×

First and second order rotatability of experimental designs, moment matrices, and information surfaces. (English) Zbl 0738.62077

Authors’ summary: We place the well-known notion of rotatable experimental designs into the more general context of invariant design problems. Rotatability is studied as it pertains to the experimental designs themselves, as well as to moment matrices, or to information surfaces. The distinct aspects become visible even in the case of first order rotatability. The case of second order rotatability then is conceptually similar, but technically more involved.
Our main result is that second order rotatability may be characterized through a finite subset of the orthogonal group, generated by sign changes, permutations, and a single reflection. This is a great reduction compared to the usual definition of rotatability which refers to the full orthogonal group. Our analysis is based on representing the second order terms in the regression function by a Kronecker power. We show that it is essentially the same as using the Schläflian powers, or the usual minimal set of second order monomials, but it allows a more transparent calculus.

MSC:

62K99 Design of statistical experiments
62A01 Foundations and philosophical topics in statistics
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Aitken AC (1949) On the Wishart distribution in statistics. Biometrika 36:59–62 · Zbl 0033.07103
[2] Aitken AC (1951) Determinants and matrices. Oliver and Boyd, Edinburgh and London
[3] BenIsrael A, Greville T (1974) Generalized inverses: theory and applications. Wiley, New York
[4] Bondar JV (1983) Universal optimality of experimental designs: definitions and a criterion. Canad J Statist 11:325–331 · Zbl 0539.62087 · doi:10.2307/3314890
[5] Bose RC, Carter RL (1959) Complex representation in the construction of rotatable designs. Ann Math Statist 30:771–780 · Zbl 0231.62091 · doi:10.1214/aoms/1177706206
[6] Bose RC, Draper NR (1959) Second order rotatable designs in three dimensions. Ann Math Statist 30:1097–1112 · Zbl 0223.62094 · doi:10.1214/aoms/1177706093
[7] Box GEP, Behnken DW (1960a) Some new three level designs for the study of quantitative variables. Technometrics 2:455–475 · doi:10.2307/1266454
[8] Box GEP, Behnken DW (1960b) Simplex-sum designs: a class of second order rotatable designs derivable from those of first order. Ann Math Statist 31:838–864 · Zbl 0103.12003 · doi:10.1214/aoms/1177705661
[9] Box GEP, Draper NR (1987) Empirical model-building and response surfaces. Wiley, New York · Zbl 0614.62104
[10] Box GEP, Hunter JS (1957) Multi-factor experimental designs for exploring response surfaces. Ann Math Statist 28:195–241 · Zbl 0080.35901 · doi:10.1214/aoms/1177707047
[11] Coxeter HSM (1963) Regular Polytopes. Dover, New York · Zbl 0118.35902
[12] Draper NR (1960) Second order rotatable designs in four or more dimensions. Ann Math Statist 31:23–33 · Zbl 0201.52504 · doi:10.1214/aoms/1177705984
[13] Draper NR, Herzberg AM (1968) Further second order rotatable designs. Ann Math Statist 39:1995–2001 · Zbl 0187.16301 · doi:10.1214/aoms/1177698027
[14] Draper NR, Pukelsheim F (1990) Another look at rotatability. Technometrics 32:195–202 · Zbl 0709.62071 · doi:10.2307/1268863
[15] Gaffke N (1987) Further characterizations of design optimality and admissibility for partial parameter estimation in linear regression. Ann Statist 15:942–957 · Zbl 0649.62070 · doi:10.1214/aos/1176350485
[16] Giovagnoli A, Wynn HP (1981) Optimum continuous block designs. Proc Roy Soc London Ser A 377:405–416 · Zbl 0463.05017 · doi:10.1098/rspa.1981.0131
[17] Giovagnoli A, Wynn HP (1985a) Schur-optimal continuous block designs for treatments with a control. In: Le Cam LM, Olshen RA (eds) Proceedings of the Berkeley Conference in honor of Jerzy Neyman and Jack Kiefer, vol 1. Wadsworth, Belmont CA, pp 418–433 · Zbl 1373.62406
[18] Giovagnoli A, Wynn HP (1985b) G-majorization with applications to matrix orderings. Linear Algebra Appl 67:111–135 · Zbl 0562.15007 · doi:10.1016/0024-3795(85)90190-9
