Editing Interval estimation (section)

=== Less common forms ===

==== Likelihood-based ====
{{main|Likelihood interval}}
Utilizes the principles of a likelihood function to estimate the parameter of interest. Utilizing the likelihood-based method, confidence intervals can be found for exponential, Weibull, and lognormal means. Additionally, likelihood-based approaches can give confidence intervals for the standard deviation. It is also possible to create a prediction interval by combining the likelihood function and the future random variable.<ref name=":0" />

==== Fiducial ====
{{main|Fiducial interval}} 
[[Fiducial inference]] utilizes a data set, carefully removes the noise and recovers a distribution estimator, Generalized Fiducial Distribution (GFD). Without the use of Bayes' Theorem, there is no assumption of a prior, much like confidence intervals.
Fiducial inference is a less common form of [[statistical inference]]. The founder, [[Ronald Fisher|R.A. Fisher]], who had been developing inverse probability methods, had his own questions about the validity of the process. While fiducial inference was developed in the early twentieth century, the late twentieth century believed that the method was inferior to the frequentist and Bayesian approaches but held an important place in historical context for statistical inference. However, modern-day approaches have generalized the fiducial interval into Generalized Fiducial Inference (GFI), which can be used to estimate discrete and continuous data sets.<ref>{{Cite journal |last=Hannig |first=Jan |last2=Iyer |first2=Hari |last3=Lai |first3=Randy C. S. |last4=Lee |first4=Thomas C. M. |date=2016-07-02 |title=Generalized Fiducial Inference: A Review and New Results |url=|journal=Journal of the American Statistical Association |language=en |volume=111 |issue=515 |pages=1346–1361 |doi=10.1080/01621459.2016.1165102 |issn=0162-1459}}</ref>

==== Tolerance ====
{{main|Tolerance interval}}
Tolerance intervals use collected data set population to obtain an interval, within tolerance limits, containing 100γ% values. Examples typically used to describe tolerance intervals include manufacturing. In this context, a percentage of an existing product set is evaluated to ensure that a percentage of the population is included within tolerance limits. When creating tolerance intervals, the bounds can be written in terms of an upper and lower tolerance limit, utilizing the sample [[mean]], <math>\mu</math>, and the sample [[standard deviation]], s.

:<math>(l_b, u_b) = \mu \pm k_2s</math> for two-sided intervals

for two-sided intervals

And in the case of one-sided intervals where the tolerance is required only above or below a critical value,

:<math>l_{b} = \mu - k_{1}s</math>

:<math>u_{b}=\mu + k_{1} s</math>

<math>k_i</math> varies by distribution and the number of sides, i, in the interval estimate. In a normal distribution, <math>k_2</math> can be expressed as <ref>{{Cite journal |last=Howe |first=W. G. |date=June 1969 |title=Two-Sided Tolerance Limits for Normal Populations, Some Improvements |url=http://dx.doi.org/10.2307/2283644 |journal=Journal of the American Statistical Association |volume=64 |issue=326 |pages=610 |doi=10.2307/2283644 |issn=0162-1459|url-access=subscription }}</ref>

:<math>k_2 = z_{\alpha/2}\sqrt{\frac{\nu(1+\frac{1}{N})}{\chi_{1-\alpha,\nu}^2}}</math>

Where,

:<math>\chi _{1-\alpha,\nu}^2</math> is the critical value of the chi-square distribution utilizing <math>\nu</math> degrees of freedom that is exceeded with probability <math>\alpha</math>.

<math> z_{\alpha/2}</math> is the critical values obtained from the normal distribution.

==== Prediction ====
{{main|Prediction interval}}

A prediction interval estimates the interval containing future samples with some confidence, γ. Prediction intervals can be used for both [[Bayesian statistics|Bayesian]] and [[Frequentist probability|frequentist]] contexts. These intervals are typically used in regression data sets, but prediction intervals are not used for extrapolation beyond the previous data's experimentally controlled parameters.<ref>{{Cite journal |last=Vardeman |first=Stephen B. |date=1992 |title=What about the Other Intervals? |url=https://www.jstor.org/stable/2685212 |journal=The American Statistician |volume=46 |issue=3 |pages=193–197 |doi=10.2307/2685212 |issn=0003-1305|url-access=subscription }}</ref>

==== Fuzzy logic ====
{{further|Fuzzy logic}}
Fuzzy logic is used to handle decision-making in a non-binary fashion for artificial intelligence, medical decisions, and other fields. In general, it takes inputs, maps them through [[Fuzzy control system|fuzzy inference systems]], and produces an output decision. This process involves fuzzification, fuzzy logic rule evaluation, and defuzzification. When looking at fuzzy logic rule evaluation, [[Membership function (mathematics)|membership functions]] convert our non-binary input information into tangible variables. These membership functions are essential to predict the uncertainty of the system.