Abstract
To model an hypothesis of double monotone dependence between two ordinal categorical variables A and B usually a set of symmetric odds ratios defined on the joint probability function is subject to linear inequality constraints. Conversely in this paper two sets of asymmetric odds ratios defined, respectively, on the conditional distributions of A given B and on the conditional distributions of B given A are subject to linear inequality constraints. If the joint probabilities are parameterized by a saturated log-linear model, these constraints are nonlinear inequality constraints on the log-linear parameters. The problem here considered is a non-standard one both for the presence of nonlinear inequality constraints and for the fact that the number of these constraints is greater than the number of the parameters of the saturated log-linear model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agresti A, Coull BA (1998) Order-restricted inference for monotone trend alternatives in contincengy tables. Comput Stat Data Anal 28:139–155
Andrews DWK (1999) Estimation when a parameter is on a boundary. Econometrica 67:1341–1383
Bartolucci F, Scaccia L (2004) Testing for positive association in contingency tables with fixed margins. Comput Stat Data Anal 47(1):195–210
Bartolucci F, Forcina A, Dardanoni V (2001) Positive quadrant dependence and marginal modelling in two-way tables with ordered margins. J Am Stat Assoc 96:1497–1505
Bazaraa MS, Sherali HD, Shetty CM (1993) Nonlinear programming: theory and algorithms. Wiley, New York
Chernoff H (1954) On the distribution of the likelihood ratio. Ann Math Stat 25(3):573–578
Colombi R, Forcina A (2000) Modelizzazione di dati discreti con vincoli di uguaglianza e diseguaglianza. Statistica 60:195–214
Colombi R, Forcina A (2001) Marginal regression models for the analysis of positive association of ordinal response variables. Biometrika 88:1007–1019
Dardanoni V, Forcina A (1998) A unified approach to likelihood inference on stochastic orderings in a nonparametric context. J Am Stat Assoc 93:1112–1123
Douglas R, Fienberg SE, Lee MT, Sampson AR, Whitaker LR (1990) Positive dependence concepts for ordinal contingency tables. In: Topics in statistical dependence. Block HW, Sampson AR, Sanits TH (eds) Institute of mathematical statistics, lecture notes, monograph series. Haywar, California 16:189–202
Geyer CJ (1994) On the asymptotics of constrained M-estimation. Ann Stat 22(4):1993–2010
Glonek GFV, McCullagh P (1995) Multivariate logistic models. J R Stat Soc B 57:533–546
Mangasarian OL (1994) Nonlinear programming. SIAM, Philadelphia
Self SG, Liang KY (1987) Asymptotic properties of maximum likelihood ratio test under nonstandard conditions. J Am Stat Assoc 82:605–610
Shaked M, Shanthikumar JG (1994) Stochastic orders and their applications. Academic, San Diego
Shapiro A (1987) On differentiability of metric projections in \(\mathbb{R}^n\), 1: boundary case. Proc Am Math Soc 99(1):123–128
Shapiro A (1988) Towards a unified theory of inequality constrained testing in multivariate analysis. Int Stat Rev 56:49–62
Silvapulle MJ, Sen PK (2005) Constrained statistical inference. Wiley, New Jersey
Author information
Authors and Affiliations
Corresponding author
Additional information
This work has been supported by the COFIN 2002 project, references 2002133957_002, 2002133957_004. Preliminary findings have been presented at SIS (Società Italiana di Statistica) Annual Meeting, Bari, 2004.
Rights and permissions
About this article
Cite this article
Cazzaro, M., Colombi, R. Maximum Likelihood Inference for Log-linear Models Subject to Constraints of Double Monotone Dependence. Stat. Meth. & Appl. 15, 177–190 (2006). https://doi.org/10.1007/s10260-006-0011-y
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10260-006-0011-y