In statistics, **interval estimation** is the use of sample data to calculate an interval of possible values of an unknown population parameter; this is in contrast to point estimation, which gives a single value. Jerzy Neyman (1937) identified interval estimation ("estimation by interval") as distinct from point estimation ("estimation by unique estimate"). In doing so, he recognized that then-recent work quoting results in the form of an estimate plus-or-minus a standard deviation indicated that interval estimation was actually the problem statisticians really had in mind.

The most prevalent forms of interval estimation are:

- confidence intervals (a frequentist method); and
- credible intervals (a Bayesian method).

Other forms include:

- likelihood intervals (a likelihoodist method); and
- fiducial intervals (a fiducial method).

Other forms of statistical intervals, which do not estimate parameters, include:

- tolerance intervals (an estimate of a population); and
- prediction intervals (an estimate of a future observation, used mainly in regression analysis).

Non-statistical methods that can lead to interval estimates include fuzzy logic. An interval estimate is one type of outcome of a statistical analysis. Some other types of outcome are point estimates and decisions.

The scientific problems associated with interval estimation may be summarised as follows:

- When interval estimates are reported, they should have a commonly held interpretation in the scientific community and more widely. In this regard, credible intervals are held to be most readily understood by the general public
^{[ citation needed ]}. Interval estimates derived from fuzzy logic have much more application-specific meanings. - For 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.
- For 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 are differences:

- credible intervals can readily deal with prior information, while confidence intervals cannot.
- 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 non-parametric models (see non-parametric statistics).

- When interval estimates are reported, they should have a commonly held interpretation in the scientific community and more widely. In this regard, credible intervals are held to be most readily understood by the general public

- 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 simulations 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 (1991) discusses conditions under which credible intervals and confidence intervals will produce similar results, and also discusses both the coverage probabilities of credible intervals and the posterior probabilities associated with confidence intervals.

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.

- 68–95–99.7 rule
- Algorithmic inference
- Coverage probability
- Estimation statistics
- Induction (philosophy)
- Multiple comparisons
- Philosophy of statistics
- Predictive inference
- Behrens–Fisher problem This has played an important role in the development of the theory behind applicable statistical methodologies.

**Frequentist probability** or **frequentism** is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in many trials. Probabilities can be found by a repeatable objective process. This interpretation supports the statistical needs of many experimental scientists and pollsters. It does not support all needs, however; gamblers typically require estimates of the odds without experiments.

**Statistics** is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.

**Statistical inference** is the process of using data analysis to infer properties of an underlying distribution of probability. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population.

**Statistics** is a field of inquiry that studies the collection, analysis, interpretation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities; it is also used and misused for making informed decisions in all areas of business and government.

A **statistical hypothesis** is a hypothesis that is testable on the basis of observed data modelled as the realised values taken by a collection of random variables. A set of data is modelled as being realised values of a collection of random variables having a joint probability distribution in some set of possible joint distributions. The hypothesis being tested is exactly that set of possible probability distributions. A **statistical hypothesis test** is a method of statistical inference. An alternative hypothesis is proposed for the probability distribution of the data, either explicitly or only informally. The comparison of the two models is deemed *statistically significant* if, according to a threshold probability—the significance level—the data would be unlikely to occur if the null hypothesis were true. A hypothesis test specifies which outcomes of a study may lead to a rejection of the null hypothesis at a pre-specified level of significance, while using a pre-chosen measure of deviation from that hypothesis. The pre-chosen level of significance is the maximal allowed "false positive rate". One wants to control the risk of incorrectly rejecting a true null hypothesis.

In statistics, **point estimation** involves the use of sample data to calculate a single value which is to serve as a "best guess" or "best estimate" of an unknown population parameter. More formally, it is the application of a point estimator to the data to obtain a point estimate.

In statistics, a **confidence interval** (**CI**) is a type of estimate computed from the statistics of the observed data. This gives a range of values for an unknown parameter. The interval has an associated **confidence level** that gives the probability with which the estimated interval will contain the true value of the parameter. The confidence level is chosen by the investigator. For a given estimation in a given sample, using a higher confidence level generates a wider confidence interval. In general terms, a confidence interval for an unknown parameter is based on sampling the distribution of a corresponding estimator.

In statistical inference, specifically predictive inference, a **prediction interval** is an estimate of an interval in which a future observation will fall, with a certain probability, given what has already been observed. Prediction intervals are often used in regression analysis.

**Mathematical statistics** is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory.

The following is a glossary of terms used in the mathematical sciences statistics and probability.

In statistics, the **Behrens–Fisher problem**, named after Walter Behrens and Ronald Fisher, is the problem of interval estimation and hypothesis testing concerning the difference between the means of two normally distributed populations when the variances of the two populations are not assumed to be equal, based on two independent samples.

