9 Day 9 (June 13)

9.1 Announcements

• Comments and plan for assignment 2

  • Guide is available
  • Learning opportunities
  • I should have grades done in the next day or two. Please submit a revised assignment 2 by Tuesday June 18

• Questions about assignment 3

• Plan for class tomorrow

9.2 Loss function approach

• Live example
  • Research publication
  • Data
  • R Script

9.3 Maximum Likelihood Estimation

• Maximum likelihood estimation for the linear model
  • Assume that \(\mathbf{y}\sim \text{N}(\mathbf{X}\boldsymbol{\beta},\sigma^2\mathbf{I})\)
  • Write out the likelihood function \[\mathcal{L}(\boldsymbol{\beta},\sigma^2)=\prod_{i=1}^n\frac{1}{\sqrt{2\pi\sigma^2}}\textit{e}^{-\frac{1}{2\sigma^2}(y_{i} - \mathbf{x}_{i}^{\prime}\boldsymbol{\beta})^2}.\]
  • Maximize the likelihood function
    1. First take the natural log of \(\mathcal{L}(\boldsymbol{\beta},\sigma^2)\), which gives \[\mathcal{l}(\boldsymbol{\beta},\sigma^2)=-\frac{n}{2}\text{ log}(2\pi\sigma^2) - \frac{1}{2\sigma^2}\sum_{i=1}^{n}(y_{i} - \mathbf{x}_{i}^{\prime}\boldsymbol{\beta})^2.\]
    2. Recall that \(\sum_{i=1}^{n}(y_{i} - \mathbf{x}_{i}^{\prime}\boldsymbol{\beta})^2 = (\mathbf{y} - \mathbf{X}\boldsymbol{\beta})^{\prime}(\mathbf{y} - \mathbf{X}\boldsymbol{\beta})\)
    3. Maximize the log-likelihood function \[\underset{\boldsymbol{\beta}, \sigma^2}{\operatorname{argmax}} -\frac{n}{2}\text{ log}(2\pi\sigma^2) - \frac{1}{2\sigma^2}(\mathbf{y} - \mathbf{X}\boldsymbol{\beta})^{\prime}(\mathbf{y} - \mathbf{X}\boldsymbol{\beta})\]
  • Visual for data \(\mathbf{y}=(0.16,2.82,2.24)'\) and \(\mathbf{x}=(1,2,3)'\)
• Three ways to do it in program R
  • Using matrix calculus and algebra

  # Data 
  y <- c(0.16,2.82,2.24)
  X <- matrix(c(1,1,1,1,2,3),nrow=3,ncol=2,byrow=FALSE)
  n <- length(y)
  
  # Maximum likelihood estimation using closed from estimators
  beta <- solve(t(X)%*%X)%*%t(X)%*%y 
  sigma2 <- (1/n)*t(y-X%*%beta)%*%(y-X%*%beta)
  
  beta

  ##       [,1]
  ## [1,] -0.34
  ## [2,]  1.04

  sigma2

  ##        [,1]
  ## [1,] 0.5832

  • Using modern (circa 1970’s) optimization techniques

  # Maximum likelihood estimation estimation using the Nelder-Mead algorithm
  y <- c(0.16,2.82,2.24)
  X <- matrix(c(1,1,1,1,2,3),nrow=3,ncol=2,byrow=FALSE)
  
  negloglik <- function(par){
  beta <- par[1:2]
  sigma2 <- par[3]
  -sum(dnorm(y,X%*%beta,sqrt(sigma2),log=TRUE))
  }
  optim(par=c(0,0,1),fn=negloglik,method = "Nelder-Mead")

  ## $par
  ## [1] -0.3397162  1.0398552  0.5831210
  ## 
  ## $value
  ## [1] 3.447978
  ## 
  ## $counts
  ## function gradient 
  ##      150       NA 
  ## 
  ## $convergence
  ## [1] 0
  ## 
  ## $message
  ## NULL

  • Using modern and user friendly statistical computing software

  # Maximum likelihood estimation using lm function
  # Note: estimate of sigma2 is not the MLE! 
  df <- data.frame(y = c(0.16,2.82,2.24),x = c(1,2,3))
  m1 <- lm(y~x,data=df)
  
  coef(m1)

  ## (Intercept)           x 
  ##       -0.34        1.04

  summary(m1)$sigma^2

  ## [1] 1.7496

  summary(m1)

  ## 
  ## Call:
  ## lm(formula = y ~ x, data = df)
  ## 
  ## Residuals:
  ##     1     2     3 
  ## -0.54  1.08 -0.54 
  ## 
  ## Coefficients:
  ##             Estimate Std. Error t value Pr(>|t|)
  ## (Intercept)  -0.3400     2.0205  -0.168    0.894
  ## x             1.0400     0.9353   1.112    0.466
  ## 
  ## Residual standard error: 1.323 on 1 degrees of freedom
  ## Multiple R-squared:  0.5529, Adjusted R-squared:  0.1057 
  ## F-statistic: 1.236 on 1 and 1 DF,  p-value: 0.4663