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Errors of Misclassification Associated with Edgeworth Series Distribution (ESD)

Received: 2 March 2018     Accepted: 30 March 2018     Published: 8 November 2019
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Abstract

This study investigates the errors of misclassification associated with Edgeworth Series Distribution (ESD) with a view to assessing the effects of sampling from non-normality. The effects of applying a normal classificatory rule when it is actually a persistent non-normal distribution were examined. These were achieved by comparing the errors of misclassification for ESD with ND using small sample sizes at every level of skewness factor. The simulation procedure for the experiment of the study was implemented using numerical inverse interpolation method in R program to generate a uniformly distributed random variable N. A configuration size of 1000 was obtained for the two training samples drawn at every level of skewness factor (λ3), in the range (0.00625, 0.4). This was repeated for different small sample sizes by comparing errors of misclassification of ESD with ND. The simulation results showed that the optimum probabilities of misclassification by ESD: (E12E) decreases and (E12E) increases, as the skewness factor (λ3) increases. The optimum total probability of misclassification is stable as λ3 also increases. The probability of misclassification E12EE12N and E21EE21N at every level of λ3. Thus, the total probabilities of misclassification are not greatly affected by the skewness factor. This asserts that the normal classification procedure is robust against departure from normality.

Published in American Journal of Theoretical and Applied Statistics (Volume 8, Issue 6)
DOI 10.11648/j.ajtas.20190806.12
Page(s) 203-213
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), 2019. Published by Science Publishing Group

Keywords

Errors of Misclassification, Edgeworth Series Distribution, Skewness Factor, Classificatory Rule, Optimum Probability of Misclassification

References
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[2] Anderson, T. W. (2003). An Introduction to Multivariate Statistical Methods, New York: John Wiley, pp. 126-127.
[3] Barton, D. E. and Dennis, K. E. (1952). The Condition under which Gram-Charlier and Edgeworth Curves are Positive Definite and Unimodal, Biometrika, 39, pp. 425- 427.
[4] Bhuyan, C. K. (2010). Probability Theory and Statistical Inference, 1st ed., New Central Book Agency (Ltd), London, pp. 203.
[5] Broffit, J. D. and Williams, J. S. (1973). Minimum Variance Estimators for Misclassification Probabilities in Discriminant Analysis, Journal of Multivariate Analysis, (3), pp. 311-327.
[6] Chingada, E. F. and Kocherlakota, S. (1978). Robustness of the Linear Discriminant Function to nonnormality, Journal of Statistical planning and Inference, 3, pp. 74.
[7] Draper, N, and Tierny, D. (1972). Region of Positive and Unimodal Series Expansion of the Edgeworth and Gramcharlier Approximations. Biometrika, 59 (2), pp. 463-465.
[8] Fisher, F. A. (1936). The Use of Multiple Measurements in Taxonomic Problems, Annals of Eugenics, 7, pp. 179-188.
[9] Geisser, S. (1967). Estimation Associated with Linear Discriminants. Annals of Mathematical Statistics, (38) pp. 807-817.
[10] John, P. B (2010). Measurement Error, Model, Methods and Applications, CRC Press, Taylor and Francis, pp. 1-10.
[11] Kendall, M. G. and Stuart, A. (1958). The Advanced Theory of Statistics, Vol. 3, Hafner: New York. pp. 155.
[12] Kocherlakota, S., Kocherlakota, K. and Balakrishnan, N (1987). Expansions of Errors of Misclassification, Advances in Multivariate Statistical Analysis, pp. 191-211.
[13] Lachenbruch, P. A., Clarke, B. B. & Lin, L. (1977). The Effect of Non-normality on Quadratic Discriminant Function, MEDINFO 77 Shives/Wolf IFEP, North Holland Publishing Co. pp. 101-104.
[14] Lachenbruch, P. A., Sneeringer, C. &Revo, L. T. (1973). Robustness of the Linear and Quadratic Discriminant Function to Certain Types of Non-normality. Journal of Communication Statistics, 1, pp. 39–57.
[15] Mahmoud, M. A. and Moustafa, H. M. (1995). Errors of Misclassification Associated with Gamma Distribution, Journal of Mathematical Computing Modeling, Vol. 22, No. 3, pp. 105-119.
[16] Ogum, G. E. O. (2002). Introduction to Methods of Multivariate Analysis, Afri-Towers Ltd., Aba, Nigeria. pp. 107.
[17] Peter, H. (1997). The Bootstrap and Edgeworth Expansions, Springer Series in Statistics, pp. 145.
[18] Richard, A. J. and Dean, W. W. (1988). Applied Multivariate Statistical analysis, 3rd ed., Prentice Hall, Inc. New Jersey, pp. 497.
[19] Ruby, C. W. (2010). A Bayesian Edgeworth Expansion by Stein’s Identity, Bayesian Analysis, 5 (4), pp. 741-764.
[20] Sedransk, N. and Okamato, M. C. (1971). Estimation of the Probabilities of Misclassification for the Linear Discriminant Function in the Univariate Normal Case. Annals of Statistics, 23 (3), pp. 419-427.
[21] Toussaint, G. T. (1974). Bibliography on Estimation of Misclassification. IEEE Transactions on Information Theory. Vol. 2 (4), pp. 472–479.
[22] William, R. D. and Matthew, G. (1984). Multivariate Analysis, Methods and Applications. John Wiley & Sons, Inc. N. Y. pp. 360-361.
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  • APA Style

