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Informally, in frequentist statistics, a confidence interval (CI) is an interval which is expected to typically contain the parameter being estimated. More specifically, given a confidence level ${\displaystyle \gamma }$ (95% and 99% are typical values), a CI is a random interval which contains the parameter being estimated ${\displaystyle \gamma }$% of the time.^[1][2] The confidence level, degree of confidence or confidence coefficient represents the long-run proportion of CIs (at the given confidence level) that theoretically contain the true value of the parameter; this is tantamount to the nominal coverage probability. For example, out of all intervals computed at the 95% level, 95% of them should contain the parameter's true value.^[3]

Factors affecting the width of the CI include the sample size, the variability in the sample, and the confidence level.^[4] All else being the same, a larger sample produces a narrower confidence interval, greater variability in the sample produces a wider confidence interval, and a higher confidence level produces a wider confidence interval.^[5]

Definition

Let ${\displaystyle X}$ be a random sample from a probability distribution with statistical parameter ${\displaystyle \theta }$, which is a quantity to be estimated, and ${\displaystyle \varphi }$, representing quantities that are not of immediate interest. A confidence interval for the parameter ${\displaystyle \theta }$, with confidence level or coefficient ${\displaystyle \gamma }$, is an interval ${\displaystyle (u(X),v(X))}$ determined by random variables ${\displaystyle u(X)}$ and ${\displaystyle v(X)}$ with the property:

${\displaystyle P(u(X)<\theta <v(X))=\gamma \quad {\text{ for every }}(\theta ,\varphi ).}$

The number ${\displaystyle \gamma }$, whose typical value is close to but not greater than 1, is sometimes given in the form ${\displaystyle 1-\alpha }$ (or as a percentage ${\displaystyle 100\%\cdot (1-\alpha )}$), where ${\displaystyle \alpha }$ is a small positive number, often 0.05.

It is important for the bounds ${\displaystyle u(X)}$ and ${\displaystyle v(X)}$ to be specified in such a way that as long as ${\displaystyle X}$ is collected randomly, every time we compute a confidence interval, there is probability ${\displaystyle \gamma }$ that it would contain ${\displaystyle \theta }$, the true value of the parameter being estimated. This should hold true for any actual ${\displaystyle \theta }$ and ${\displaystyle \varphi }$.^[2]

Approximate confidence intervals

In many applications, confidence intervals that have exactly the required confidence level are hard to construct, but approximate intervals can be computed. The rule for constructing the interval may be accepted as providing a confidence interval at level ${\displaystyle \gamma }$ if

${\displaystyle P(u(X)<\theta <v(X))\approx \ \gamma \quad {\text{ for every }}(\theta ,\varphi )}$

to an acceptable level of approximation. Alternatively, some authors^[6] simply require that

${\displaystyle P(u(X)<\theta <v(X))\geq \ \gamma \quad {\text{ for every }}(\theta ,\varphi ),}$

which is useful if the probabilities are only partially identified or imprecise, and also when dealing with discrete distributions. Confidence limits of the form

${\displaystyle P(u(X)<\theta )\geq \ \gamma }$   and   ${\displaystyle P(\theta <v(X))\geq \gamma }$

are called conservative;^[7](p 210) accordingly, one speaks of conservative confidence intervals and, in general, regions.

Desired properties

When applying standard statistical procedures, there will often be standard ways of constructing confidence intervals. These will have been devised so as to meet certain desirable properties, which will hold given that the assumptions on which the procedure relies are true. These desirable properties may be described as: validity, optimality, and invariance.

Of the three, "validity" is most important, followed closely by "optimality". "Invariance" may be considered as a property of the method of derivation of a confidence interval, rather than of the rule for constructing the interval. In non-standard applications, these same desirable properties would be sought:

Validity

This means that the nominal coverage probability (confidence level) of the confidence interval should hold, either exactly or to a good approximation.

Optimality

This means that the rule for constructing the confidence interval should make as much use of the information in the data-set as possible.

Recall that one could throw away half of a dataset and still be able to derive a valid confidence interval. One way of assessing optimality is by the length of the interval so that a rule for constructing a confidence interval is judged better than another if it leads to intervals whose lengths are typically shorter.

Invariance

In many applications, the quantity being estimated might not be tightly defined as such.

For example, a survey might result in an estimate of the median income in a population, but it might equally be considered as providing an estimate of the logarithm of the median income, given that this is a common scale for presenting graphical results. It would be desirable that the method used for constructing a confidence interval for the median income would give equivalent results when applied to constructing a confidence interval for the logarithm of the median income: Specifically the values at the ends of the latter interval would be the logarithms of the values at the ends of former interval.

Methods of derivation

For non-standard applications, there are several routes that might be taken to derive a rule for the construction of confidence intervals. Established rules for standard procedures might be justified or explained via several of these routes. Typically a rule for constructing confidence intervals is closely tied to a particular way of finding a point estimate of the quantity being considered.

Summary statistics

This is closely related to the method of moments for estimation. A simple example arises where the quantity to be estimated is the population mean, in which case a natural estimate is the sample mean. Similarly, the sample variance can be used to estimate the population variance. A confidence interval for the true mean can be constructed centered on the sample mean with a width which is a multiple of the square root of the sample variance.

Likelihood theory

Estimates can be constructed using the maximum likelihood principle, the likelihood theory for this provides two ways of constructing confidence intervals or confidence regions for the estimates.

Estimating equations

The estimation approach here can be considered as both a generalization of the method of moments and a generalization of the maximum likelihood approach. There are corresponding generalizations of the results of maximum likelihood theory that allow confidence intervals to be constructed based on estimates derived from estimating equations.^{[citation needed]}

Hypothesis testing

If hypothesis tests are available for general values of a parameter, then confidence intervals/regions can be constructed by including in the 100 p % confidence region all those points for which the hypothesis test of the null hypothesis that the true value is the given value is not rejected at a significance level of (1 − p).^{[7](§ 7.2 (iii))}

Bootstrapping

In situations where the distributional assumptions for the above methods are uncertain or violated, resampling methods allow construction of confidence intervals or prediction intervals. The observed data distribution and the internal correlations are used as the surrogate for the correlations in the wider population.

Central limit theorem

The central limit theorem is a refinement of the law of large numbers. For a large number of independent identically distributed random variables ${\displaystyle X_{1},...,X_{n},}$ with finite variance, the average ${\displaystyle {\overline {X}}_{n}}$ approximately has a normal distribution, no matter what the distribution of the ${\displaystyle X_{i}}$ is, with the approximation roughly improving in proportion to ${\displaystyle {\sqrt {n}}}$.^[2]

Example

Suppose ${\displaystyle {X_{1},\ldots ,X_{n}}}$ is an independent sample from a normally distributed population with unknown parameters mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}.}$ Let

${\displaystyle {\bar {X}}={\frac {X_{1}+\cdots +X_{n}}{n}},}$

${\displaystyle S^{}}$Zdroj:https://en.wikipedia.org?pojem=Confidence_interval
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