In Bayesian statistics, a **credible interval** is an interval within which an unobserved parameter value falls with a particular probability. It is an interval in the domain of a posterior probability distribution or a predictive distribution. The generalisation to multivariate problems is the **credible region**. Credible intervals are analogous to confidence intervals in frequentist statistics, although they differ on a philosophical basis: Bayesian intervals treat their bounds as fixed and the estimated parameter as a random variable, whereas frequentist confidence intervals treat their bounds as random variables and the parameter as a fixed value. Also, Bayesian credible intervals use knowledge of the situation-specific prior distribution, while the frequentist confidence intervals do not.

**Fiducial inference** is one of a number of different types of statistical inference. These are rules, intended for general application, by which conclusions can be drawn from samples of data. In modern statistical practice, attempts to work with fiducial inference have fallen out of fashion in favour of frequentist inference, Bayesian inference and decision theory. However, fiducial inference is important in the history of statistics since its development led to the parallel development of concepts and tools in theoretical statistics that are widely used. Some current research in statistical methodology is either explicitly linked to fiducial inference or is closely connected to it.

The **foundations of statistics** concern the epistemological debate in statistics over how one should conduct inductive inference from data. Among the issues considered in statistical inference are the question of Bayesian inference versus frequentist inference, the distinction between Fisher's "significance testing" and Neyman–Pearson "hypothesis testing", and whether the likelihood principle should be followed. Some of these issues have been debated for up to 200 years without resolution.

**Frequentist inference** is a type of statistical inference that draws conclusions from sample data by emphasizing the frequency or proportion of the data. An alternative name is **frequentist statistics**. This is the inference framework in which the well-established methodologies of statistical hypothesis testing and confidence intervals are based. Other than frequentistic inference, the main alternative approach to statistical inference is Bayesian inference, while another is fiducial inference.

**Exact statistics**, such as that described in exact test, is a branch of statistics that was developed to provide more accurate results pertaining to statistical testing and interval estimation by eliminating procedures based on asymptotic and approximate statistical methods. The main characteristic of exact methods is that statistical tests and confidence intervals are based on exact probability statements that are valid for any sample size. Exact statistical methods help avoid some of the unreasonable assumptions of traditional statistical methods, such as the assumption of equal variances in classical ANOVA. They also allow exact inference on variance components of mixed models.

In statistical inference, the concept of a **confidence distribution** (**CD**) has often been loosely referred to as a distribution function on the parameter space that can represent confidence intervals of all levels for a parameter of interest. Historically, it has typically been constructed by inverting the upper limits of lower sided confidence intervals of all levels, and it was also commonly associated with a fiducial interpretation, although it is a purely frequentist concept. A confidence distribution is NOT a probability distribution function of the parameter of interest, but may still be a function useful for making inferences.

In Bayesian inference, the **Bernstein-von Mises theorem** provides the basis for using Bayesian credible sets for confidence statements in parametric models. It states that under some conditions, a posterior distribution converges in the limit of infinite data to a multivariate normal distribution centered at the maximum likelihood estimator with covariance matrix given by , where is the true population parameter and is the Fisher information matrix at the true population parameter value.

In particle physics, **CLs** represents a statistical method for setting *upper limits* on model parameters, a particular form of interval estimation used for parameters that can take only non-negative values. Although CLs are said to refer to Confidence Levels, "The method's name is ... misleading, as the CLs exclusion region is not a confidence interval." It was first introduced by physicists working at the LEP experiment at CERN and has since been used by many high energy physics experiments. It is a frequentist method in the sense that the properties of the limit are defined by means of error probabilities, however it differs from standard confidence intervals in that the stated confidence level of the interval is not equal to its coverage probability. The reason for this deviation is that standard upper limits based on a most powerful test necessarily produce empty intervals with some fixed probability when the parameter value is zero, and this property is considered undesirable by most physicists and statisticians.

- Neyman, J. (1937), "Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability",
*Philosophical Transactions of the Royal Society of London A,***236**, 333–380. - Severini, T.A. (1991), "On the relationship between Bayesian and Non-Bayesian interval estimates",
*Journal of the Royal Statistical Society, Series B*,**53**(3), 611–618

- Kendall, M.G. and Stuart, A. (1973).
*The Advanced Theory of Statistics. Vol 2: Inference and Relationship*(3rd Edition). Griffin, London.

- In the above Chapter 20 covers confidence intervals, while Chapter 21 covers fiducial intervals and Bayesian intervals and has discussion comparing the three approaches. Note that this work predates modern computationally intensive methodologies. In addition, Chapter 21 discusses the Behrens–Fisher problem.

- Meeker, W.Q., Hahn, G.J. and Escobar, L.A. (2017).
*Statistical Intervals: A Guide for Practitioners and Researchers*(2nd Edition). John Wiley & Sons.

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