2. Consider a continuous random variable Y whose probability density function is given by = = f(y; 7) = m(y)74 exp(-ry),

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2 Consider A Continuous Random Variable Y Whose Probability Density Function Is Given By F Y 7 M Y 74 Exp Ry 1 (148.89 KiB) Viewed 108 times

2 Consider A Continuous Random Variable Y Whose Probability Density Function Is Given By F Y 7 M Y 74 Exp Ry 2 (86.32 KiB) Viewed 108 times

2. Consider a continuous random variable Y whose probability density function is given by = = f(y; 7) = m(y)74 exp(-ry), y>0,7>0 (1) where m(y) = y3/6 and f(y; v) = 0) if y < 0. (a) (i) Demonstrate that f(y; 7) may be written in the form of a Generalised Linear Model (GLM), i.e. show that S 10; 9) = exp{ 19 = b[0) + (0, 0)) 9,0} f(y = ув — ( φ y where 0, 0, b(0) and c(y,0) should be determined explicitly. [4 Marks] (ii) Find the mean u = E[Y] in terms of 0. Also express j in terms of 7 [2 marks] (iii) What is the canonical link function for this GLM? Discuss a potential drawback with the canonical link function. Suggest another link function that would avoid this (potential) problem. [4 Marks] (iv) Find the variance function V (u) for the GLM associated with (1). [4 Marks) (v) Suppose now that we have response data y1, ... , Yn. After fitting a particular GLM with response distribution of the form (1), the fitted values corresponding to y1, ... , Yn, were found to be û1, ...,În, respctively. Find an expression for the deviance residual for observation i. (5 marks] (b) In Chapter 2 of the lecture notes it was pointed out that in general estimation of the parameter vector B in GLMs requires an iter- ative procedure ("Iteratively weighted least squares”); and that each step in the iteration can be formulated as a weighted least squares problem. Specifically, if 6m) denotes the value of ß after m iterations, then the weighted least squares problem to be solved is: find 6(m) to solve xTw(m-1) Xb(m) = x'W(m–1)z(m-1), (2)

where X is the n * p matrix of covariates, W(m-1) is a diagonal nxn weight matrix which depends on B = b(m-1) and z(m-1) is an nx1 vector which also depends on b(m-1). The diagonal elements of W(m-1) are given by 2 = (m-1) 1 dpi Var[y] dni where Ni is the linear predictor and all terms on the right-hand side are evaluated at B = b[m-1); and the components of the vector z(m-1) are given by 24m-1) = x;} blm=1) + (yi {m-1))g'(4&m=1)), = (m-1) where g'() is the first derivative of the link function g() and " is pli evaluated at B=bm-1). Answer the following questions, assuming that the relevant prob- ability density function is given by (1) and that the canonical link function is used. (i) Calculate w.n-1), expressing your answer in as simple a form (m-) as possible. [4 Marks] 9 (ii) Express the updating formula for b(m) in vector-matrix form, i.e. make b(m) the subject of (2). [2 Marks]