Here you find an overview of the literature about constrained statistical Inference, robust estimation or a combination of both. All formulas used in restriktor are coming from these articles.


Bartholomew, D.J. 1959a. “A Test of Homogeneity for Ordered Alternatives.” Biometrika 46: 36–48.

———. 1959b. “A Test of Homogeneity for Ordered Alternatives. II.” Biometrika 46: 328–35.

———. 1961a. “A Test of Homogeneity of Means Under Restricted Alternatives.” Journal of the Royal Statistical Society. Series B (Methodological) 23: 239–81.

———. 1961b. “Ordered Tests in the Analysis of Variance.” Biometrika 48: 325–32.

Dykstra, R. 1991. “Asymptotic Normality for Chi-Bar-Square Distributions.” The Canadian Journal of Statistics 19: 297–306.

Gouriéroux, C., A. Holly, and A. Monfort. 1982. “Likelihood Ratio Test, Wald Test, and Kuhn-Tucker Test in Linear Models with Inequality Constraints on the Regression Parameters.” Econometrica 50: 63–80.

Grömping, U. 2010. “Inference with Linear Equality and Inequality Constraints Using R: The Package Ic.infer.” Journal of Statistical Software 33: 1–31. doi:10.18637/jss.v033.i10.

Huber, P.J. 1964. “Robust Estimation of a Location Parameter.” The Annals of Mathematical Statistics 35: 73–101.

———. 1973. “Robust Regression: Asymptotics, Conjectures and Monte Carlo.” The Annals of Statistics 1: 799–821.

Kudô, A. 1963. “A Multivariate Analogue of the One-Sided Test.” Biometrika 50: 403–18.

Kudô, A., and J.R. Choi. 1975. “A Generalized Multivariate Analogue of the One Sided Test.” Memoirs of the Faculty of Science 29: 303–28.

Kuiper, R.M., H. Hoijtink, and M.J. Silvapulle. 2011. “Generalization of the Order-Restricted Information Criterion for Multivariate Normal Linear Models.” Biometrica 2: 495–501.

Kuiper, R.M., I.G. Klugkist, and H.J.A. Hoijtink. 2010. “A Fortran 90 Program for Confirmatory Analysis of Variance.” Journal of Statistical Software 34: 1–31.

Nüesch, P.E. 1966. “On the Problem of Testing Location in Multivariate Populations for Restricted Alternatives.” The Annals of Mathematical Statistics 37: 113–9.

Perlman, M.D. 1969. “One-Sided Testing Problems in Multivariate Analysis.” The Annals of Mathematical Statistics 40: 549–67.

Salibián-Barrera, M. 2005. “Estimating the p-Values of Robust Tests for the Linear Model.” Journal of Statistical Planning and Inference 128: 241–57. doi:doi:10.1016/j.jspi.2003.09.033.

Salibián-Barrera, M., S. Van Aelst, and V.J. Yohai. 2014. “Robust Tests for Linear Regression Models Based on τ-Estimates.” Computational Statistics and Data Analysis. doi:10.1016/j.csda.2014.09.012v.

Schoenberg, R. 1997. “Constrained Maximum Likelihood.” Computational Economics 10: 251–66.

Schrader, R.M., and T.P. Hettmansperger. 1980. “Robust Analysis of Variance Based Upon a Likelihood Ratio Criterion.” Biometrika 67: 93–101.

Self, S.G., and K.Y. Liang. 1987. “Asymptotic Properties of Maximum Likelihood Estimators and the Likelihood Ratio Tests Under Nonstandard Conditions.” Journal of the American Statistical Association, no. 82: 605–10.

Shapiro, A. 1988. “Towards a Unified Theory of Inequality Constrained Testing in Multivariate Analysis.” International Statistical Review, 49–62. doi:10.2307/1403361.

Shi, N.Z., S.R. Zheng, and J. Guo. 2005. “The Restricted EM Algorithm Under Inequality Restrictions on the Parameters.” Journal of Multivariate Analysis, no. 92: 53–76.

Silvapulle, M. 1992a. “Robust Tests of Inequality Constraints and One-Sided Hypotheses in the Linear Model.” Biometrika 79: 621–30.

———. 1992b. “Robust Wald-Type Tests of One-Sided Hypotheses in the Linear Model.” Journal of the American Statistical Association 87 (417): 156–61.

———. 1996a. “On an F-type Statistic for Testing One-Sided Hypotheses and Computation of Chi-Bar-Squared Weights.” Statistics & Probability Letters 28: 137–41.

———. 1996b. “Robust Bounded Influence Tests Against One-Sided Hypotheses in General Parametric Models.” Statistics & Probability Letters, no. 31: 45–50.

———. 1997. “On Order Restricted Inference in Some Mixed Linear Models.” Statistics & Probability Letters 36: 23–27.

Silvapulle, M., and P. Silvapulle. 1995. “A Score Test Against One-Sided Alternatives.” American Statistical Association 90 (429): 342–49.

Silvapulle, M.J. 1994. “On Tests Against One-Sided Hypotheses in Some Generalized Linear Models.” Biometrics, no. 50: 853–58.

Vanbrabant, L., R. Van de Schoot, and Y Rosseel. 2015. “Constrained Statistical Inference: Sample-Size Tables for ANOVA and Regression.” Frontiers in Psychology 5: 1–8. doi:10.3389/fpsyg.2014.01565.

Wolak, F. 1987. “An Exact Test for Multiple Inequality and Equality Constraints in the Linear Regression Model.” Journal of the American Statistical Association 82 (399): 782–93.

Wolak, F.A. 1989. “Testing Inequality Constraints in Linear Econometric Models.” Journal of Econometrics 41: 205–35.

Wright, F.T. 1988. “The One-Way Analysis of Variance with Ordered Alternatives: A Modification of Bartholomew’s $\bar{E}^2$ Test.” The Canadian Journal of Statistics.

Yancey, T.A., G.G. Judge, and M.E. Bock. 1981. “Testing Multiple Equality and Inequality Hypothesis Is Economics.” Economics Letters 7: 249–55.

Yohai, V.J. 1987. “High Breakdown-Point and High Efficiency Robust Estimates for Regression.” The Annals of Statistics 15: 642–56.

Zelazo, P.R., N.A. Zelazo, and S. Kolb. 1972. “‘Walking’ in the Newborn.” Science.


Barlow, R. E., D. J. Bartholomew, H. M. Bremner, and H. D. Brunk. 1972. Statistical Inference Under Order Restrictions. New York: Wiley.

Efron, E., and R.J. Tibshirani. 1993. An Introduction to the Bootstrap. Chapman & Hall.

Hampel, F., Ronchetti E., Rousseeuw P., and W. Stahel. 1986. Robust Statistics. John Willey; Sons, New York.

Huber, P.J. 1981. Robust Statistics. New York: Wiley.

Huber, P.J., and E.M. Ronchetti. 2009. Robust Statistics. New York: Wiley.

Robertson, T., F. T. Wright, and R. L. Dykstra. 1988. Order Restricted Statistical Inference. New York: Wiley.

Rousseeuw, P., and V. Yohai. 1984. “Robust Regression by Means of S-Estimators.” In Robust and Nonlinear Time Series Analysis, edited by J. Franke, W. Härdle, and D. Martin, 256–72. Spring-Verlag: New York.

Silvapulle, M.J., and P.K. Sen. 2005. Constrained Statistical Inference: Order, Inequality, and Shape Constraints. Hoboken, NJ: Wiley.