Least Squares Adjustment Concepts

This section analyzes the truthfulness of statements concerning the principles of least squares adjustment for observations.

Statement Analysis

• Statement 1: Observations have a Chi-square distribution
  This statement is FALSE. In least squares theory, the observations ($y_i$) are typically assumed to follow a normal distribution. The Chi-square ($\chi^2$) distribution typically arises when considering the distribution of the sum of squared residuals (or weighted sum of squared residuals), which is used in statistical testing and variance estimation, not the distribution of the observations themselves.
• Statement 2: Random errors in the observations are assumed to have a symmetrical distribution
  This statement is TRUE. A core assumption in the method of least squares is that the random errors ($\epsilon_i$) affecting the observations are symmetrically distributed around zero. This implies that the probability of a positive error is the same as the probability of a negative error of the same magnitude. Common distributions like the normal distribution satisfy this condition.
• Statement 3: The positive and negative random observation errors are equally likely
  This statement is TRUE. This is a direct consequence of assuming a symmetrical distribution for the random errors centered at zero. A symmetrical distribution means $P(\epsilon_i = +x) = P(\epsilon_i = -x)$ for any error magnitude $x$.
• Statement 4: The adjusted parameters are independent of a priori reference variance
  This statement is TRUE. The adjustment process minimizes the weighted sum of squares of residuals. The parameter estimates ($\hat{x}$) are derived using the normal equations, which depend on the weight matrix ($P$) representing the inverse covariance matrix ($\Sigma^{-1}$) of the observations. If the covariance matrix is expressed as $\Sigma = \sigma^2 Q$, where $Q$ contains the known structure of variances and covariances and $\sigma^2$ is an unknown variance factor, the parameter estimates $\hat{x} = (A^T Q^{-1} A)^{-1} (A^T Q^{-1} l)$ are independent of the specific value of $\sigma^2$. The variance factor $\sigma^2$ influences the computed variances and standard errors of the adjusted parameters, but not the parameter values themselves, assuming the relative precisions (structure $Q$) are known.

Summary of True Statements

The statements identified as TRUE for the least squares adjustment of observations are:

• Random errors in the observations are assumed to have a symmetrical distribution.
• The positive and negative random observation errors are equally likely.
• The adjusted parameters are independent of a priori reference variance.