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Abstract We derive, via the Hardy–Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of nonsingular local solubility. Our polynomials satisfy the condition that . Our conclusions improve on those that would follow from a direct application of the methods of Birch. For example, we show that in many circumstances the expected asymptotic formula holds when and .
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Usable and Precise Asymptotics for Generalized Linear Mixed Model Analysis and Design
Abstract We derive precise asymptotic results that are directly usable for confidence intervals and Wald hypothesis tests for likelihood-based generalized linear mixed model analysis. The essence of our approach is to derive the exact leading term behaviour of the Fisher information matrix when both the number of groups and number of observations within each group diverge. This leads to asymptotic normality results with simple studentizable forms. Similar analyses result in tractable leading term forms for the determination of approximate locally D-optimal designs.
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- Award ID(s):
- 1934568
- PAR ID:
- 10398625
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Journal of the Royal Statistical Society Series B: Statistical Methodology
- Volume:
- 84
- Issue:
- 1
- ISSN:
- 1369-7412
- Format(s):
- Medium: X Size: p. 55-82
- Size(s):
- p. 55-82
- Sponsoring Org:
- National Science Foundation
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