Nigerian Statistical Association Logo - A Galaxy of Professional Statisticians
Close
The third edition of the Nigerian Statistical Association Competition for Undergraduate students has been postponed indefinitely due to the current pandemic outbreak (COVID-19) facing the world and Nigeria. A new date shall be communicated to you as soon as things get back to normal.We sincerely regret any inconveniences this would cost you.Thank you while staying safe.

Effect of power transformations on a Weibull-distributed error component of a multiplicative error model

AUTHOR(S):

Chris Uchechi Onyemachi ;Johnson Ohakwe ; Ifeanyi Sidney Onyeagu

JOURNAL: Journal of Nigerian Statistical Association Vol.33, 2021
YEAR: 2021

ABSTRACT

The error component of a Multiplicative Error Model (MEM) can possibly be a Weibull distribution ( ;   and   are shape and scale parameters respectively). Data transformation is a popular remedial measure to stabilize the variance of a data set prior to statistical modeling. Therefore, in this paper the effects of power transformations on the mean and variance of a Weibull distributed error component of a MEM are investigated. The popular transformations - inverse, square-root, inverse-square-root, square, inverse-square, cube root, inverse cube root, cube and inverse cube transformations were studied. The probability density function (pdf) and the kth raw moment of the p-th power–transformed Weibull random variable are obtained. The mean and variance of     and those of the power – transformed distributions are calculated for  = 6, 7, . ., 99, 100 with the corresponding values of n for which the mean of the untransformed distribution is equal to one. The relative changes in mean and variance are used for the investigations. For all the transformations, the means of the power transformed distributions are not different from 1. For variances, it was found that there are relative increases for the inverse, square, inverse square, cube and inverse cube transformations. However, the square-root, inverse square root, cube root and inverse-cube-root transformations decreased the variance relative to the variance of the untransformed distribution. This paper concludes that the square-root, inverse square root, cube root and inverse-cube-root transformations would yield better results as they reveal constancy in variance when using MEM with a Weibull distributed error component and where data transformation is deemed necessary to stabilize the variance of the data set.




BIBLIOGRAPHY

Adejumo, A.O. and Ahmadu, A.O. (2016). A Study of the Slope of Cox Proportional Hazard and Weibull Models: Simulated and Real Life Data Approach, Science World Journal, 11 (3), 2016.

Akpanta, A.C. and Iwueze, I.S. (2009). On Applying the Bartlett Transformation Methods to Time Series Data, Journal of Mathematical Sciences, 20, (3), 171 – 199.

Andersen, T.G., Bollerslev, T., Diebold, F.X. and Ebens, H. (2001). The distribution of realized stock return volatility, Journal of Financial Economics, 61, 43-76.

Bandi, F.M. and Russell, J.R. (2006). Separating microstructure noise from volatility, Journal of Financial Economics, 79, 655 – 692.

Bartlett, M.S. (1947).The Use of Transformations, Biometrics, 3, 39 – 52.

Box, G.E.P. and Cox, D.R. (1964). An Analysis of Transformation, Journal of Royal Statistical Society – Series B, 26, 211 – 252.

Brownlees, C.T. Cipollini, F. and Gallo G.M. (2011). Multiplicative Error Models, Working paper, http:ssrn.com/abstract=1852285.

Chartfield, C. (2004). The Analysis of Time Series: An Introduction, 6th ed., Chapman and Hall, CRC Press, Boca Raton.

Dike, A.O., Otuonye, E.L. and Chikezie, D.C. (2016). The nth Power Transformation of the Error Component of the Multiplicative Time Series Model, British Journal of Mathematics and Computer Science, 18(1), 1 – 15.

Engle, R. and Russell, J.R. (1998). Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data, Econometrica, 66 (5), 1127 – 1162.

Engle, R. (2002). Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models, Journal of Business & Economic Statistics, 20, 339 – 350. DOI: https://doi.org/10.1198/073500102288618487.

Healy, M.J.R. and Taylor, L.R. (1962). Tables for Power-Law Transformations, Biometrika, 49, 557 – 559.

