silveredition
Newbie level 6
This is probably way too specialized of a question to ask and expect an answer on, but I'm using "The Elements of Statistical Learning" by Hastie, Tibshirani, and Friedman, and have been working on one question for about 4 days now and still can't get it, so if anyone out there has the text and knows how to do Exercise 4.2b, I'd be eternally grateful if you'd impart some hints about it.
If anyone doesn't have the text but may be able to help, the question is:
"Suppose we have features x\in R^p, a two-class response, with class sizes N_1 and N_2, and the target coded -\frac{N}{N_1}, \frac{N}{N_2}. Consider minimization of the least squares criterion: \sum_{i=1}^N{(y_i-\beta_0-\beta^Tx_i)^2}. Show that the solution \hat{\beta} satisfies: [(N-2)\hat{\Sigma}+\frac{N_1N_2}{N}\hat{\Sigma}_B]\beta=N(\hat{\mu}_2-\hat{\mu}_1)
where \hat{\Sigma}_B=(\hat{\mu}_2-\hat{\mu}_1)(\hat{\mu}_2-\hat{\mu}_1)^T
If the LaTeX causes any problems, I can put it in plain text. Also, the chapter defines:
\hat{\mu}_2=\sum_{g_i=2}{\frac{x_i}{N_2}}
\hat{\mu}_2=\sum_{g_i=1}{\frac{x_i}{N_1}}
\hat{\Sigma}=\sum_{k=1}^2{\frac{(x_i-\hat{\mu}_k)(x_i-\hat{\mu}_k)^T}{N-2}}
I'm not really expecting anybody to answer this, but this is sort of a last resort.
If anyone doesn't have the text but may be able to help, the question is:
"Suppose we have features x\in R^p, a two-class response, with class sizes N_1 and N_2, and the target coded -\frac{N}{N_1}, \frac{N}{N_2}. Consider minimization of the least squares criterion: \sum_{i=1}^N{(y_i-\beta_0-\beta^Tx_i)^2}. Show that the solution \hat{\beta} satisfies: [(N-2)\hat{\Sigma}+\frac{N_1N_2}{N}\hat{\Sigma}_B]\beta=N(\hat{\mu}_2-\hat{\mu}_1)
where \hat{\Sigma}_B=(\hat{\mu}_2-\hat{\mu}_1)(\hat{\mu}_2-\hat{\mu}_1)^T
If the LaTeX causes any problems, I can put it in plain text. Also, the chapter defines:
\hat{\mu}_2=\sum_{g_i=2}{\frac{x_i}{N_2}}
\hat{\mu}_2=\sum_{g_i=1}{\frac{x_i}{N_1}}
\hat{\Sigma}=\sum_{k=1}^2{\frac{(x_i-\hat{\mu}_k)(x_i-\hat{\mu}_k)^T}{N-2}}
I'm not really expecting anybody to answer this, but this is sort of a last resort.
