"Challenging Research Issues in Statistics and Survey Methodology at the BLS"Topic Statement: Stability of Linearization-Based Variance Estimators Computed from Potentially Unstable Estimators of First Derivatives Key words: Asymptotics; Balanced repeated replication; Degrees of freedom; Inference; Nonlinear function of means; Replication-based variance estimation; t distribution approximation; Wishart distribution approximation. Contact for further discussion: Background, Definitions and Notation: In the analysis of complex survey data, we often need to estimate the variance of the approximate distribution of a random vector In the complex-survey literature, regularity conditions on the sample design and population lead to results on the consistency of Furthermore, under additional regularity conditions (e.g., Korn and Graubard, 1990), Now consider variance estimator for
Under regularity conditions, one can show that Issue: In samples of moderate size, the estimated matrix of first derivatives, Questions on Properties of Standard Variance Estimators for Nonlinear Functions of Estimated Means, and Modifications of Said Variance Estimators:
Acknowledgements The author thanks Moon Jung Cho, Alan Dorfman, Partha Lahiri and Michael Sverchkov for helpful comments on earlier versions of this topic statement. References: Bickel, P. J., C.A.J. Klaassen, Y. Ritov, Y. and J.A. Wellner (1993). Efficient and Adaptive Estimation for Semiparametric Models. Baltimore: John Hopkins University Press. Binder, D.A. (1983). On the variances of asymptotically normal estimators from complex surveys. International Statistical Review 51, 279-292. Binder, D.A. and Z. Patak (1994). Use of estimating functions for estimation from complex surveys, Journal of the American Statistical Association 89 1035-1043. Casady, R.J., A.H. Dorfman and S. Wang (1998). Confidence intervals for domain parameters when the domain sample size is random, Survey Methodology 24, 57-67. Francisco, C.A. and W.A. Fuller (1991). Quantile estimation with a complex survey design, The Annals of Statistics 19, 454-469. Korn, E.L. and Graubard, B.I. (1990). Simultaneous testing of regression coefficients with complex survey data: Use of Bonferroni statistics, The American Statistician 44, 270-276 Krewski, D. and J.N.K. Rao (1981). Inference from stratified samples: Properties of the linearization, jackknife and balanced repeated replication methods, The Annals of Statistics 19 , 1010-1019. Ritov, Y. (1991). Estimating functions in semi-parametric models. Pp. 319-336 in Estimating Functions (V.P. Godambe, ed.), Oxford: Clarendon Press. Shao, J. (1996). Resampling methods in sample surveys (with discussion), Statistics 27, 203-254. Last Modified Date: January 06, 2006
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, where 
is an estimator of a k -dimensional population mean 
computed from complex survey data involving n sample elements, and 
is a continuously differentiable m -dimensional real function of the k -dimensional real argument y.
and the convergence in law of 
to a normal distribution with mean 0 and 
-dimensional variance-covariance matrix 
. Additional regularity conditions then lead to development of a consistent estimator 
of 
where 
is the inverse of the symmetric square root of 
is distributed approximately as a Wishart 
random matrix, where d is a known "degrees of freedom" term computed from the number of primary sample units and the number of strata. Korn and Graubard (1990) also consider extensions of this Wishart approximation for cases in which 
is replaced by a corresponding difference between an estimator and true value for a vector of regression coefficients.
, defined to be the 
-dimensional variance-covariance matrix of the limiting distribution of 
. Under a standard linearization approach, we define the 
matrices
, 
and 
. In practical applications, these matrices often are functions of additional variables, the presence of which is suppressed in the current notation.
, and one commonly defines the corresponding random matrix 
and uses it as an estimator of 
the same degrees of freedom term, d , that was previously attributed to 
as if it were distributed approximately as a Wishart 
random matrix. In addition, under conditions on the function 
and its derivatives, and additional regularity conditions, one can establish that 
converges in law to a multivariate standard normal distribution. For some general background on such asymptotic approaches, see, e.g., Krewski and Rao (1981), Binder (1983), Francisco and Fuller (1991), Binder and Patak (1994) and Shao (1996).
, may itself demonstrate nontrivial random variability.
and 
are of the same order in probability. In a standard asymptotic setting for complex surveys, this generally would occur when the degrees-of-freedom term d is increasing at the same rate as n. Thus, we are excluding from consideration the case in which d is fixed, as would occur with a fixed number of strata and primary sample units, and increasing numbers of sample elements within each primary sample unit. In addition, although we are using an asymptotic framework, we are implicitly excluding from consideration the cases in which d and n are so large that the errors differences 
based on 
, 
? To what extent, if at all, would such a "smoothing" approach be related to methods of variance estimation derived from estimating equations? Also, to what extent can results on variance-estimator stability in the literature on semi-parametric estimation (e.g., Ritov, 1991; and Bickel et al., 1993) be extended to work with complex survey data?
