Affiliations: [a] Department of Statistics, The University of Burdwan, Burdwan, West Bengal, India | [b] Department of Statistics and Information Science, Dongguk University, Gyeongju, Korea | [c] Department of Mathematics, NSHM Knowledge Campus, Durgapur 713 212, WB, India
Correspondence:
[*]
Corresponding author: Rabindra Nath Das, Department of Statistics, The University of Burdwan, Burdwan, West Bengal, India. E-mail:rabin.bwn@gmail.com
Abstract: The Gaussian (normal) distribution is most
often assumed to describe the random variation that occurs in the
data from many scientific disciplines. However, it is rarely
observed that a real data set exactly follows normal distribution.
Generally, positive observations with constant variance from a
continuous random variable are analyzed either by the log-normal
or the gamma models. In practice, the variance is not always
constant. For handling non-constant variance in the log-normally
distributed continuous response random variable, some concomitant
variables are included as explanatory variables in the regression
models. In the present article, the response distribution is
assumed to have the log-normal with compound symmetry errors. The
log-normal model with composite compound symmetry errors is
developed. The best linear unbiased estimators of all the
regression coefficients have been derived except the intercept
which is often unimportant in practice. Both the correlation
coefficients (within and between groups) have been estimated. A
robust (free of correlation coefficients) testing procedure for
any set of linear hypotheses regarding the unknown regression
coefficients has been introduced. Confidence intervals of an
estimable function and confidence ellipsoids of a set of estimable
functions of regression coefficients have been derived. An index
of fit for the fitted regression model has also been developed. An
example (with simulated data) illustrates the results derived in
this report. A real example with replicated observations has been
modelled based on the present developed theories. Some new data
analysis theories are developed in the present report to analyze
positive, skewed, correlated and also replicated data.
Keywords: Composite errors, compound symmetry structure, confidence ellipsoid, correlated errors, non-linear models, log-normal distribution, index of fit
