 Solution

Assume $S_{xx}={\sum }_i{x}_i^2>0$, so the slope in this linear regression through the origin is identifiable. Expanding the residual sum of squares givesDifferentiation gives the unique least-squares estimator and its normal distribution:For $n\ge 2$, estimate the unknown variance byThe fitted component and orthogonal residual of the Gaussian vector are independent, and $(n-1)s^2/{\sigma }^2$ has the chi-squared distribution with $n-1$ degrees of freedom. Under the null hypothesis,The Student t test for regression through the origin rejects at significance level $\alpha$ when $|T|>t_{n-1,1-\alpha /2}$, where the subscript denotes the corresponding quantile of Student's t-distribution. If all $x_i=0$, the data contain no information about $\beta$; if $n=1$, no residual degree of freedom is available for this unknown-variance test.