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A statistical hypothesis test is a decision rule calibrated to distinguish a null hypothesis $\theta \in {\Theta }_0$ from a specified alternative hypothesis. A randomized statistical test gives a rejection probability $\varphi (x)\in [0,1]$ for possible data $x$. Its power function of a statistical test is ${\mathbb{E}}_{\theta }\varphi (X)$, and a level-$\alpha$ test satisfiesThe supremum is the size of a statistical test. Controlling it limits type I error; a useful test also has high statistical power against scientifically relevant alternatives.

For ten independent Bernoulli trials with success probability $p$, test $H_0:p\le 1/2$ against $H_1:p>1/2$ by rejecting for at least nine successes. The upper-tail probability increases with $p$, so the largest null rejection probability occurs at $p=1/2$ and equalsIt is therefore a conservative level-0.05 test. Rejecting at eight successes would have size $56/1024>0.05$. This illustrates why discrete calibration need not attain the advertised level exactly. A p-value is a null-tail probability for the observation, not a probability that the null hypothesis is true. The decision also does not measure the size or practical importance of an effect.

The statistical power of a test at an alternative parameter value is the probability that the test rejects its Null hypothesis at that value.

A statistical test is a measurable rule for deciding whether to reject a null hypothesis. Allowing randomisation, it is represented by a function $\phi (x)\in [0,1]$ giving the conditional rejection probability after observing $x$. A deterministic test takes only values zero and one. Under a simple null, $E_0\phi$ is its size of a statistical test; under an alternative, $E_1\phi$ is its statistical power. Randomisation on a likelihood ratio equality set allows a Neyman-Pearson lemma threshold test to attain a prescribed size even for discrete data.

The first Wasserstein distance measures the least expected separation achievable when two random variables are coupled with specified marginal probability measures. For a metric space $(E,\rho )$ and measures $\mu ,\nu$ with finite first moments,A transport plan $\pi$ says how much probability mass moves from each starting location to each destination; the distance pays the corresponding metric cost. For point masses on the real line, $W_1({\delta }_a,{\delta }_b)=|a-b|$, because there is only one coupling. In contrast, their Total variation distance is one whenever $a\ne b$. Wasserstein distance therefore records that nearby displacements are small even when the two laws give mass to disjoint sets.

Every coupling gives an upper bound by its expected separation. Conversely, the Kantorovich–Rubinstein theorem gives lower bounds from differences of expectations of Lipschitz functions. The p-Wasserstein distance generalizes the construction by using the $p$th power of the ground metric and then taking a $p$th root.