Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 13J / v

The fitted Poisson regression hasso that ${\mu }_i=e^{{\beta }_0}{x}_i^{{\beta }_1}$. Hypothesis $H_2$ requires both restrictionsas well as the adequacy of the assumed Poisson distribution. The displayed interval addresses only the single slope restriction ${\beta }_1=1$. Even if the separate interval for the intercept also contains zero, separate one-parameter intervals do not implement the relevant joint hypothesis test, because the two coefficient estimators can be correlated. One should test $({\beta }_0,{\beta }_1)=(0,1)$ jointly, for example by a likelihood-ratio test, and separately assess the Poisson goodness of fit. Thus the slope interval alone cannot decide whether $H_2$ is rejected at the $5\%$ level.