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
SN - 0003-1305
VL - 38
SP - 283
EP - 287
JO - American Statistician
JF - American Statistician
IS - 4
ER -