    Awogbemi Clement Adeyeye, Onyeagu Sidney Iheanyi. (2019). Errors of Misclassification Associated with Edgeworth Series Distribution (ESD). American Journal of Theoretical and Applied Statistics, 8(6), 203-213. https://doi.org/10.11648/j.ajtas.20190806.12

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    ACS Style

    Awogbemi Clement Adeyeye; Onyeagu Sidney Iheanyi. Errors of Misclassification Associated with Edgeworth Series Distribution (ESD). Am. J. Theor. Appl. Stat. 2019, 8(6), 203-213. doi: 10.11648/j.ajtas.20190806.12

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    AMA Style

    Awogbemi Clement Adeyeye, Onyeagu Sidney Iheanyi. Errors of Misclassification Associated with Edgeworth Series Distribution (ESD). Am J Theor Appl Stat. 2019;8(6):203-213. doi: 10.11648/j.ajtas.20190806.12

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  • @article{10.11648/j.ajtas.20190806.12,
      author = {Awogbemi Clement Adeyeye and Onyeagu Sidney Iheanyi},
      title = {Errors of Misclassification Associated with Edgeworth Series Distribution (ESD)},
      journal = {American Journal of Theoretical and Applied Statistics},
      volume = {8},
      number = {6},
      pages = {203-213},
      doi = {10.11648/j.ajtas.20190806.12},
      url = {https://doi.org/10.11648/j.ajtas.20190806.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20190806.12},
      abstract = {This study investigates the errors of misclassification associated with Edgeworth Series Distribution (ESD) with a view to assessing the effects of sampling from non-normality. The effects of applying a normal classificatory rule when it is actually a persistent non-normal distribution were examined. These were achieved by comparing the errors of misclassification for ESD with ND using small sample sizes at every level of skewness factor. The simulation procedure for the experiment of the study was implemented using numerical inverse interpolation method in R program to generate a uniformly distributed random variable N. A configuration size of 1000 was obtained for the two training samples drawn at every level of skewness factor (λ3), in the range (0.00625, 0.4). This was repeated for different small sample sizes by comparing errors of misclassification of ESD with ND. The simulation results showed that the optimum probabilities of misclassification by ESD: (E12E) decreases and (E12E) increases, as the skewness factor (λ3) increases. The optimum total probability of misclassification is stable as λ3 also increases. The probability of misclassification E12E ≥E12N and E21E ≥E21N at every level of λ3. Thus, the total probabilities of misclassification are not greatly affected by the skewness factor. This asserts that the normal classification procedure is robust against departure from normality.},
     year = {2019}
    }
    

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  • TY  - JOUR
    T1  - Errors of Misclassification Associated with Edgeworth Series Distribution (ESD)
    AU  - Awogbemi Clement Adeyeye
    AU  - Onyeagu Sidney Iheanyi
    Y1  - 2019/11/08
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    N1  - https://doi.org/10.11648/j.ajtas.20190806.12
    DO  - 10.11648/j.ajtas.20190806.12
    T2  - American Journal of Theoretical and Applied Statistics
    JF  - American Journal of Theoretical and Applied Statistics
    JO  - American Journal of Theoretical and Applied Statistics
    SP  - 203
    EP  - 213
    PB  - Science Publishing Group
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    UR  - https://doi.org/10.11648/j.ajtas.20190806.12
    AB  - This study investigates the errors of misclassification associated with Edgeworth Series Distribution (ESD) with a view to assessing the effects of sampling from non-normality. The effects of applying a normal classificatory rule when it is actually a persistent non-normal distribution were examined. These were achieved by comparing the errors of misclassification for ESD with ND using small sample sizes at every level of skewness factor. The simulation procedure for the experiment of the study was implemented using numerical inverse interpolation method in R program to generate a uniformly distributed random variable N. A configuration size of 1000 was obtained for the two training samples drawn at every level of skewness factor (λ3), in the range (0.00625, 0.4). This was repeated for different small sample sizes by comparing errors of misclassification of ESD with ND. The simulation results showed that the optimum probabilities of misclassification by ESD: (E12E) decreases and (E12E) increases, as the skewness factor (λ3) increases. The optimum total probability of misclassification is stable as λ3 also increases. The probability of misclassification E12E ≥E12N and E21E ≥E21N at every level of λ3. Thus, the total probabilities of misclassification are not greatly affected by the skewness factor. This asserts that the normal classification procedure is robust against departure from normality.
    VL  - 8
    IS  - 6
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Author Information
  • Statistics Department, Nnamdi Azikiwe University, Awka, Nigeria

  • Statistics Department, Nnamdi Azikiwe University, Awka, Nigeria

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