[19] Giovagnoli A, Pukelsheim F, Wynn HP (1987) Group invariant orderings and experimental designs. J Statist Plann Inference 17:159–171 · Zbl 0648.62078 · doi:10.1016/0378-3758(87)90109-1
[20] Henderson HV, Pukelsheim F, Searle SR (1983) On the history of the Kronecker product. Linear and Multilinear Algebra 14:113–120 · Zbl 0517.15017 · doi:10.1080/03081088308817548
[21] Henderson HV, Searle SR (1981) The vec-permutation matrix, the vec operator and Kronecker products: A review. Linear and Multilinear Algebra 9:271–288 · Zbl 0458.15006 · doi:10.1080/03081088108817379
[22] Herzberg AM (1967) A method for the construction of second order rotatable designs ink dimensions. Ann Math Statist 38:177–180 · Zbl 0171.17101 · doi:10.1214/aoms/1177699068
[23] Huda S (1981) A method for constructing second order rotatable designs. Calcutta Statist Assoc Bull 30:139–144 · Zbl 0486.62076
[24] Kiefer JC (1985) Jack Carl Kiefer collected papers, vol III. Brown RD, Olkin I, Sacks J, Wynn HP (eds). Springer, New York
[25] Koll K (1980) Drehbare Versuchspläne erster und zweiter Ordnung. Diplomarbeit, RWTH Aachen
[26] Marcus M, Minc H (1964) A survey of matrix theory and matrix inequalities. Prindle, Weber and Schmidt, Boston MA, London, Syndney · Zbl 0126.02404
[27] Minc H (1978) Permanents. Encyclopedia of mathematics and its applications, vol 6. Addison-Wesley, Reading MA
[28] Muir T (1911) The theory of determinants in the historical order of development, vol II. The Period 1841–1860. Macmillan, London · JFM 42.0168.06
[29] Neumaier A, Seidel JJ (1990) Measures of strength 2e, and optimal designs of degreee. Sankhya, forthcoming
[30] Nigam AK (1977) A note on four and six level second order rotatable designs. J Indian Soc Agric Statist 29:89–91
[31] Nigam AK, Das MN (1986) On a method of construction of rotatable designs with smaller number of points controlling the number of levels. Calcutta Statist Assoc Bull 15:153–174
[32] Nigam AK, Dey A (1970) Four and six level second order rotatable designs. Calcutta Statist Assoc Bull 19:155–157 · Zbl 0224.62037
[33] Pukelsheim F (1977) On Hsu’s model in regression analysis. Math Operationsforsch Statist Ser Statist 8:323–331 · Zbl 0375.62069
[34] Pukelsheim F (1987a) Majorization orderings for linear regression designs. In: Pukkila T, Puntanen S (eds) Proceedings of the second international Tampere Conference in statistics. Department of Mathematical Sci, Tampere, pp 261–274
[35] Pukelsheim F (1987b) Information increasing orderings in experimental design theory. Internat Statist Rev 55:203–219 · Zbl 0625.62061 · doi:10.2307/1403196
[36] Pukelsheim F (1987c) Ordering experimental designs. In: Prohorov Yu A, Sazonov VV (eds) Proceedings of the 1st World Congress of the Bernoulli Society, vol 2. Tashkent, USSR, 8–14 Sept 1986. VNU Science Press, Utrecht, pp 157–165
[37] Raghavarao D (1963) Construction of second order rotatable designs through incomplete block designs. J Ind Statist Assoc 1:221–225
[38] Schläfli L (1851) Über die Resultante eines Systems mehrerer algebraischer Gleichungen. Ein Beitrag zur Theorie der Elimination. Denkschriften der Kaiserlichen Akademie der Wissenschaften, mathematisch-naturwissenschaftliche Klasse, 4. Band (1852) Wien. Reprinted in: Ludwig Schläfli (1814–1895) Gesammelte Mathematische Abhandlungen, Band II, herausgegeben vom Steiner-Schläfli-Komitee der Schweizerischen Naturforschenden Gesellschaft, Birkhäuser, Basel 1953
[39] Searle SR (1982) Matrix algebra useful for statistics. Wiley, New York · Zbl 0555.62002
[40] Seely J (1971) Quadratic subspaces and completeness. Ann Math Statist 42:710–721 · Zbl 0249.62067 · doi:10.1214/aoms/1177693420
[41] Seymour PD, Zaslavski T (1984) Averaging sets: a generalization of mean values and spherical designs. Adv in Math 52:213–240 · Zbl 0596.05012 · doi:10.1016/0001-8708(84)90022-7
[42] Singh M (1979) Group divisible second order rotatable designs. Biometrical J 21:579–589 · Zbl 0434.62059 · doi:10.1002/bimj.4710210607
[43] Wedderburn JHM (1934) Lectures on Matrices. Colloquium Publ vol XVII. American Math Society, Providence RI · Zbl 0010.09904
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.