TY - JOUR

T1 - An elementary introduction to maximum likelihood estimation for multinomial models

T2 - Birch’s theorem and the delta method

AU - Cox, Christopher

N1 - Funding Information:
*Christopher Cox is Assistant Professor of Biostatistics in the Division of Biostatistics, University of Rochester School of Medicine and Dentistry, Box 630, 601 Elmwood Avenue, Rochester, NY 14642. Work was supported by National Institute of Environmental Health Sciences Grant ES-01248, by National Institute on Drug Abuse Grant DA-00623, and in part by U.S. Department of Energy Contract DE-AC02-76EV03490 with the University of Rochester, Department of Radiation, Biology, and Biophysics. The author would like to thank Christy Chuang and Charles Odoroff for helpful discussions and the referees and editor for constructive comments.

PY - 1984/11

Y1 - 1984/11

N2 - A fairly complete introduction to the large sample theory of parametric multinomial models, suitable for a second- year graduate course in categorical data analysis, can be based on Birch’s theorem (1964) and the delta method (Bishop, Fienberg, and Holland 1975). I present an elementary derivation of a version of Birch’s theorem using the implicit function theorem from advanced calculus, which allows the presentation to be relatively self-contained. The use of the delta method in deriving asymptotic distributions is illustrated by Rao’s (1973) result on the distribution of standardized residuals, which complements the presentation in Bishop, Fienberg, and Holland. The asymptotic theory is illustrated by two examples.

AB - A fairly complete introduction to the large sample theory of parametric multinomial models, suitable for a second- year graduate course in categorical data analysis, can be based on Birch’s theorem (1964) and the delta method (Bishop, Fienberg, and Holland 1975). I present an elementary derivation of a version of Birch’s theorem using the implicit function theorem from advanced calculus, which allows the presentation to be relatively self-contained. The use of the delta method in deriving asymptotic distributions is illustrated by Rao’s (1973) result on the distribution of standardized residuals, which complements the presentation in Bishop, Fienberg, and Holland. The asymptotic theory is illustrated by two examples.

KW - Asymptotic distributions

KW - Birch’s theorem

KW - Delta method

KW - Maximum likelihood

KW - Multinomial models

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U2 - 10.1080/00031305.1984.10483226

DO - 10.1080/00031305.1984.10483226

M3 - Article

AN - SCOPUS:84954889681

VL - 38

SP - 283

EP - 287

JO - American Statistician

JF - American Statistician

SN - 0003-1305

IS - 4

ER -