British Journal of Mathematics & Computer Science, ISSN: 2231-0851,Vol.: 4, Issue.: 2 (16-31 January)

## Original-research-article

Power Transformations and Unit Mean and Constant Variance Assumptions of the Multiplicative Error Model: The Generalized Gamma Distribution

J. Ohakwe1* and D. C. Chikezie2

1Department of Mathematics/Computer Science/Physics, Faculty of Science, Federal University, Otuoke, P.M.B. 126, Yenagoa, Bayelsa State, Nigeria.
2Department of Statistics, Faculty of Biological and Physical Sciences, Abia State University, P.M.B. 2000, Uturu, Abia State, Nigeria.

Article Information

Editor(s):

(1) Carlo Bianca, Professor, Laboratoire de Physique Théorique de la Matière Condensée, Sorbonne Universités, France.

Reviewers:

(1) Mehmet Hakan Satman, Istanbul University, Turkey.

Complete Peer review History:http://www.sciencedomain.org/review-history/2358

Abstracts

Aims: To study the implications of power transformations namely; inverse-square-root, inverse, inverse-square and square transformations on the error component of the multiplicative error and determine whether the unit-mean and constant variance assumptions of the model are either retained or violated after the transformation.
Methodology: We studied the distributions of the error component under the various distributional forms of the generalized gamma distribution namely; Gamma (a, b, 1), Chi-square, Exponential, Weibull, Rayleigh and Maxwell distributions. We first established the functions describing the distributional characteristics of interest for the generalized power transformed error component and secondly applied the unit-mean conditions of the untransformed distributions to the established functions.
Results: We established the following important results in modeling using a multiplicative error model, where data transformation is absolutely necessary;(i) For the inverse-square-root transformation, the unit-mean and constant variance assumptions are approximately maintained for all the distributions under study except the Chi-square distribution where it was violated. (ii) For the inverse transformation, the unit-mean assumptions are violated after the transformation except for the Rayleigh and Maxwell distributions. (iii) For the inverse-square transformation, the unit-mean assumption is violated for all the distributions under study. (iv) For the square transformation, it is only the Maxwell distribution that maintained the unit-mean assumption. (v) For all the studied transformations the variances of the transformed distributions were found to be constant but greater than those of the untransformed distribution.
Conclusion: The results of this study though restricted to the distributional forms of the generalized gamma distribution, however they provide a useful framework in modeling for determining where a particular power transformation is successful for a model whose error component has a particular distribution.

Keywords :

Error component; mean; multiplicative error model; power transformation; variance.

## Our Contacts

Guest House Road, Street no - 1/6,
Hooghly, West Bengal,
India

+91 8617752708

Third Floor, 207 Regent Street
London, W1B 3HH,
UK

+44 20-3031-1429