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Explore the people, places, events, achievements, struggles and stories that shaped our journey.
Movements, leaders, victories and the continuing fight for equality.
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Meet the people whose lives, choices and achievements shaped the journey.
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In 1968 Shirley Chisholm became the first Black woman elected to the United States Congress. In 1972 she launched a campaign for the Democratic presidential nomination, breaking another political barrier.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
In statistics, universal hypothesis testing is a special case of binary simple hypothesis testing. The universal problem is to distinguish between a simple null hypothesis , and the most general composite alternative , using independent and identically distributed samples from . The setting is sometimes referred to as goodness of fit testing, or one-sample testing. Note that "universal hypothesis testing" is sometimes used to refer to composite hypothesis testing problems using universal inference estimators, which are statistics that work without regularity assumptions.[1]
A simple binary hypothesis testing problem involves distinguishing between and , using samples . In the traditional setting of hypothesis testing are known apriori. A composite version of this problem involves sets of probability distributions , and asks to distinguish between and . In contrast, the universal setting corresponds to the special case of composite hypothesis testing, where the null hypothesis is simple, and the alternative hypothesis is the set of all distributions other than , . For example, someone might want to know if a particular coin was fair, i.e. or not, i.e. , where denote the coin coming up heads or tails.
The asymptotics of universal hypothesis testing were first discussed in Hoeffding's work on optimal tests for multinomial distributions.[2] There have been many subsequent works on the topic[3][4][5] in many directions. While Hoeffding's initial results were restricted to distributions with finite supports, later results developed solutions for continuous distributions using extensions of the Kullback-Leibler Divergence,[6] or kernel methods.[7][8]
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In statistics, universal hypothesis testing is a special case of binary simple hypothesis testing. The universal problem is to distinguish between a simple null hypothesis H 0 : Q = P {\displaystyle H_{0}:Q=P} , and the most general composite alternative H 1 : Q ≠ P {\displaystyle H_{1}:Q\neq P} , using independent and identically distributed samples from Q {\displaystyle Q} . The setting is sometimes referred to as goodness of fit testing, or one-sample testing. Note that "universal hypothesis testing" is sometimes used to refer to composite hypothesis testing problems using universal inference estimators, which are statistics that work without regularity assumptions. A simple binary hypothesis testing problem involves distinguishing between H 0 : Q = P 0 {\displaystyle H_{0}:Q=P_{0}} and H 1 : Q = P 1 {\displaystyle H_{1}:Q=P_{1}} , using samples X 1 , … , X n ∼ i.i.d. Q {\displaystyle X_{1},\dots ,X_{n}{\overset {\text{i.i.d.}}{\sim }}Q} . In the traditional setting of hypothesis testing P 0 , P 1 {\displaystyle P_{0},P_{1}} are known apriori. A composite version of this problem involves sets of probability distributions Ω 0 , Ω 1 {\displaystyle \Omega _{0},\Omega _{1}} , and asks to distinguish between H 0 : Q ∈ Ω 0 {\displaystyle H_{0}:Q\in \Omega _{0}} and H 1 : Q ∈ Ω 1 {\displaystyle H_{1}:Q\in \Omega _{1}} . In contrast, the universal setting corresponds to the special case of composite hypothesis testing, where the null hypothesis is simple, Ω 0 = P {\displaystyle \Omega _{0}=P} and the alternative hypothesis is the set of all distributions other than P {\displaystyle P} , Ω 1 = { F : F ≠ P } {\displaystyle \Omega _{1}=\{F:F\neq P\}} . For example, someone might want to know if a particular coin was fair, i.e. P [ X = H ] = 1 2 = P [ X = T ] {\displaystyle \mathbb {P} [X=H]={\frac {1}{2}}=\mathbb {P} [X=T]} or not, i.e. P [ X = H ] ≠ P [ X = T ] {\displaystyle \mathbb {P} [X=H]\neq \mathbb {P} [X=T]} , where H , T {\displaystyle H,T} denote the coin coming up heads or tails. The asymptotics of universal hypothesis testing were first discussed in Hoeffding's work on optimal tests for multinomial distributions. There have been many subsequent works on the topic in many directions. While Hoeffding's initial results were restricted to distributions with finite supports, later results developed solutions for continuous distributions using extensions of the Kullback-Leibler Divergence, or kernel methods.
In statistical significance testing, a one-tailed test and a two-tailed test are alternative ways of computing the statistical significance of a parameter inferred from a data set, in terms of a test statistic. A two-tailed test is appropriate if the estimated value is greater or less than a certain range of values, for example, whether a test taker may score above or below a specific range of scores. This method is used for null hypothesis testing and if the estimated value exists in the critical areas, the alternative hypothesis is accepted over the null hypothesis. A one-tailed test is appropriate if the estimated value may depart from the reference value in only one direction, left or right, but not both. An example can be whether a machine produces more than one-percent defective products. In this situation, if the estimated value exists in one of the one-sided critical areas, depending on the direction of interest (greater than or less than), the alternative hypothesis is accepted over the null hypothesis. Alternative names are one-sided and two-sided tests; the terminology "tail" is used because the extreme portions of distributions, where observations lead to rejection of the null hypothesis, are small and often "tail off" toward zero as in the normal distribution, colored in yellow, or "bell curve", pictured on the right and colored in green.
Rhipsalis baccifera, commonly known as the mistletoe cactus, is an epiphytic cactus which originates in Central and South America, the Caribbean, and Florida. It is also found throughout the tropics of Africa and into Sri Lanka. This is the only cactus species naturally occurring outside the Americas. One hypothesis is that it was introduced to the Old World by migratory birds, long enough ago for the Old World populations to be regarded as distinct subspecies. An alternative hypothesis holds that the species initially crossed the Atlantic Ocean on European ships trading between South America and Africa, after which birds may have spread it more widely.
In frequentist statistics, power is the probability of detecting an effect (i.e. rejecting the null hypothesis) given that some prespecified effect actually exists using a given test in a given context. In typical use, it is a function of the specific test that is used (including the choice of test statistic and significance level), the sample size (more data tends to provide more power), and the effect size (effects or correlations that are large relative to the variability of the data tend to provide more power). More formally, in the case of a simple hypothesis test with two hypotheses, the power of the test is the probability that the test correctly rejects the null hypothesis ( H 0 {\displaystyle H_{0}} ) when the alternative hypothesis ( H 1 {\displaystyle H_{1}} ) is true. It is commonly denoted by 1 − β {\displaystyle 1-\beta } , where β {\displaystyle \beta } is the probability of making a type II error (a false negative) conditional on there being a true effect or association.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →The Greenwood District.
![{\displaystyle \mathbb {P} [X=H]={\frac {1}{2}}=\mathbb {P} [X=T]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/614e79b700ef028b7e4a83b20af173857bd23fe9)
![{\displaystyle \mathbb {P} [X=H]\neq \mathbb {P} [X=T]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ac97d4d32bf88ba5b77cfeb50ad7e655300337)
