Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD)

Awogbemi Clement Adeyeye

doi:doi:10.11648/j.engmath.20200401.11

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Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD)

Awogbemi Clement Adeyeye

Published in Engineering Mathematics (Volume 4, Issue 1)

Received: 16 October 2019 Accepted: 12 November 2019 Published: 28 May 2020

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Abstract

The exact distribution of the test statistics in multivariate case is quite complicated in many situations, even when the underlying distribution is multivariate normal. This is due to the complex nature of the expression and therefore, there is a need to derive the asymptotic expression for the distribution. In this study, the asymptotic distribution of errors of misclassification for Edgeworth Series is derived by using Taylor’s expansion. The error of misclassification for the conditional probability of misclassification was expanded around the means emanating from populations one and two using approximated mean and variance of the errors of misclassification. The distribution of error of misclassification of the conditional probability of misclassification for ESD is approximately normal with mean zero and variance one.

Published in	Engineering Mathematics (Volume 4, Issue 1)
DOI	10.11648/j.engmath.20200401.11
Page(s)	1-9
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Asymptotic Distribution, Probability of Misclassification, Edgeworth Series Distribution, Approximate Mean, Approximate Variance

References

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[2]	Bhuyan, C. K. (2010), Probability Theory and Statistical Inference, 1^st ed. New Central Book Agency (Ltd), London.
[3]	Does, R. J. M. M., Helmers, R. and Klaassen, C. A. J. (1987). Approximating the Distribution of Greenwood’s Statistics. Statistica Neerlandica, 42: 153-162.
[4]	Ekezie, D. D. and Onyeagu, S. I. (2013). Comparison of Seven Asymptotic Expansion for the Sample Linear Discriminant. Function, Canadian Journal of Computations in Mathematics, Natural Sciences, Engineering and Medicine, 4 (1): 93-115.
[5]	Ghosh, K. and Jammalamadaka, S. R. (1998). Small Sample Approximations for Spacing Statistics. Journal of Statistical Planning and Inference, 69: 245-261.
[6]	Goncalves, S. and Meddahi, N. (2009), Bootstrap Realized Volatility. Econometrica, 77 (1): 283-306.
[7]	Gudhinder, K. and Paul, R. (2012). Edgeworth Expansions for GEL Estimator. Journal of Multivariate Analysis, 106: 118-146.
[8]	Kendall, M. G. and Stuart, A. (1958). The Advanced Theory of Statistics, Hafner: New York.
[9]	Kocherlakota, S. and Chinganda, E. F. (1978). Robustness of the Linear Discriminant Function to Non-normality: Edgeworth Series Distribution. Journal of Statistical Planning and Inference, 2: 79-91.
[10]	Meyer, P. L. (1966). Introductory Probability and Statistical Applications. Addison-Wesley: Reading, Mass.
[11]	Muhammad, N. (2017). On the Edgeworth Type Expansions for the Distribution of Extreme Spacing Statistics. Pakistan Journal of Statistics, 33 (2): 117-128.
[12]	Peter, H. (1997). The Bootstrap and Edgeworth Expansions. Springer Series in Statistics, 145.
[13]	Rani, B. (2010). Edgeworth Expansion and Saddle Point Approximation for Discrete Data with Application to Chance Game. M. Sc. Thesis, University of Linnaeus, School of Computer, Physics and Mathematics, 1-39.
[14]	Ruby, C. W. (2010). A Bayesian Edgeworth Expansion by Stein’s Identity. Bayesian Analysis, 5 (4): 741-764.
[15]	Ulrich, H. and Bezirgen, V. (2015). Validity of Edgeworth Expansions for Realized Volatility Estimators. CREATES Research Papers 2015-21, Centre for Research in Econometrics Analysis of Time Series, Department of Economics and Business Aarhus University, Denmark, 1-31.
[16]	Wu, X. and Wang, S. (2011). Maximum Entropy Approximations for Asymptotic Distributions of Smooth Functions of Sample. Scandinavian Journal of Statistics, 38: 130-146.
[17]	Yonatan, R. and Monika, P. (2014). Edgeworth Expansion Based Model for the Convolutional Noise pdf. Mathematical Problems in Engineering, 20 (14): 1-19.
[18]	Zarkar, K., Alenxader, M. & Henning, S. (2016). Mode and Edgeworth Expansion for the Ewens Distribution and the Stirling Number., Journal of Integer Sequences, 19: 1-7.

