Sankhya: The Indian Journal of Statistics

1999, Volume 61, Series B, Pt. 1, pp. 106--132

CONFIDENCE BOUNDS FOR SURVEY-WEIGHTED QUANTILE PLOTS AND OFFSET-FUNCTION PLOTS

By

S.R. LEE and J.L. ELTINGE *Texas A&M University, College Station*

*SUMMARY.*This paper discusses construction and interpretation of quantile
plots using data obtained through a complex sample design.
Previous quantile-plotting methods are extended
through the use of survey-weighted quantile point estimators.
The resulting graphical methods include normal quantile plots and
related normal offset-function plots; and quantile and offset-function
plots for comparison of two subpopulations.

Confidence bounds associated with each quantile plot can be based on any of three related methods of variance estimation and pivot construction. The first method is based on the Francisco and Fuller (1991) test inversion approach to confidence interval construction. The second method is based on the Woodruff (1952) direct inversion of the quantile function. This second method can be viewed as a variant on the first method, based on a local approximation to the variance of the distribution-function estimator. The third approach is based on a direct normal approximation for the distribution of quantile point estimators. In general, each of the three approaches can be used either for construction of pointwise confidence bounds; or for construction of simultaneous confidence bounds at $k$ quantile points, e.g., the deciles of a distribution. For simultaneous inference at $k$ predetermined points, one generally will prefer to use Bonferroni-based critical points in construction of confidence bounds, but one can also consider Scheff\'e-based critical points. Preference for a given method depends on trade-offs between computational burden and specific inferential goals. The proposed plotting and inference methods are applied to medical-examination data from the Third National Health and Nutrition Examination Survey (NHANES III).

*AMS (1991) subject classification.*62D05.

*Key words and phrases. *
Normal plot; p-p
plot; q-q plot; shift function; third national health and nutrition
examination survey (NHANES III); Test inversion.