6. (15 pts) In logistic regression, the response variable is sometimes labeled y (-1,+1) rather than...
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6. (15 pts) In logistic regression, the response variable is sometimes labeled y (-1,+1) rather than y {0, 1). Using a maximum likelihood approach, show that when y (-1, +1}, the logistic regression problem is to m min log (1+ exp(-yax)) (3) where x is the vector of parameters to be estimated (i.e. the regression coefficients) and a, is the vector of predictor variables for observation i (i=1,...,m). Here are the steps. (a) Show that 1 1+exp(-yaa) P(Y=y)= is in fact a valid probability distribution when y {-1, +1}. Here, the subscript i is dropped. (b) Write the log-likelihood function. (e) Show that minimizing the negative of the log-likelihood function leads to the problem (4) shown above. 6. (15 pts) In logistic regression, the response variable is sometimes labeled y (-1,+1) rather than y {0, 1). Using a maximum likelihood approach, show that when y (-1, +1}, the logistic regression problem is to m min log (1+ exp(-yax)) (3) where x is the vector of parameters to be estimated (i.e. the regression coefficients) and a, is the vector of predictor variables for observation i (i=1,...,m). Here are the steps. (a) Show that 1 1+exp(-yaa) P(Y=y)= is in fact a valid probability distribution when y {-1, +1}. Here, the subscript i is dropped. (b) Write the log-likelihood function. (e) Show that minimizing the negative of the log-likelihood function leads to the problem (4) shown above.
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Answer rating: 100% (QA)
a To show that PY 1 1 expyat is a valid probability distribution when y 1 1 we need to verify two properties nonnegativity and a sum of probabilities ... View the full answer
Related Book For
Applied Regression Analysis and Other Multivariable Methods
ISBN: 978-1285051086
5th edition
Authors: David G. Kleinbaum, Lawrence L. Kupper, Azhar Nizam, Eli S. Rosenberg
Posted Date:
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