Ibe, G.C. and Nwosu, C.R. (2013).Study on the Error Component of Multiplicative Time Series Model Under Inverse Transformation, American Journal of Mathematics and Statistics, 3(6), 362 – 374.

Iwueze, I.S. (2007). Some Implications of Truncating the N(1, σ2) Distribution to the Left at Zero, Journal of Applied Sciences, 7(2), 189 - 195.

Manganelli, S. (2005). Duration, Volume and Volatility Impact of Trades, Journal of Financial Markets, 8, 377-399.


Moore, P.G. (1957). Transformations to Normality Using Fractional Powers of the Variables. Journal the American Statistical Association, 52, 237-246.

Nwosu, C.R., Iwueze, I.S. and Ohakwe, J. (2013). Condition for Successful Inverse Transformation of the Error Component of the Multiplicative Time Series Model, Asian Journal of Applied Sciences, 6(1), 1–15. DOI: 10.3923/ajaps.2013.1-15.

Ohakwe, J. (2013). The Effect of Inverse Transformation on the Unit Mean and Constant Variance Assumptions of a Multiplicative Error Model Whose Error Component has a Gamma Distribution, Mathematical Theory and Modeling, 3(3), 44 – 52.

Ohakwe, J. and Ajibade, F.B. (2019). The Impact of Power Transformations on the Parameters of the Gamma Distributed Error Component of a Multiplicative Error Model. To appear in Benin Journal of Statistics, 2019.

Ohakwe, J., Dike, O.A. and Akpanta, A.C. (2012). The Implication of Square Root Transformation on a Gamma Distributed Error Component of a Multiplicative Time Series Model, Proceedings of African Regional Conference on Sustainable Development, University of Calabar, Nigeria, 2012, 6 (4), 65 – 78.

Ohakwe, J, Iwueze, I.S. and Otuonye, E.L. (2013). The Exact Region for the Successful Application of Square Root Transformation in Time Series Decomposition using the Multiplicative Model, International Journal of Basic Science and Technology, 4(1), 1 – 11.

Ohakwe, J., Akpanta, A.C. and Dike, O.A. (2012). Unit Mean and Constant Variance of the Generalized Gamma Distribution after Square Root Transformation in Statistical Modeling, Mathematical Theory and Modeling, 2 (12), 22 - 34.

Osborne, J. (2002). Notes on the use of Data Transformations, Journal of Practice Assessment, Research and Evaluation, 8(6), 1 – 10.

Otuonye, E.L., Iwueze, I.S. and Ohakwe, J. (2011). The Effect of Square Root Transformation on the Error Component of the Multiplicative Time Series Model, International Journal of Statistics and Systems, 6(4), 461 – 476.

Ozdemir, O. (2017). Power Transformations for Families of Statistical Distributions to Satisfy Normality, International Journal of Economics and Statistics, 5, 1 – 4.

Patton, A. (2011). Volatility Forecast Comparison Using Imperfect Volatility Proxies, Journal of Econometrics, 160(1), 246 – 256.

Ramachandran, K.M. and Tsokos, C.P. (2009). Mathematical Statistics with Applications, Academic Press, London. 

Russell, J.R. and Engle, R.F. (2010). Analysis of high-frequency data, In: Ait-Sahalia, Y. and L.P. Hansen, (eds), Handbook of Financial Econometrics, Vol. 1, Tools and Techniques, 383 – 427, North Holland Publishing, Amsterdam.

Stanley, C.R. (2006). Numerical Transformation of Geochemical Data, Geochemistry: Exploration, Environment, Analysis, 6, 69-78.

Tukey, J.W. (1957). On the Comparative Anatomy of Transformations, Annals of Mathematical Statistics, 28, 602-632.


READING / DOWNLOAD

Our journal is now available to everyone
Download Journal

Journal of Nigerian Statistical Association Vol.33, 2021
2021

SHARE WITH OTHERS

Privacy Policy Terms of Service © 2021 Nigerian Statistical Association - All Rights Reserved Developed by Masterweb Solutions