Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 28J / b / i / Solution

The strong law of large numbers givesA continuous one-to-one real function is strictly monotone and has a continuous inverse on its image. Hencewhenever ${\hat{\mu }}_n$ lies in that image; an arbitrary extension outside the image handles the remaining finite-sample values. The continuous mapping theorem gives ${\hat{\theta }}_n\to {\theta }_0$ almost surely, so this is a consistent estimator.