Central fluctuation interval

The central fluctuation interval is a term from mathematical statistics . It says something about the precision of the position estimation of a parameter (for example a mean value). The fluctuation interval includes a range around the true value of the parameter in the population, which - to put it simply - contains the parameter estimated from the sample with a previously determined certainty probability.

idea

An estimation function is a random variable for an unknown true parameters of a population . Hence it has a distribution and we can give intervals with respect to the realization with the probability . ${\displaystyle \theta (X_{1},\dots ,X_{n})}$ ${\displaystyle \vartheta }$ ${\displaystyle 1-\alpha }$

In other words, if we take a sample of the values , we can calculate an estimate and, with a given probability, specify an interval in which we expect the estimate . ${\displaystyle x_{1},\dots ,x_{n}}$ ${\displaystyle {\hat {\vartheta }}=\theta (x_{1},\dots ,x_{n})}$ ${\displaystyle {\hat {\vartheta }}}$

The central fluctuation intervals have a disadvantage: the interval limits contain the unknown parameter (in contrast to the confidence interval ). Nevertheless, the central fluctuation interval provides valuable information, namely the size of the deviation of a parameter estimated from the sample from the true parameter. ${\displaystyle \theta }$

parameter	condition	Central fluctuation interval
${\displaystyle \mu }$	${\displaystyle X_{i}\sim N(\mu ,\sigma )}$, known ${\displaystyle \sigma }$	${\displaystyle \left[\mu -z_{1-\alpha /2}\sigma /{\sqrt {n}};\mu +z_{1-\alpha /2}\sigma /{\sqrt {n}}\right]}$
${\displaystyle \mu }$	${\displaystyle X_{i}\sim N(\mu ,\sigma )}$, unknown ${\displaystyle \sigma }$	${\displaystyle \left[\mu -t_{n-1;1-\alpha /2}S/{\sqrt {n}};\mu +t_{n-1;1-\alpha /2}S/{\sqrt {n}}\right]}$
${\displaystyle \mu }$	${\displaystyle X_{i}\sim (\mu ,\sigma )}$ arbitrarily distributed, ${\displaystyle n>30}$	${\displaystyle \left[\mu -z_{1-\alpha /2}\sigma /{\sqrt {n}};\mu +z_{1-\alpha /2}\sigma /{\sqrt {n}}\right]}$( known) ${\displaystyle \sigma }$ ${\displaystyle \left[\mu -z_{1-\alpha /2}S/{\sqrt {n}};\mu +z_{1-\alpha /2}S/{\sqrt {n}}\right]}$( unknown) ${\displaystyle \sigma }$
${\displaystyle \sigma ^{2}}$	${\displaystyle X_{i}\sim N(\mu ,\sigma )}$, known ${\displaystyle \mu }$	${\displaystyle \left[\chi _{n;\alpha /2}^{2}{\frac {\sigma ^{2}}{n}};\chi _{n;1-\alpha /2}^{2}{\frac {\sigma ^{2}}{n}}\right]}$
${\displaystyle \sigma ^{2}}$	${\displaystyle X_{i}\sim N(\mu ,\sigma )}$, unknown ${\displaystyle \mu }$	${\displaystyle \left[\chi _{n-1;\alpha /2}^{2}{\frac {\sigma ^{2}}{n-1}};\chi _{n-1;1-\alpha /2}^{2}{\frac {\sigma ^{2}}{n-1}}\right]}$
${\displaystyle \pi }$	${\displaystyle X_{i}}$ Bernoulli distributed with parameters ${\displaystyle \pi }$	${\displaystyle \left[\pi -z_{1-\alpha /2}{\sqrt {p(1-p)/n}};\pi +z_{1-\alpha /2}{\sqrt {p(1-p)/n}}\right]}$ or. ${\displaystyle \left[\pi -{\frac {z_{1-\alpha /2}}{\sqrt {4n}}};\pi +{\frac {z_{1-\alpha /2}}{\sqrt {4n}}}\right]}$

Are there

• ${\displaystyle 1-\alpha }$ the probability of security,
• ${\displaystyle z_{q}}$, and the quantiles of the standard normal , t and chi-square distributions with degrees of freedom, ${\displaystyle t_{m;q}}$ ${\displaystyle \chi _{m;q}^{2}}$ ${\displaystyle q}$ ${\displaystyle m}$
• ${\displaystyle S^{2}={\frac {1}{n-1}}\sum _{i=1}(X_{i}-{\bar {X}})^{2}}$ the corrected sample variance and
• ${\displaystyle p}$ the estimated proportional value from the sample.

