Traditional linear model, Advanced Statistics

What is a Generalized Linear Model? A traditional linear model is of the form

691_Traditional linear model.png

where Yi is the response variable for the ith observation, xi is a column vector of explanatory variables for the ith response. Note that the p-dimensional vector xi is usually considered to be ?xed, or nonrandom. The p-dimensional vector β of unknown coe?cients is to be estimated on the basis of n observations. The errors i are assumed to be independent normally distributed zero-mean random variables with a constant variance. Hence, 1925_Traditional linear model1.png holds, that is, the expected value of the output random variable is a linear transformation of the input. All these assumptions are limitations and may not hold in some cases. In particular:

  • We may have non-normally distributed errors;
  • The mean µi may be restricted by the nature of the problem (for example, if it has a meaning of probability, it would be in (0,1)) but the linear predictor x0iβ of this mean is not restricted to this range;
  • It may be unrealistic to assume that the variance of the data is constant for all observations. (For example, the variance of Poisson data increases with the mean.)

These limitations are dealt with in the setting of the generalized linear model (GLM) or their generalization, the generalized additive model (GAM). GLM have been introduced in Statistics by Nelder and Wedderburn (1972).

Posted Date: 2/27/2013 12:33:56 AM | Location : United States

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