On the Markov Chain Binomial Model

Affiliation(s)

Department of Mathematics and Statistics, Grant MacEwan University, Edmonton, Canada.

Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada.

Department of Mathematics and Statistics, Grant MacEwan University, Edmonton, Canada.

Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada.

ABSTRACT

Rudolfer [1] studied properties and estimation of a state Markov chain binomial (MCB) model of extra-binomial variation. The variance expression in Lemma 4 is stated without proof but is incorrect, resulting in both Lemma 5 and Theorem 2 also being incorrect. These errors were corrected in Rudolfer [2]. In Sections 2 and 3 of this paper, a new derivation of the variance expression in a setting involving the natural parameters is presented and the relation of the MCB model to Edwards’ [3] probability generating function (pgf) approach is discussed. Section 4 deals with estimation of the model parameters. Estimation by the maximum likelihood method is difficult for a larger number n of Markov trials due to the complexity of the calculation of probabilities using Equation (3.2) of Rudolfer [1]. In this section, the exact maximum likelihood estimation of model parameters is obtained utilizing a sequence of Markov trials each involving n observations from a {0,1}- state MCB model and may be used for any value of n. Two examples in Section 5 illustrate the usefulness of the MCB model. The first example gives corrected results for Skellam’s Brassica data while the second applies the “sequence approach” to data from Crouchley and Pickles [4].

KEYWORDS

Extrabinomial Variation; Markov Chain Binomial Model; Maximum Likelihood Estimation; Sequence Data

Extrabinomial Variation; Markov Chain Binomial Model; Maximum Likelihood Estimation; Sequence Data

Cite this paper

Islam, M. and O’shaughnessy, C. (2013) On the Markov Chain Binomial Model.*Applied Mathematics*, **4**, 1726-1730. doi: 10.4236/am.2013.412236.

Islam, M. and O’shaughnessy, C. (2013) On the Markov Chain Binomial Model.

References

[1] S. M. Rudolfer, “A Markov Chain Model of Extrabinomial Variation,” Biometrika, Vol. 77, No. 2, 1990, pp. 255-264. http://dx.doi.org/10.1093/biomet/77.2.255

[2] S. M. Rudolfer, “Correction to a Markov Chain Model of Extrabinomial Variation,” Biometrika, Vol. 78, No. 4, 1991, p. 935.

http://dx.doi.org/10.2307/2336950

[3] A. W. F. Edwards, “The Meaning of Binomial Distribution,” Nature (London), Vol. 186, 1960, p. 1074.

http://dx.doi.org/10.1038/1861074a0

[4] R. Crouchley and A. R. Pickles, “Methods for the Identification of Lexian, Poisson and Markovian Variations in the Secondary Sex Ratio,” Biometrics, Vol. 40, No. 1, 1984, pp. 165-175.

http://dx.doi.org/10.2307/2530755

[5] J. L. Devore, “A Note on the Estimation of Parameters in a Bernoulli Model with Dependence,” Annals of Statistics, Vol. 4, No. 5, 1976, pp. 990-992.

http://dx.doi.org/10.1214/aos/1176343597

[6] A. W. F. Edwards, “Estimation of the Parameters in Short Markov Sequences,” Journal of the Royal Statistical Society, Series B, Vol. 25, No. 1, 1963, pp. 206-208.

[7] J. G. Skellam, “A Probability Distribution Derived from the Binomial Distribution by Regarding the Probability of Success as Variable between the Sets of Trials,” Journal of the Royal Statistical Society, Series B, Vol. 10, No. 2, 1948, pp. 257-261.

[1] S. M. Rudolfer, “A Markov Chain Model of Extrabinomial Variation,” Biometrika, Vol. 77, No. 2, 1990, pp. 255-264. http://dx.doi.org/10.1093/biomet/77.2.255

[2] S. M. Rudolfer, “Correction to a Markov Chain Model of Extrabinomial Variation,” Biometrika, Vol. 78, No. 4, 1991, p. 935.

http://dx.doi.org/10.2307/2336950

[3] A. W. F. Edwards, “The Meaning of Binomial Distribution,” Nature (London), Vol. 186, 1960, p. 1074.

http://dx.doi.org/10.1038/1861074a0

[4] R. Crouchley and A. R. Pickles, “Methods for the Identification of Lexian, Poisson and Markovian Variations in the Secondary Sex Ratio,” Biometrics, Vol. 40, No. 1, 1984, pp. 165-175.

http://dx.doi.org/10.2307/2530755

[5] J. L. Devore, “A Note on the Estimation of Parameters in a Bernoulli Model with Dependence,” Annals of Statistics, Vol. 4, No. 5, 1976, pp. 990-992.

http://dx.doi.org/10.1214/aos/1176343597

[6] A. W. F. Edwards, “Estimation of the Parameters in Short Markov Sequences,” Journal of the Royal Statistical Society, Series B, Vol. 25, No. 1, 1963, pp. 206-208.

[7] J. G. Skellam, “A Probability Distribution Derived from the Binomial Distribution by Regarding the Probability of Success as Variable between the Sets of Trials,” Journal of the Royal Statistical Society, Series B, Vol. 10, No. 2, 1948, pp. 257-261.