Binary Hypothesis Testing Task

Example(s):
- …
Counter-Example(s):
- Univariate Hypothesis Testing.
- N-ary Hypothesis Testing.
See: Random Variable, Null Hypothesis, Alternative Hypothesis.

References

(Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/Power_of_a_test Retrieved:2021-4-22.
- The power of a binary hypothesis test is the probability that the test rejects the null hypothesis ([math]\displaystyle{ H_0 }[/math] ) when a specific alternative hypothesis ([math]\displaystyle{ H_1 }[/math] ) is true — i.e., it indicates the probability of avoiding a type II error. The statistical power ranges from 0 to 1, and as statistical power increases, the probability of making a type II error (wrongly failing to reject the null hypothesis) decreases.

(Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/Error_exponents_in_hypothesis_testing#Error_exponents_in_binary_hypothesis_testing Retrieved:2021-4-22.
- Consider a binary hypothesis testing problem in which observations are modeled as independent and identically distributed random variables under each hypothesis. Let [math]\displaystyle{ Y_1, Y_2, \ldots, Y_n }[/math] denote the observations. Let [math]\displaystyle{ f_0 }[/math] denote the probability density function of each observation [math]\displaystyle{ Y_i }[/math] under the null hypothesis [math]\displaystyle{ H_0 }[/math] and let [math]\displaystyle{ f_1 }[/math] denote the probability density function of each observation [math]\displaystyle{ Y_i }[/math] under the alternate hypothesis [math]\displaystyle{ H_1 }[/math] .
  In this case there are two possible error events. Error of type 1, also called [[False positives and false negatives|false positive]], occurs when the null hypothesis is true and it is wrongly rejected. Error of type 2, also called false negative, occurs when the alternate hypothesis is true and null hypothesis is not rejected. The probability of type 1 error is denoted [math]\displaystyle{ P (\mathrm{error}\mid H_0) }[/math] and the probability of type 2 error is denoted [math]\displaystyle{ P (\mathrm{error}\mid H_1) }[/math] .

Stack Exchange. (2014). “https://stats.stackexchange.com/q/109350 Is binary hypothesis testing a better statistical term than A/B testing?]". Stack Exchange (version: 2014-07-26):
- QUOTE:
  - “Binary hypothesis testing” is hypothesis testing when one wants to decide between two hypotheses.
  - “Two-sample hypothesis testing” is what is known colloquially as A/B testing.
  - “Paired hypothesis testing” when you compare the same sample before and after an event to find if it had an effect. Similar to A/B testing but not A/B testing.

(Naghshvar & Javidi, 2010) ⇒ Mohammad Naghshvar, and Tara Javidi. (2010). “Active M-ary Sequential Hypothesis Testing.” In: IEEE International Symposium on Information Theory Proceedings (ISIT 2010).
- QUOTE: … The most well known instance of our problem is the case of binary hypothesis testing (M = 2) with passive sensing (K = 1), first studied by Wald [1]. In this instance of the problem, the optimal action at any given time is given by a sequential probability ratio test (SPRT). …

(Johnson, 2003) ⇒ Don Johnson. (2003). “The Likelihood Ratio Test."
- QUOTE:In a binary hypothesis testing problem, four possible outcomes can result. Model M0 did in fact represent the best model for the data and the decision rule said it was (a correct decision) or said it wasn't (an erroneous decision).

(Wald & Wolfowitz, 1948) ⇒ Abraham Wald, and Jacob Wolfowitz. (1948). “Optimum character of the sequential probability ratio test.” In: The Annals of Mathematical Statistics 19,3 (1948).