# zbMATH — the first resource for mathematics

Covariance chains. (English) Zbl 1134.62031
Summary: Covariance matrices which can be arranged in tridiagonal form are called covariance chains. They are used to clarify some issues of parameter equivalence and of independence equivalence for linear models in which a set of latent variables influences a set of observed variables. For this purpose, orthogonal decompositions for covariance chains are derived first in explicit form. Covariance chains are also contrasted to concentration chains, for which estimation is explicit and simple. For this purpose, maximum-likelihood equations are derived first for exponential families when some parameters satisfy zero value constraints. From these equations explicit estimates are obtained, which are asymptotically efficient, and they are applied to covariance chains. Simulation results confirm the satisfactory behaviour of the explicit covariance chain estimates also in moderate-sized samples.

##### MSC:
 62H12 Estimation in multivariate analysis 62J05 Linear regression; mixed models 15A99 Basic linear algebra
ggm
Full Text:
##### References:
  Aitchison, J. and Silvey, S.D. (1958) Maximum-likelihood estimation of parameters subject to restraints. Ann. Math. Statist, 29, 813-828. · Zbl 0092.36704  Anderson, T.W. (1958) An Introduction to Multivariate Statistical Analysis. New York: Wiley. · Zbl 0083.14601  Anderson, T.W. (1969) Statistical inference for covariance matrices with linear structure. In P.R. Krishnaiah (ed.), Multivariate Analysis II, pp. 55-66, New York: Academic Press.  Anderson, T.W. (1973) Asymptotically efficient estimation of covariance matrices with linear structure. Ann. Statist., 1, 135-141. · Zbl 0296.62022  Anderson, T.W. and Olkin, I. (1986) Maximum-likelihood estimation of the parameters of a multivariate normal distribution. Linear Algebra Appl., 70, 147-171. · Zbl 0586.62074  Christensen, S. (1989) Statistical properties of I-projections within exponential families. Scand. J. Statist., 16, 307-318. · Zbl 0679.62004  Cochran, W.G. (1938) The omission or addition of an independent variate in multiple linear regression. J. Roy. Statist. Soc., Suppl., 5, 171-176. · Zbl 0019.31902  Cox, D.R. and Wermuth, N. (1990) An approximation to maximum-likelihood estimates in reduced models. Biometrika, 77, 747-761. JSTOR: · Zbl 0709.62050  Cox, D.R. and Wermuth, N. (1993) Linear dependencies represented by chain graphs (with discussion). Statist. Sci., 8, 204-218; 247-277. · Zbl 0955.62593  Cox, D.R. and Wermuth, N. (2000) On the generation of the chordless 4-cycle. Biometrika, 87, 206-212. JSTOR: · Zbl 0976.62054  Cramér, H. (1946) Mathematical Methods of Statistics. Princeton, NJ: Princeton University Press. · Zbl 0063.01014  Dempster, A.P. (1969) Elements of Continuous Multivariate Analysis. Reading, MA: Addison-Wesley. · Zbl 0197.44904  Dempster, A.P. (1972) Covariance selection. Biometrics, 28, 157-175.  Drton, M. and Richardson, T.S. (2003) A new algorithm for maximum-likelihood estimation in Gaussian graphical models for marginal independence. In U. Kjærulff and C. Meek (eds), Uncertainty in Artificial Intelligence: Proceedings of the Nineteenth Conference, pp. 181-191.  