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For a concrete isotonic regression model, observe $Y_i={\theta }_i+{\epsilon }_i$ with independent Gaussian distribution errors of expected value zero and fixed variance ${\sigma }^2>0$. Let ${\Theta }_V=\{0\le {\theta }_1\le \cdots \le {\theta }_n\le V\}$ for fixed $V>0$, and use normalized squared-error loss $n^{-1}{\sum }_i({\hat{\theta }}_i-{\theta }_i{)}^2$. The minimax risk over ${\Theta }_V$ has orderas $n\to \infty$ with $V,\sigma$ fixed. This states the model and loss behind the shorthand rate $n^{-2/3}$.

The isotonic least squares estimator satisfies an upper bound of order $({\sigma }^{2}V/n{)}^{2/3}+{\sigma }^2\log (en)/n$. Locally increasing sequences supply matching lower bounds for the first term. The primary analysis also shows why a worst-case rate does not describe every signal: when the true sequence has few constant blocks, the risk can be substantially smaller.^[ref]

A heuristic for the exponent is to approximate a changing monotone signal by $k$ blocks: squared approximation error scales as $(V/k{)}^2$, while estimating $k$ block levels costs approximately ${\sigma }^{2}k/n$. Balancing these gives $k\asymp (n{V}^2/{\sigma }^2{)}^{1/3}$ and risk $({\sigma }^{2}V/n{)}^{2/3}$. This balance motivates the rate; it does not replace the lower-bound argument. Uniformly constant signals or a variation bound shrinking with $n$ require their own regime analysis.