Existence and Uniqueness of the Maximum Likelihood Estimator for a Multivariate Probit Model

References

• Albert , A. and Anderson , J. A. 1981 . Probit and Logistic Discriminant Functions . Communications in Statistics, Series A—Theory and Methods , : 641 – 657 .
   Google Scholar
• Albert , A. and Anderson , J. A. 1984 . On the Existence of Maximum Likelihood Estimates in Logistic Regression Models . Biometrika , 71 : 1 – 10 .
   Web of Science ®Google Scholar
• Anderson , J. A. and Pemberton , J. D. 1985 . The Grouped Continuous Model for Multivariate Ordered Categorical Variables and Covariate Adjustment . Biometrics , 41 : 875 – 885 .
   PubMed Web of Science ®Google Scholar
• Ashford , J. R. and Sowden , R. R. 1970 . Multivariate Probit Analysis . Biometrics , 26 : 535 – 546 .
   PubMed Web of Science ®Google Scholar
• Bahadur , R. R. 1961 . “ A Representation of the Joint Distribution of Responses to n Dichotomous Items ” . In Studies in Item Analysis and Prediction, Stanford Mathematical Studies in the Social Sciences VI , Edited by: Solomon , H. 169 – 176 . Palo Alto , CA : Stanford University Press .
   Google Scholar
• Billingsley , P. 1968 . Convergence of Probability Measures , New York : John Wiley .
   Google Scholar
• Burridge , J. 1981 . A Note on Maximum Likelihood Estimation for Regression Models Using Grouped Data . Journal of the Royal Statistical Society , 43 : 41 – 45 . Ser. B
   Google Scholar
• Chung , K. L. 1974 . A Course in Probability Theory , New York : Academic Press .
   Google Scholar
• Daganzo , C. 1979 . Multinomial Probit: The Theory and Its Application to Demand Forecasting , New York : Academic Press .
   Google Scholar
• Dale , J. R. 1986 . Global Cross-Ratio Models for Bivariate, Discrete, Ordered Responses . Biometrics , 42 : 909 – 917 .
   PubMed Web of Science ®Google Scholar
• Edlefsen , L. E. and Jones , S. D. 1986 . GAUSS Programming Language , Kent , WA : Aptech Systems .
   Google Scholar
• Hauck , W. W. and Donner , A. 1977 . Wald's Test as Applied to Hypotheses in Logit Analysis . Journal of the American Statistical Association , 72 : 851 – 853 .
   Web of Science ®Google Scholar
• Kaufmann , H. 1988a . On Directions of Strictness, Affinity and Constancy and the Minimum Set of a Proper Convex Function . Optimization , 19 : 157 – 167 .
   Google Scholar
• Kaufmann , H. 1988b . On Existence and Uniqueness of Maximum Likelihood Estimates in Quantal and Ordinal Response Models . Metrika , 35 : 291 – 313 .
   Google Scholar
• Kaufmann , H. 1988c . On Existence and Uniqueness of a Vector Minimizing a Convex Function . ZOR-Methods and Models of Operations Research , 32 : 357 – 373 .
   Google Scholar
• Kiefer , N. M. 1982 . Testing for Dependence in Multivariate Probit Models . Biometrika , 69 : 161 – 166 .
   Web of Science ®Google Scholar
• Lesaffre , E. and Albert , A. 1989 . Partial Separation in Logistic Discrimination . Journal of the Royal Statistical Society , 51 : 109 – 116 . Ser. B
   Google Scholar
• Lesaffre , E. and Molenberghs , G. 1991 . Multivariate Probit Analysis: A Neglected Procedure in Medical Statistics . Statistics in Medicine , 10 : 1391 – 1403 .
   PubMed Web of Science ®Google Scholar
• Liang , K-Y. and Zeger , S. L. 1986 . Longitudinal Data Analysis Using Generalized Linear Models . Biometrika , 73 : 13 – 22 .
   Web of Science ®Google Scholar
• McCullagh , P. and Nelder , J. A. 1989 . Generalized Linear Models , London : Chapman and Hall .
   Google Scholar
• Morimune , K. 1979 . Comparisons of Normal and Logistic Models in the Bivariate Dichotomous Analysis . Econometrica , 47 : 957 – 975 .
   Web of Science ®Google Scholar
• Plackett , R. L. 1965 . A Class of Bivariate Distributions . Journal of the American Statistical Association , 60 : 516 – 522 .
   Web of Science ®Google Scholar
• Pratt , J. W. 1981 . Concavity of the Log Likelihood . Journal of the American Statistical Association , 76 : 103 – 106 .
   Web of Science ®Google Scholar
• Prékopa , A. 1973 . On Logarithmic Concave Measures and Functions . Acta Scientiarum Mathematicarum (Szeged) , 34 : 335 – 343 .
   Web of Science ®Google Scholar
• Prentice , R. L. 1988 . Correlated Binary Regression With Covariates Specific to Each Binary Observation . Biometrics , 44 : 1033 – 1048 .
   PubMed Web of Science ®Google Scholar
• Rao , C. R. and Mitra , S. K. 1971 . Generalized Inverse of Matrices and its Applications , New York : John Wiley .
   Google Scholar
• Rockafellar , R. T. 1970 . Convex Analysis , Princeton , NJ : Princeton University Press .
   Google Scholar
• Santner , T. J. and Duffy , D. E. 1987 . A Note on A. Albert and J. A. Anderson's Conditions for the Existence of Maximum Likelihood Estimates in Logistic Regression Models . Biometrika , 73 : 755 – 758 .
   Web of Science ®Google Scholar
• Silvapulle , M. J. 1981 . On the Existence of Maximum Likelihood Estimates for the Binomial Response Models . Journal of the Royal Statistical Society , 43 : 310 – 313 . Ser. B
   Google Scholar
• Silvapulle , M. J. and Burridge , J. 1986 . Existence of Maximum Likelihood Estimates in Regression Models for Grouped and Ungrouped Data . Journal of the Royal Statistical Society , 48 : 100 – 106 . Ser. B
   Google Scholar
• Verloove-Vanhorick , S. P. , Verwey , R. A. , Brand , R. , Bennebroeck Gravenhorst , J. , Keirse , M. J. N. C. and Ruys , J. H. 1986 . Neonatal Mortality Risk in Relation to Gestational Age and Birthweight . Lancet , i : 55 – 57 .
   Google Scholar
• Zeger , S. L. and Liang , K-Y. 1986 . Longitudinal Analysis for Discrete and Continuous Outcomes . Biometrics , 42 : 121 – 130 .
   PubMed Web of Science ®Google Scholar