20 / 25 Distributions

Weighted Least Squares II

For 1≤i≤n1 \leq i \leq n , let Yi∣β,σ2∼N(βXi,σ2/wi)Y_i \mid \beta, \sigma^2 \sim N(\beta X_i,\sigma^2/w_i) independently, where X1,…,Xn,w1,…,wnX_1,\dots, X_n, w_1,\dots, w_n are known constants. Note that β\beta and σ2\sigma^2 are not known. Let β^\hat{\beta} be the value of β\beta that maximizes the likelihood of observing the data Y1,…,YnY_1,\dots, Y_n . Which of the below is the correct expression for β^\hat{\beta} ?

A

β^=∑i=1nYi∑i=1nXi\hat{\beta}=\frac{\sum_{i=1}^{n} Y_i}{\sum_{i=1}^{n} X_i}

B

β^=∑i=1nwiXiYi∑i=1nwiXi2\hat{\beta}=\frac{\sum_{i=1}^{n} w_i X_i Y_i}{\sum_{i=1}^{n} w_i X_i^2}

C

β^=∑i=1nXiYi∑i=1nXi2\hat{\beta}=\frac{\sum_{i=1}^{n} X_i Y_i}{\sum_{i=1}^{n} X_i^2}

D

β^=∑i=1nwiYi∑i=1nwiXi\hat{\beta}=\frac{\sum_{i=1}^{n} w_i Y_i}{\sum_{i=1}^{n} w_i X_i}