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Concept
Frequentist inference
Summary
Frequentist inference is a type of statistical inference based in frequentist probability, which treats “probability” in equivalent terms to “frequency” and draws conclusions from sample-data by means of emphasizing the frequency or proportion of findings in the data. Frequentist-inference underlies frequentist statistics, in which the well-established methodologies of statistical hypothesis testing and confidence intervals are founded.
The primary formulation of frequentism stems from the presumption that statistics could be perceived to have been a probabilistic frequency. This view was primarily developed by Ronald Fisher and the team of Jerzy Neyman and Egon Pearson. Ronald Fisher contributed to frequentist statistics by developing the frequentist concept of "significance testing", which is the study of the significance of a measure of a statistic when compared to the hypothesis. Neyman-Pearson extended Fisher's ideas to multiple hypotheses by conjecturing that the ratio of probabilities of hypotheses when maximizing the difference between the two hypotheses leads to a maximization of exceeding a given p-value, and also provides the basis of type I and type II errors. For more, see the foundations of statistics page.
For statistical inference, the statistic about which we want to make inferences is , where the random vector is a function of an unknown parameter, . The parameter is further partitioned into (), where is the parameter of interest, and is the nuisance parameter. For concreteness, might be the population mean, , and the nuisance parameter the standard deviation of the population mean, .
Thus, statistical inference is concerned with the expectation of random vector , .
To construct areas of uncertainty in frequentist inference, a pivot is used which defines the area around that can be used to provide an interval to estimate uncertainty. The pivot is a probability such that for a pivot, , which is a function, that is strictly increasing in , where is a random vector.
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