Drton, M. and Richardson, T.S. (2004) Multimodality of the likelihood in the bivariate seemingly unrelated regression model. Biometrika, 91, 383-392. · Zbl 1079.62060  Fisher, R.A. (1922) On the mathematical foundations of theoretical statistics. Philos. Trans. Roy. Soc. Lond. Ser. A, 222, 309-368. · JFM 48.1280.02  Frydenberg, M. (1990) The chain graph Markov property. Scand. J. Statist., 17, 333-353. · Zbl 0713.60013  Goldberger, A.S. (1964) Econometric Theory. New York: Wiley. · Zbl 0124.12102  Gram, J.P. (1883) Über die Entwicklung reeller Funktionen in Reihen mittelst der Methode der kleinsten Quadrate. J. Reine Angew. Math., 94, 41-73. · JFM 15.0321.03  Heywood, H.B. (1931) On finite sequences of real numbers. Proc. Roy. Soc. Lond. Ser. A, 134, 486-501. · Zbl 0003.16404  Isserlis, L. (1918) Formulae for determining the near values of products of deviations of mixed moment coefficients. Biometrika, 12, 183-184.  Kauermann, G. (1996) On a dualization of graphical Gaussian models. Scand. J. Statist., 23, 105-116. · Zbl 0912.62006  Kiiveri, H.T. (1987) An incomplete data approach to the analysis of covariance structures. Psychometrika, 52, 539-554. · Zbl 0718.62102  Koster, J. (1999) On the validity of the Markov interpretation of path diagrams of Gaussian structural equation systems of simultaneous equations. Scand. J. Statist., 26, 413-431. · Zbl 0947.60057  Marchetti, G.M. (2006) Independencies induced from a graphical Markov model after marginalization and conditioning: the R package ggm. J. Statist. Software, 15(6).  Markov, A.A. (1912) Wahrscheinlichkeitsrechnung (German translation of 2nd Russian edition). Leipzig: Teubner.  McDonald, R.P. (2002) What can we learn from the path equations? Identifiability, constraints, equivalence. Psychometrika, 67, 225-249. · Zbl 1297.62236  McCullagh, P. (1987) Tensor Methods in Statistics. London: Chapman & Hall. · Zbl 0732.62003  Pearl, J. and Brito, C. (2002) A new identification condition for recursive models with correlated errors. Structural Equation Modeling, 9, 459-474.  Pearl, J. and Wermuth, N. (1994) When can association graphs admit a causal interpretation? In P. · Zbl 0828.05060  Cheeseman and W. Oldford (eds.), Selecting Models from Data: Artificial Intelligence and Statistics IV, Lecture Notes in Statist. 89, pp. 205-214. New York: Springer-Verlag.  Press, S.J. (1972) Applied Multivariate Analysis: Using Bayesian and Frequentist Methods of Inference. New York: Holt, Rinehart and Winston. · Zbl 0276.62051  Richardson, T.S. and Spirtes, P. (2002) Ancestral Markov graphical models. Ann. Statist., 30, 962-1030. · Zbl 1033.60008  Roverato, A. and Whittaker, J. (1998) The Isserlis matrix and its application to non-decomposable graphical models. Biometrika, 85, 711-725. JSTOR: · Zbl 0946.62031  Sargan, J.D. (1958) The estimation of economic relationships using instrumental variables. Econometrica, 26, 393-415. JSTOR: · Zbl 0101.36804  Schmidt, E. (1907) Zur Theorie der linearen und nichtlinearen Integralgleichungen. I. Teil: Entwicklungen willkürlicher Funktionen nach Systemen vorgeschriebener, Math. Ann., 63, 433-476. · JFM 38.0377.02  Speed, T.P. and Kiiveri, H.T. (1986) Gaussian Markov distributions over finite graphs. Ann. Statist., 14, 138-150. · Zbl 0589.62033  Stanghellini, E. and Wermuth, N. (2005) On the identification of path analysis models with one hidden variable. Biometrika, 92, 332-350. · Zbl 1093.62064  Wermuth, N. (1980) Linear recursive equations, covariance selection, and path analysis. J. Amer. Statist. Assoc., 75, 963-997. · Zbl 0475.62056  Wermuth, N. and Cox, D.R. (1998) On association models defined over independence graphs. Bernoulli, 4, 477-495. · Zbl 1037.62054  Wermuth, N. and Cox, D.R. (2004) Joint response graphs and separation induced by triangular systems. J. Roy. Statist. Soc. Ser. B, 66, 687-717. JSTOR: · Zbl 1050.05116  Wilks, S.S. (1932) Certain generalisations in the analysis of variance. Biometrika, 24, 471-494. · Zbl 0006.02301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.