Abstract
We propose a new type of stochastic ordering which imposes a monotone tendency in differences between one multinomial probability and a known standard one. An estimation procedure is proposed for the constrained maximum likelihood estimate, and then the asymptotic null distribution is derived for the likelihood ratio test statistic for testing equality of two multinomial distributions against the new stochastic ordering. An alternative test is also discussed based on Neyman modified minimum chi-square estimator. These tests are illustrated with a set of heart disease data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agresti A (1990) Categorical data analysis. Wiley, New York
Cornfield J (1962) Joint dependence of risk of coronary heart disease on serum cholesterol and systolic blood proessure: a discriminant function analysis. Fed Proc 21: 58–61
Cressie N, Read TRC (1984) Multinomial goodness of fit models. J Roy Stat Soc (B) 46: 440–464
Dykstra RL, Lee CIC (1991) Multinomial estimation procedures for isotonic cones. Stat Probab Lett 11: 155–160
Kuhn HW, Tucker AW (1951) Nonlinear programming. 2nd Berkeley Symp., Math Stat probab. University of California Press, Berkeley. pp 481–492
Lee CIC (1987) Chi-squared tests for and against an order restriction on multinomial parameters. J Am Stat Assoc 82: 611–618
Menendez ML, Pardo L, Zografos K (2002) Tests of hypotheses for and against order restrictions on multinomial parameters based on \({\phi}\) -divergences. Utilitas Mathematica 61: 209–223
Robertson T (1978) Testing for and against an order restriction on multinomial parameters. J Am Stat Assoc 73: 197–202
Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley, New York
Serfling RJ (1980) Approximation theorems of mathematical statistics. Wiley, New York
Shaked M, Shanthikumar JG (1994) Stochastic orders and their applications. Academic Press, New York
Shapiro A (1985) Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika 72: 133–144
Silvapulle MJ, Sen PK (2005) Constrained statistical inference. Wiley, New Jersey
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lee, CI.C., Liu, W., Park, C.G. et al. Inference for comparing a multinomial distribution with a known standard. Stat Papers 53, 775–788 (2012). https://doi.org/10.1007/s00362-011-0380-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-011-0380-7