Question: Maximum Likelihood Estimator in Logistic Regression Suppose we have data in pairs (xi,yi)(xi,yi) for i=1,2,...,30i=1,2,...,30. Conditional on xixi, yiyi is Bernoulli with success probability pi=P[yi=1|xi]=exp(0+1xi)/(1+exp(0+1xi)).pi=P[yi=1|xi]=exp(0+1xi)/(1+exp(0+1xi)).
Maximum Likelihood Estimator in Logistic Regression Suppose we have data in pairs (xi,yi)(xi,yi) for i=1,2,...,30i=1,2,...,30. Conditional on xixi, yiyi is Bernoulli with success probability
pi=P[yi=1|xi]=exp(0+1xi)/(1+exp(0+1xi)).pi=P[yi=1|xi]=exp(0+1xi)/(1+exp(0+1xi)).
The aim is to compute the maximum likelihood estimate ^^ of the parameter vector =(0,1)T=(0,1)T.
The log-likelihood is
()=i=1n[yilog(pi)+(1yi)log(1pi)].()=i=1n[yilog(pi)+(1yi)log(1pi)].
The data are given below:
X values:
1.34 -1.38 -0.19 -0.44 1.90 -0.80 0.91 0.26 1.37 -1.62 -0.96 1.90 0.99 1.96 -1.57 1.51 -1.61 -1.02 -0.92 -1.87 1.73 -1.23 -1.24 0.22 1.42 1.40 1.23 -0.75 1.47 -0.93
Y values:
1 0 0 1 1 0 1 1 1 0 1 1 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0
(a) Use the function optim()to compute ^^ using initial value (.25,.75).
(b) Again, starting with (.25,.75) find the next value when using the Newton-Raphson algorithm.
(c) Assume that 0=00=0, and plot the likelihood function L(1)L(1) as a function of 11.
(d) Again, assume 0=00=0 and compute ^1^1 using uniroot(), a grid search, and by the Newton-Raphson algorithm. You can use the plot in part (c) to find a good initial value.
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