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:
John L. Eltinge
Office of Survey Methods Research, PSB 1950 Bureau of Labor Statistics
2 Massachusetts Avenue NE
Washington, DC 20212
Telephone: (202) 691-7404
Fax: (202) 691-7426
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 , 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.
In the complex-survey literature, regularity conditions on the sample design and population lead to results on the consistency of for 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 , and to results on the limiting multivariate standard normal distribution of where is the inverse of the symmetric square root of .
Furthermore, under additional regularity conditions (e.g., Korn and Graubard, 1990), 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.
Now consider variance estimator for . In formal terms, we wish to estimate , 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.
Under regularity conditions, one can show that , and one commonly defines the corresponding random matrix and uses it as an estimator of . Furthermore, one generally attributes to the same degrees of freedom term, d , that was previously attributed to , and thus treat 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).
Issue: In samples of moderate size, the estimated matrix of first derivatives, , may itself demonstrate nontrivial random variability.
Questions on Properties of Standard Variance Estimators for Nonlinear Functions of Estimated Means, and Modifications of Said Variance Estimators:
The author thanks Moon Jung Cho, Alan Dorfman, Partha Lahiri and Michael Sverchkov for helpful comments on earlier versions of this topic statement.
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Last Modified Date: January 06, 2006
Last Modified Date: July 19, 2008