<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 13 Issue 3" %>
Inverting Random Functions III: Discrete MLE Revisited
Mike A. Steel1 and Laszlo A. Szekely2
Biomathematics Research Centre, Mathematics and Statistics Department, University of Canterbury, Private Bag 4800, Christchurch 8041, New Zealand
m.steel@math.canterbury.ac.nz
Department of Mathematics, University of South Carolina, LeConte College, 1523 Greene Street, Columbia, SC 29208, USA
szekely@math.sc.edu
Annals of Combinatorics 13 (3) pp.365-382 September, 2009
AMS Subject Classification: 60C05, 62B10, 92B10, 94A17
Abstract:
This paper continues our earlier investigations into the inversion of random functions in a general (abstract) setting. In Section~\ref{two}, we investigate a concept of invertibility and the invertibility of the composition of random functions defined on finite sets. In Section~\ref{three}, we resolve some questions concerning the number of samples required to ensure the accuracy of maximum likelihood estimation (MLE) in the presence of `nuisance' parameters. A direct application to phylogeny reconstruction is given.
Keywords: random function, maximum likelihood estimation, phylogeny reconstruction

References:

1. Allman, E.S., An´e, C., Rhodes, J.A.: Identiability of a Markovian model of molecular evolution with gamma-distributed rates. Adv. in Appl. Probab. 40, 229-–249 (2008)

2. Casella, G., Berger, R.L.: Statistical Inference. Duxbury Press, Belmont (1990)

3. Chang, J.T.: Full reconstruction of Markov models on evolutionary trees: identiability and consistency. Math. Biosci. 137, 51-–73 (1996)

4. Cover, T.M., Thomas, J.A.: Elements of Information Theory. John Wiley & Sons, Inc., New York (1991)

5. Erd¨os, P.L., Steel, M.A., Sz´ekely, L.A., Warnow, T.: A few logs sufce to build (almost) all trees (Part 1). Random Structures Algorithms 14, 153-–184 (1999)

6. Everitt, B.S.: The Cambridge Dictionary of Statistics. Cambridge University Press, Cambridge, UK (1998)

7. Farris, J.S.: Likelihood and inconsistency. Cladistics 15, 199–-204 (1999)

8. Felsenstein, J.: Inferring Phylogenies. Sinauer Associates Inc., Sunderland, MA. (2004)

9. Rogers, J.S.: On the consistency of maximum likelihood estimation of phylogenetic trees from nucleotide sequences. Syst. Biol. 46, 354-–357 (1997)

10. Rogers, J.S.: Maximum likelihood estimation of phylogenetic trees is consistent when substitution rates vary according to the invariable sites plus gamma distribution. Syst. Biol. 50, 713-–722 (2001)

11. Schrijver, A.: Theory of Linear and Integer Programming. JohnWiley&Sons Ltd., Chichester (1986)

12. Semple, C., Steel, M.: Phylogenetics. Oxford University Press, Oxford (2003)

13. Siddall, M.E.: Success of parsimony in the four-taxon case: long-branch repulsion by likelihood in the Farris zone. Cladistics 14, 209-–220 (1998)

14. Steel, M.A., Sz´ekely, L.A.: Inverting random functions. Ann. Combin. 3, 103-–113 (1999)

15. Steel, M.A., Sz´ekely, L.A.: Inverting random functions II: explicit bounds for the discrete maximum likelihood estimation, with applications. SIAM J. Discrete Math. 15, 562-–575 (2002)

16. Steel, M.A., Sz´ekely, L.A.: Teasing apart two trees. Combin. Probab. Comput. 16, 903–- 922 (2007)

17. Wald, A.: Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20, 595–-601 (1949)

18. Yang, Z.: Statistical properties of the maximum likelihood method of phylogenetic estimation and comparison with distance matrix methods. Syst. Biol. 43, 329-342 (1994)

19. Yang, Z.: Phylogenetic analysis using parsimony and likelihood methods. J. Mol. Evol. 42, 294-–307 (1996)