Hastie, Tibshirani and Friedman (2009), The Elements of Statistical
Learning, 2nd ed., Springer, Section 2.6.3, book p. 31 (PDF p. 50):
L(theta) = sum_i log Pr_theta(y_i) (2.33) with
Pr(Y|X, theta) = N(f_theta(X), sigma^2) (2.34), giving
L = -(N/2) log(2 pi) - N log sigma - (1/(2 sigma^2)) sum (y_i - f_i)^2 (2.35).
Usage
Gausslik(y, mu, sigma = NULL)
Arguments
- y
N-vector of observations.
- mu
fitted values f_theta(x_i); a scalar is recycled.
- sigma
optional standard deviation.
Value
list: estimate, loglik, rss, sigma, n, aic, method.
Details
When sigma is not supplied it is profiled at its maximum-likelihood value
sqrt(RSS/N), which is what makes least squares and maximum likelihood
coincide here.
Examples
Gausslik(c(1, 2, 3), c(1.1, 2.1, 2.9), sigma = 1)$loglik
#> [1] -2.771816