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Awogbemi Clement Adeyeye. (2020). Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD). Engineering Mathematics, 4(1), 1-9. https://doi.org/10.11648/j.engmath.20200401.11

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Awogbemi Clement Adeyeye. Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD). Eng. Math. 2020, 4(1), 1-9. doi: 10.11648/j.engmath.20200401.11

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Awogbemi Clement Adeyeye. Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD). Eng Math. 2020;4(1):1-9. doi: 10.11648/j.engmath.20200401.11

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@article{10.11648/j.engmath.20200401.11,
  author = {Awogbemi Clement Adeyeye},
  title = {Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD)},
  journal = {Engineering Mathematics},
  volume = {4},
  number = {1},
  pages = {1-9},
  doi = {10.11648/j.engmath.20200401.11},
  url = {https://doi.org/10.11648/j.engmath.20200401.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20200401.11},
  abstract = {The exact distribution of the test statistics in multivariate case is quite complicated in many situations, even when the underlying distribution is multivariate normal. This is due to the complex nature of the expression and therefore, there is a need to derive the asymptotic expression for the distribution. In this study, the asymptotic distribution of errors of misclassification for Edgeworth Series is derived by using Taylor’s expansion. The error of misclassification for the conditional probability of misclassification was expanded around the means emanating from populations one and two using approximated mean and variance of the errors of misclassification. The distribution of error of misclassification of the conditional probability of misclassification for ESD is approximately normal with mean zero and variance one.},
 year = {2020}
}

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TY  - JOUR
T1  - Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD)
AU  - Awogbemi Clement Adeyeye
Y1  - 2020/05/28
PY  - 2020
N1  - https://doi.org/10.11648/j.engmath.20200401.11
DO  - 10.11648/j.engmath.20200401.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 1
EP  - 9
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20200401.11
AB  - The exact distribution of the test statistics in multivariate case is quite complicated in many situations, even when the underlying distribution is multivariate normal. This is due to the complex nature of the expression and therefore, there is a need to derive the asymptotic expression for the distribution. In this study, the asymptotic distribution of errors of misclassification for Edgeworth Series is derived by using Taylor’s expansion. The error of misclassification for the conditional probability of misclassification was expanded around the means emanating from populations one and two using approximated mean and variance of the errors of misclassification. The distribution of error of misclassification of the conditional probability of misclassification for ESD is approximately normal with mean zero and variance one.
VL  - 4
IS  - 1
ER  -

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Author Information

Awogbemi Clement Adeyeye

Statistics Department, National Mathematical Centre, Abuja, Nigeria

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APA Style

Awogbemi Clement Adeyeye. (2020). Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD). Engineering Mathematics, 4(1), 1-9. https://doi.org/10.11648/j.engmath.20200401.11

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ACS Style

Awogbemi Clement Adeyeye. Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD). Eng. Math. 2020, 4(1), 1-9. doi: 10.11648/j.engmath.20200401.11

Copy | Download

AMA Style

Awogbemi Clement Adeyeye. Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD). Eng Math. 2020;4(1):1-9. doi: 10.11648/j.engmath.20200401.11

Copy | Download

@article{10.11648/j.engmath.20200401.11,
  author = {Awogbemi Clement Adeyeye},
  title = {Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD)},
  journal = {Engineering Mathematics},
  volume = {4},
  number = {1},
  pages = {1-9},
  doi = {10.11648/j.engmath.20200401.11},
  url = {https://doi.org/10.11648/j.engmath.20200401.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20200401.11},
  abstract = {The exact distribution of the test statistics in multivariate case is quite complicated in many situations, even when the underlying distribution is multivariate normal. This is due to the complex nature of the expression and therefore, there is a need to derive the asymptotic expression for the distribution. In this study, the asymptotic distribution of errors of misclassification for Edgeworth Series is derived by using Taylor’s expansion. The error of misclassification for the conditional probability of misclassification was expanded around the means emanating from populations one and two using approximated mean and variance of the errors of misclassification. The distribution of error of misclassification of the conditional probability of misclassification for ESD is approximately normal with mean zero and variance one.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD)
AU  - Awogbemi Clement Adeyeye
Y1  - 2020/05/28
PY  - 2020
N1  - https://doi.org/10.11648/j.engmath.20200401.11
DO  - 10.11648/j.engmath.20200401.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 1
EP  - 9
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20200401.11
AB  - The exact distribution of the test statistics in multivariate case is quite complicated in many situations, even when the underlying distribution is multivariate normal. This is due to the complex nature of the expression and therefore, there is a need to derive the asymptotic expression for the distribution. In this study, the asymptotic distribution of errors of misclassification for Edgeworth Series is derived by using Taylor’s expansion. The error of misclassification for the conditional probability of misclassification was expanded around the means emanating from populations one and two using approximated mean and variance of the errors of misclassification. The distribution of error of misclassification of the conditional probability of misclassification for ESD is approximately normal with mean zero and variance one.
VL  - 4
IS  - 1
ER  -

Copy | Download