Editing Multinomial distribution (section)

===Equivalence tests for multinomial distributions===
The goal of equivalence testing is to establish the agreement between a theoretical multinomial distribution and  observed counting frequencies. The theoretical distribution may be a fully specified multinomial distribution or a parametric family of multinomial distributions.

Let <math>q</math> denote a theoretical multinomial distribution and let <math>p</math> be a true underlying distribution. The distributions <math>p</math> and <math>q</math> are considered equivalent if <math>d(p,q)<\varepsilon</math>  for a distance <math>d</math> and a tolerance parameter <math>\varepsilon>0</math>. The equivalence test problem is <math>H_0=\{d(p,q)\geq\varepsilon\}</math> versus  <math>H_1=\{d(p,q)<\varepsilon\}</math>. The true underlying distribution <math>p</math> is unknown. Instead, the counting frequencies <math>p_n</math> are observed, where <math>n</math> is a sample size. An equivalence test uses <math>p_n</math> to reject <math>H_0</math>. If <math>H_0</math> can be rejected then the equivalence between <math>p</math> and <math>q</math> is shown at a given significance level. The equivalence test for Euclidean distance can be found in text book of Wellek (2010).<ref>{{Cite book|title=Testing statistical hypotheses of equivalence and noninferiority|last=Wellek|first=Stefan|publisher=Chapman and Hall/CRC|year=2010|isbn=978-1439808184}}</ref> The equivalence test for the total variation distance is developed in Ostrovski (2017).<ref>{{cite journal|last1=Ostrovski|first1=Vladimir|date=May 2017|title=Testing equivalence of multinomial distributions|journal=Statistics & Probability Letters|volume=124|pages=77–82|doi=10.1016/j.spl.2017.01.004|s2cid=126293429}}[http://dx.doi.org/10.1016/j.spl.2017.01.004 Official web link (subscription required)]. [https://www.researchgate.net/publication/312481284_Testing_equivalence_of_multinomial_distributions Alternate, free web link].</ref> The exact equivalence test for the specific cumulative distance is proposed in Frey (2009).<ref>{{cite journal|last1=Frey|first1=Jesse|date=March 2009|title=An exact multinomial test for equivalence|journal=The Canadian Journal of Statistics|volume=37|pages=47–59|doi=10.1002/cjs.10000|s2cid=122486567 }}[http://www.jstor.org/stable/25653460 Official web link (subscription required)].</ref>

The distance between the true underlying distribution <math>p</math> and a family of the multinomial distributions <math>\mathcal{M}</math> is defined by <math>d(p, \mathcal{M})=\min_{h\in\mathcal{M}}d(p,h)  </math>. Then the equivalence test problem is given by <math>H_0=\{d(p,\mathcal{M})\geq \varepsilon\}</math> and <math>H_1=\{d(p,\mathcal{M})< \varepsilon\}</math>. The distance <math>d(p,\mathcal{M})</math> is usually computed using numerical optimization. The tests for this case are developed recently in Ostrovski (2018).<ref>{{cite journal|last1=Ostrovski|first1=Vladimir|date=March 2018|title=Testing equivalence to families of multinomial distributions with application to the independence model|journal=Statistics & Probability Letters|volume=139|pages=61–66|doi=10.1016/j.spl.2018.03.014|s2cid=126261081}}[https://doi.org/10.1016/j.spl.2018.03.014 Official web link (subscription required)]. [https://www.researchgate.net/publication/324124605_Testing_equivalence_to_families_of_multinomial_distributions_with_application_to_the_independence_model Alternate, free web link].</ref>