Time: Thursday, August 15, 2002 : 10:30 AM to 12:20 PM
• Sponsor: Section on Survey Research Methods*
• Author(s): Babubhai Shah*+
• Affiliation(s): RTI International
• Keywords: Taylor deviation ; linearized jackknife ; sampe survey ; Asymptotic variance ; robust variance

Abstract

In large samples, the approximate variance of a complex statistic is obtained through Taylor series. In sample survey applications, the approach has been to derive a pseudo variate for each observation corresponding to each statistics, so that the variance of the statistic is approximated by the variance of the sum of the pseudo variable. The pseudo variables are commonly known as the Taylor (or Taylorized) deviations, or Jackknife linearization, depending on how they are derived. In this paper, we define an operator, capital delta, with the suffix "I." The DELTA operator, when applied to a function of a statistic, produces the function for the Taylor deviation with respect to the i-th observation for that statistic. The rules are similar to differential calculus and can be applied to statistics that are defined explicitly or implicitly or any combination. The method correctly produces the appropriate Taylor deviation for all the known cases. We illustrate the usefulness of this approach by applying to some complex statistics.