Question

For the weighted least squares adjustment, which of the following statements is/are correct?

• A. Weighted sum of the squares of the residuals is minimized
• B. The expected value of the residuals is equal to zero
• C. Redundancy of observations is maximized
• D. Weights are taken inversely proportional to the variance of the observations

Hint:

In weighted least squares, always relate weight to confidence. More variance → less confidence → smaller weight. Also, residuals always balance out to zero in expectation.

Answer 1

The Correct Option is A, B, D

In Weighted Least Squares (WLS) adjustment:

Statement (A) is correct because the core idea of WLS is to minimize the weighted sum of squared residuals, unlike ordinary least squares which minimizes the unweighted sum.
Statement (B) is correct. The residuals in least squares estimation have an expected value of zero under the assumption of unbiased estimators.
Statement (C) is incorrect. Redundancy refers to the extra observations beyond the minimum required and is not directly maximized by WLS.
Statement (D) is correct. In WLS, the weights are typically taken as the inverse of the variance of the observations.

\[ w_i = \frac{1}{\sigma_i^2} \]

giving less weight to less reliable (more uncertain) observations.