Alloy Inference: Tests of a Single Null

Abstract: This paper presents a new joint testing framework that fuses multiple test statistics into a single, more powerful inference tool. Using the probability integral transform to confine the support to the unit-hypercube, I use simulated null cases and Archimedean copulas to approximate the underlying joint null distribution of two or more statistics. Analogous to an alloy in metallurgy, where the final product has [typically] stronger properties than its constituent parts, I show how two or more tests can be combined to outperform a single test statistic in finite samples. To illustrate the performance of this approach, I provide a stylized example using the game of craps such that trade-offs can be be assessed in economic terms. Under potential uncertainty in the fairness of game dice, the proposed method–a combination of the Student-t and $\chi^2$ statistic–provides increased power, producing a revenue distribution which second-order stochastically dominates its constituent parts.