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=== E-value ===
In [https://en.wikipedia.org/wiki/Statistical_hypothesis_testing statistical hypothesis testing], '''e-values''' quantify the evidence in the data against a [https://en.wikipedia.org/wiki/Null_hypothesis null hypothesis] (e.g., "the coin is fair", or, in a medical context, "this new treatment has no effect"). They serve as a more robust alternative to [https://en.wikipedia.org/wiki/P-value p-values], addressing some shortcomings of the latter. In contrast to p-values, e-values can deal with optional continuation: e-values of subsequent experiments (e.g. clinical trials concerning the same treatment) may simply be multiplied to provide a new, "product" e-value that represents the evidence in the joint experiment. This works even if, as often happens in practice, the decision to perform later experiments may depend in vague, unknown ways on the data observed in earlier experiments, and it is not known beforehand how many trials will be conducted: the product e-value remains a meaningful quantity, leading to tests with [https://en.wikipedia.org/wiki/Type_I_and_type_II_errors Type-I error control]. For this reason, e-values and their sequential extension, the ''e-process'', are the fundamental building blocks for anytime-valid statistical methods (e.g. confidence sequences). Another advantage over p-values is that any weighted average of e-values remains an e-value, even if the individual e-values are arbitrarily dependent. This is one of the reasons why e-values have also turned out to be useful tools in [https://en.wikipedia.org/wiki/Multiple_comparisons_problem multiple testing].<sup id="cite_ref-1">[https://en.wikipedia.org/wiki/E-values#cite_note-1 [1]]</sup> E-values can be interpreted in a number of different ways: first, the reciprocal of any e-value is itself a p-value, but a special, conservative one, quite different from p-values used in practice. Second, they are broad generalizations of [https://en.wikipedia.org/wiki/Likelihood_function likelihood ratios] and are also related to, yet distinct from, [https://en.wikipedia.org/wiki/Bayes_factors Bayes factors]. Third, they have an interpretation as bets. Finally, in a sequential context, they can also be interpreted as increments of nonnegative [https://en.wikipedia.org/wiki/Martingale_(probability_theory) supermartingales]. Interest in e-values has exploded since 2019, when the term 'e-value' was coined and a number of breakthrough results were achieved by several research groups. The first overview article appeared in 2023.<sup id="cite_ref-:2_2-0">[https://en.wikipedia.org/wiki/E-values#cite_note-:2-2 [2]]</sup><br/> <br/> Let the [https://en.wikipedia.org/wiki/Statistical_hypothesis_testing null hypothesis] π»0<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/43910602a221b7a4c373791f94793e3008622070> be given as a set of distributions for data π<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f>. Usually π=(π1,β¦,ππ)<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/232436e83a6551876d9ea98a759d4e9681e80975> with each ππ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d> a single outcome and π<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c> a fixed sample size or some stopping time. We shall refer to such π<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f>, which represent the full sequence of outcomes of a statistical experiment, as a ''sample'' or ''batch of outcomes.'' But in some cases π<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f> may also be an unordered bag of outcomes or a single outcome. An '''e-variable''' or '''e-statistic''' is a ''nonnegative'' random variable πΈ=πΈ(π)<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/2dff3e1e32d9281280efa425fc381f4348711c47> such that under all πβπ»0<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/975a55140fa8854be1a02c263d0dfdb74edff76a>, its expected value is bounded by 1: πΈπ[πΈ]β€1<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/1b5decf85768b772fff63e687c162ea50d8c3445>. The value taken by e-variable πΈ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b> is called the '''e-value'''''.'' In practice, the term ''e-value'' (a number) is often used when one is really referring to the underlying e-variable (a random variable, that is, a measurable function of the data).<br/> <br/> full text link : [https://en.wikipedia.org/wiki/E-values https://en.wikipedia.org/wiki/E-values]
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