Abstract
We consider one-way analysis of variance (ANOVA) model when the error terms have skew- normal distribution. We obtain the estimators of the model parameters by using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies (see, Tiku 1967). In the ML method, iteratively reweighting algorithm (IRA) is used to solve the likelihood equations. The MML approach is a non-iterative method used to obtain the explicit estimators of model parameters. We also propose new test statistics based on these estimators for testing the equality of treatment effects. Simulation results show that the proposed estimators and the tests based on them are more efficient and robust than the corresponding normal theory solutions. Also, real data is analysed to show the performance of the proposed estimators and the tests.
Highlights
Consider the following one-way ANOVA model, yij = μ + αi + ij, i = 1, 2, . . . , a; j = 1, 2, . . . , n (1)where, yij are the responses corresponding to jth observation in the ith treatment, μ is the overall mean, αi is the effect of ith treatment and ij are the independent and identically distributed random error terms.In general, normality assumption is made for the random error terms and the well known least squares (LS) method is used for estimating model parameters
Normality assumption is made for the random error terms and the well known least squares (LS) method is used for estimating model parameters
It is known that LS estimators of the parameters and the test statistics based on them lose their efficiency when the normality assumption is not satisfied
Summary
Normality assumption is made for the random error terms and the well known least squares (LS) method is used for estimating model parameters. See Senoglu & Tiku (2001) and the references therein These conclusions are true for non normal distributions having skewness in different directions (Senoglu & Tiku 2002). We assume that the distribution of the error terms in one-way ANOVA model in (1) is Azzalini’s skew-normal (Azzalini 1985, 1986) and obtain the ML and the MML estimators of the model parameters. SN (λ) distribution is considered to be an extension of normal distribution This provides us flexibility for modeling the data with normal − like shape but with skewness and heavy tails.
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