adjusters

In game-theoretic statistics we often have an e-process $(E_t)$ that we’re using to collect evidence against the null. We might want to ensure that this process never decreases. But most e-processes are not nondecreasing. And while it’s tempting to just consider the running maximum of an e-process, $E_t^{\ast }={\max }_{s\le t}{E}_s$ the result is not an e-process. In fact, the running maximum can have expectation tending to infinity under the null; see Appendix D in this paper for an example.

Adjusters solve this problem. They take the running maximum $(E_t^{\ast })$ and shift it so that the result is an e-process. Formally, an adjuster $A$ is a function $A:[1,\infty ]\to [0,\infty ]$ such that

$${\int }_1^{\infty }\frac{A(y)}{y^2}\text{d}y\le 1.$$

They relate to e-processes as follows: For any test supermartingale $(X_t)$ (meaning a supermartingale with initial value 1) there exists a test supermartingale $(Y_t)$ such that $A(X_t^{\ast })\le Y_t$ almost surely, where $X_t^{\ast }={\max }_{0\le s\le t}{X}_s$. This together with the optional stopping theorem imply that $A(X_t^{\ast })$ is an e-process. Using results which upper bound e-processes in terms of supermartingales, the same conclusion holds if take $(X_t)$ to be an e-process.

Adjusters originated here, here, and here. Examples are $A(y)=\kappa y^{1-\kappa }$ for any $\kappa \in (0,1)$, $A(y)=\sqrt{y}-1$,

$$A(y)={\int }_0^1\kappa y^{1-\kappa }\text{d}y=\frac{y-1-\log (y)}{{\log }^2(y)},$$

with $A(y)=1/2$, or

$$A(y)=\frac{y^2\log 2}{(1+y){\log }^2(1+y)}.$$

See here for credit for all of these, and an overview.

One way to think about the definition of an adjuster is as follows. Suppose we define the function $h(w)=A(e^w)/e^w$. By a change of variables, write

$$1\ge {\int }_1^{\infty }\frac{A(y)}{y^2}\text{d}y={\int }_0^{\infty }h(t)\text{d}t.$$

Now, $A(y)=yh(\log y)$, so the adjuster can be seen as adjusting $y$ downwards by a multiplicative factor of $h(\log y)$. And this function $h$ must act like a density on the log scale of $y$. Different adjusters will spread out this mass differently. Moreover, admissible adjusters are precisely those such that ${\int }_1^{\infty }A(y)/y^2\text{d}y=1$, implying that $\int h(t)\text{d}t=1$. So as long as the adjuster spreads all its mass out, it is admissible.

Reading

• Test martingales, Bayes factors, and p-values
• Combining evidence across filtrations