Date of Award
Doctor of Philosophy
Herbert T. David
The asymptotic relative efficiency (ARE) of the rounded sample median M[subscript][epsilon] with respect to the rounded sample mean [macron] X[subscript][epsilon] has been considered in the literature for estimating of a Normal population mean restricted to a uniform grid of mesh size 2[epsilon]. This dissertation extends that work using large deviations, to a certain class of symmetric "two-sided extended increasing failure rate" (TEIFR) distributions;Even within our somewhat narrow class, we find the ARE of M[subscript][epsilon] with respect to [macron] X[subscript][epsilon] to be surprisingly sensitive to distribution shape, as well as to grid mesh size and to the actual definition of the ARE. Among our findings is the fact that, in the symmetric TEIFR class, the ARE of M[subscript][epsilon] with respect to [macron] X[subscript][epsilon] is continuous in [epsilon] at [epsilon] = 0 under a definition of the ARE closely related to the commonly used limiting ratio of equivalent sample sizes. A related finding that, within the TEIFR class, the "asymptotic effective variance" of the sample median M equals its asymptotic variance as usually defined;Another finding is that, in the case of the Laplace distribution, M[subscript][epsilon] is asymptotically more efficient that [macron] X[subscript][epsilon], as estimator of the grid-valued population center, when the grid is fine ([epsilon] small), but an asymptotically less efficient estimator when the grid is course ([epsilon] large). This finding bears on certain tests of hypotheses;By extending the theory of univariate large deviations to the bivariate case, the asymptotic variances and covariances of rounded sample means and rounded sample medians can be analyzed for the class of "two-sided extended multivariate IFR" (TEMIFR) distributions;Furthermore when least-squares estimators of the grid-valued parameters in a p-variate regression model, with normal iid disturbance items, are rounded to uniform grids, their asymptotic variances and covariances can be evaluated in terms of appropriate univariate and bivariate large-deviation rates. Also, their joint asymptotic efficiency (JAE), defined as the determinant of the asymptotic variance-covariance matrix, is considered. The JAE turns out to be the sum of the large-deviation rates corresponding to their asymptotic variances.
Digital Repository @ Iowa State University, http://lib.dr.iastate.edu/
Chong Sun Hong
Hong, Chong Sun, "Granularity and efficiency " (1988). Retrospective Theses and Dissertations. 8771.