Editing Interval estimation (section)

==Discussion==
{{see also|Tolerance interval#Relation to other intervals}}

When determining the [[statistical significance]] of a parameter, it is best to understand the data and its collection methods. Before collecting data, an experiment should be planned such that the [[sampling error]] is [[statistical variability]] (a [[random error]]), as opposed to a [[Bias (statistics)|statistical bias]] (a [[systematic error]]).<ref>{{Cite journal |last=Hahn |first=Gerald J. |last2=Meeker |first2=William Q. |date=1993 |title=Assumptions for Statistical Inference |url=https://www.jstor.org/stable/2684774 |journal=The American Statistician |volume=47 |issue=1 |pages=1–11 |doi=10.2307/2684774 |issn=0003-1305|url-access=subscription }}</ref> After experimenting, a typical first step in creating interval estimates is [[exploratory analysis]] plotting using various [[Statistical graphics|graphical methods]]. From this, one can determine the distribution of samples from the data set. Producing interval boundaries with incorrect assumptions based on distribution makes a prediction faulty.<ref>{{Cite journal |last=Hahn |first=Gerald J. |last2=Doganaksoy |first2=Necip |last3=Meeker |first3=William Q. |date=2019-08-01 |title=Statistical Intervals, Not Statistical Significance |url=https://academic.oup.com/jrssig/article/16/4/20/7038025 |journal=Significance |language=en |volume=16 |issue=4 |pages=20–22 |doi=10.1111/j.1740-9713.2019.01298.x |issn=1740-9705|doi-access=free }}</ref>

When interval estimates are reported, they should have a commonly held interpretation within and beyond the scientific community. Interval estimates derived from fuzzy logic have much more application-specific meanings.

In commonly occurring situations there should be sets of standard procedures that can be used, subject to the checking and validity of any required assumptions. This applies for both confidence intervals and credible intervals. However, in more novel situations there should be guidance on how interval estimates can be formulated. In this regard confidence intervals and credible intervals have a similar standing but there two differences. First, credible intervals can readily deal with prior information, while confidence intervals cannot. Secondly, confidence intervals are more flexible and can be used practically in more situations than credible intervals: one area where credible intervals suffer in comparison is in dealing with [[Nonparametric statistics|non-parametric models]].

There should be ways of testing the performance of interval estimation procedures. This arises because many such procedures involve approximations of various kinds and there is a need to check that the actual performance of a procedure is close to what is claimed. The use of [[stochastic simulation]]s makes this is straightforward in the case of confidence intervals, but it is somewhat more problematic for credible intervals where prior information needs to be taken properly into account. Checking of credible intervals can be done for situations representing no-prior-information but the check involves checking the long-run frequency properties of the procedures.

Severini (1993) discusses conditions under which credible intervals and confidence intervals will produce similar results, and also discusses both the [[Coverage probability|coverage probabilities]] of credible intervals and the posterior probabilities associated with confidence intervals.<ref>{{Cite journal |last=Severini |first=Thomas A. |date=1993 |title=Bayesian Interval Estimates which are also Confidence Intervals |url=https://www.jstor.org/stable/2346212 |journal=Journal of the Royal Statistical Society. Series B (Methodological) |volume=55 |issue=2 |pages=533–540 |issn=0035-9246}}</ref>

In [[decision theory]], which is a common approach to and justification for Bayesian statistics, interval estimation is not of direct interest. The outcome is a decision, not an interval estimate, and thus Bayesian decision theorists use a [[Bayes action]]: they minimize expected loss of a loss function with respect to the entire posterior distribution, not a specific interval.