Abstract
We propose conditional exact tests based on sufficient statistics to compare exchangeability, Markov exchangeability, and Markov exchangeability of the reversible type, when data consist of several sequences of categorical data. As particular cases, we can compare three classes of mixture models (mixtures of i.i.d. sequences, mixtures of Markov chains and of reversible Markov chains) and we can test Markovianity and reversibility of a single sequence.
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References
Aldous DJ (1985) Exchangeability and related topics. In: École d’été de probabilités de Saint-Flour, XIII–1983. Lecture notes in math., vol 1117. Springer, Berlin, pp 1–198
Beckett L, Diaconis P (1994) Spectral analysis for discrete longitudinal data. Adv Math 103(1):107–128. doi:10.1006/aima.1994.1002
Billard L, Meshkani MR (1995) Estimation of a stationary Markov chain. J Am Stat Assoc 90(429):307–315
Brualdi RA, Cvetković D (2009) A combinatorial approach to matrix theory and its applications. Discrete mathematics and its applications. CRC Press, Boca Raton
Cowan R (1991) Expected frequencies of DNA patterns using Whittle’s formula. J Appl Probab 28(4):886–892
Dawson R, Good IJ (1957) Exact Markov probabilities from oriented linear graphs. Ann Math Stat 28:946–956
Di Cecco D (2009) A class of models for multiple binary sequences under the hypothesis of Markov exchangeability. Electron J Stat 3:1113–1132. doi:10.1214/09-EJS478
Diaconis P, Freedman D (1980a) de Finetti’s generalizations of exchangeability. In: Studies in inductive logic and probability, vol II, pp 233–249. University California Press, Berkeley
Diaconis P, Freedman D (1980b) de Finetti’s theorem for Markov chains. Ann Probab 8(1):115–130
Diaconis P, Freedman D (1984) Partial exchangeability and sufficiency. In: Statistics: applications and new directions (Calcutta, 1981), pp 205–236. Indian Statist Inst, Calcutta
Diaconis P, Rolles SWW (2006) Bayesian analysis for reversible Markov chains. Ann Stat 34(3):1270–1292
Diaconis P, Sturmfels B (1998) Algebraic algorithms for sampling from conditional distributions. Ann Stat 26(1):363–397. doi:10.1214/aos/1030563990
de Finetti B (1959) La probabilità e la statistica nei rapporti con l’induzione, secondo i diversi punti di vista C I M E, Induzione e Statistica (No 1), pp 1–115
Freedman D (1962) Invariants under mixing which generalize de Finetti’s theorem. Ann Math Stat 33:916–923
Hettmansperger T, Hoben T (2000) Almost nonparametric inference for repeated measures in mixture models. J R Stat Soc, Ser B 62:811–825
Hutchinson JP, Wilf HS (1975) On Eulerian circuits and words with prescribed adjacency patterns. J Comb Theory, Ser A 18, 80–87
Janssen A, Völker D (2007) Most powerful conditional tests. Stat Decis 25(1):41–62. doi:10.1524/stnd.2007.25.1.41
McCausland WJ (2007) Time reversibility of stationary regular finite-state Markov chains. J Econom 136(1):303–318. doi:10.1016/j.jeconom.2005.09.001
Mihail M, Winkler P (1996) On the number of Eulerian orientations of a graph. Algorithmica 16(4–5):402–414. doi:10.1007/s004539900057
Quintana FA, Newton MA (1998) Assessing the order of dependence for partially exchangeable binary data. J Am Stat Assoc 93(441):194–202
Radlow R, Alf EF (1975) An alternate multinomial assessment of the accuracy of the χ 2 test of goodness of fit. J Am Stat Assoc 70(352):811–813
Richman D, Sharp WE (1990) A method for determining the reversibility of a Markov sequence. Math Geol 22(7):749–761. doi:10.1007/BF00890660
Rødland EA (2006) Exact distribution of word counts in shuffled sequences. Adv Appl Probab 38(1):116–133. doi:10.1239/aap/1143936143
Rolles SWW (2003) How edge-reinforced random walk arises naturally. Probab Theory Relat Fields 126(2):243–260
Sharifdoost M, Mahmoodi S, Pasha E (2009) A statistical test for time reversibility of stationary finite state Markov chains. Appl Math Sci (Ruse) 3(49–52), 2563–2574
Tetali P, Vempala S (2001) Random sampling of Euler tours. Algorithmica 30(3):376–385
Whittle P (1955) Some distribution and moment formulae for the Markov chain. J R Stat Soc, Ser B 17:235–242
Wood GR (1992) Binomial mixtures and finite exchangeability. Ann Probab 20(3):1167–1173
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Communicated by: Domingo Morales.
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Di Cecco, D. Conditional exact tests for Markovianity and reversibility in multiple categorical sequences. TEST 21, 170–187 (2012). https://doi.org/10.1007/s11749-011-0241-7
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DOI: https://doi.org/10.1007/s11749-011-0241-7