Indeed most of the results on hypothesis testing rely on the assumption of

Indeed most of the results on hypothesis testing rely

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tied to the normal distribution. Indeed, most of the results on hypothesis testing rely on the assumption of normality. The assumption that y 1 , . . . , y n are normal may not always be appropriate. Examples arise when y 1 , . . . , y n are binary or when they represent counts. It might therefore be nice to generalize the theory of linear models to include these other distributional assumptions for the response values. Generalized Linear Models (GLM) generalize linear models by including both of the above features. They allow more general distributional assumptions for y 1 , . . . , y n and they also allow (1). 2 Distributional Assumptions in GLMs In GLMs, the response variables y 1 , . . . , y n can be either discrete (have pmfs) or continuous (have pdfs). It is assumed that y 1 , . . . , y n are independent. We also assume that the pmf or pdf of y i can be modelled by two parameters θ i and φ i and can be written as f ( x ; θ i , φ i ) := h ( x, φ i ) exp i - b ( θ i ) a ( φ i ) . (2) 1
θ i is the main parameter (also called the canonical parameter). φ i is called the dispersion parameter and one often assumes that φ i is the same for all i . The function b ( θ i ) is called the cumulant function. This distributional form includes the normal density assumption used in the classical linear models. In classical linear models, we assume that y i N ( μ i , σ 2 ). The density of y i can then be written as f ( x ) := 1 2 πσ exp - ( x - μ i ) 2 2 σ 2 and this can be rewritten as f ( x ) := exp( - y 2 / (2 σ 2 )) 2 πσ exp - μ 2 / 2 σ 2 .

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