Formal definition

The central fluctuation interval for an estimator is the interval for which applies or , i.e. ${\displaystyle \theta }$ ${\displaystyle [\vartheta -c_{u};\vartheta +c_{o}]}$ ${\displaystyle P(\theta <\vartheta -c_{u})=\alpha /2}$ ${\displaystyle P(\theta >\vartheta -c_{o})=\alpha /2}$

${\displaystyle P(\vartheta -c_{u}\leq \theta \leq \vartheta +c_{o})=1-\alpha }$.

The central fluctuation interval can, but does not have to, be symmetrical about the unknown parameter. The values or depend ${\displaystyle c_{u}}$ ${\displaystyle c_{o}}$

• on the distribution type of the estimator (see , ) and ${\displaystyle c_{u}^{*}}$ ${\displaystyle c_{o}^{*}}$
• the variance of the estimator : ${\displaystyle Var(\theta )}$

${\displaystyle P\left(\vartheta -c_{u}^{*}{\sqrt {Var(\theta )}}\leq \theta \leq \vartheta +c_{o}^{*}{\sqrt {Var(\theta )}}\right)=1-\alpha }$.

Special central fluctuation intervals

For the population mean ${\displaystyle \mu }$

The estimator is used for the unknown mean of the population . There are two cases for the distribution : ${\displaystyle \mu }$ ${\displaystyle {\bar {X}}={\frac {X_{1}+\dots +X_{n}}{n}}}$ ${\displaystyle {\bar {X}}}$

1. ${\displaystyle X_{i}\sim N(\mu ,\sigma )\,}$, then applies (reproductive property of normal distribution) or ${\displaystyle {\bar {X}}\sim N(\mu ,\sigma /{\sqrt {n}})}$
2. ${\displaystyle X_{i}\sim (\mu ,\sigma )\,}$(arbitrarily distributed) and fulfills the requirements of the central limit value theorem , then applies . ${\displaystyle {\bar {X}}\approx N(\mu ,\sigma /{\sqrt {n}})}$

This results in three fluctuation intervals:

1a. known , then and ${\displaystyle \sigma }$ ${\displaystyle {\frac {{\bar {X}}-\mu }{\sigma /{\sqrt {n}}}}\sim N(0,1)}$

${\displaystyle P\left(\mu -z_{1-\alpha /2}\sigma /{\sqrt {n}}\leq {\bar {X}}\leq \mu +z_{1-\alpha /2}\sigma /{\sqrt {n}}\right)=1-\alpha }$

1b. unknown , then and ${\displaystyle \sigma }$ ${\displaystyle {\frac {{\bar {X}}-\mu }{S/{\sqrt {n}}}}\sim t_{n-1}}$

${\displaystyle P\left(\mu -t_{n-1;1-\alpha /2}S/{\sqrt {n}}\leq {\bar {X}}\leq \mu +t_{n-1;1-\alpha /2}S/{\sqrt {n}}\right)=1-\alpha }$

2. It applies and ${\displaystyle {\frac {{\bar {X}}-\mu }{S/{\sqrt {n}}}}\approx N(0,1)}$

${\displaystyle P\left(\mu -z_{1-\alpha /2}S/{\sqrt {n}}\leq {\bar {X}}\leq \mu +z_{1-\alpha /2}S/{\sqrt {n}}\right)\approx 1-\alpha }$.

The values or are the - quantiles of the standard normal distribution or of the Student t-distribution with degrees of freedom. ${\displaystyle z_{q}}$ ${\displaystyle t_{m;q}}$ ${\displaystyle q}$ ${\displaystyle m}$

For the population variance ${\displaystyle \sigma ^{2}}$

If the sample variables are distributed, then there are two different possible estimators for the variance : ${\displaystyle X_{i}\sim N(\mu ,\sigma )}$ ${\displaystyle \sigma ^{2}}$

1. If is known , then it arises . ${\displaystyle \mu }$ ${\displaystyle S^{*^{2}}={\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}}$
2. If is unknown , then it arises . ${\displaystyle \mu }$ ${\displaystyle S^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}}$

In the first case it is distributed and the central fluctuation interval is ${\displaystyle {\frac {nS^{*^{2}}}{\sigma ^{2}}}\sim \chi _{n}^{2}}$

${\displaystyle P\left(\chi _{n;\alpha /2}^{2}{\frac {\sigma ^{2}}{n}}\leq S^{*^{2}}\leq \chi _{n;1-\alpha /2}^{2}{\frac {\sigma ^{2}}{n}}\right)=1-\alpha }$

and in the second case it is distributed and the central fluctuation interval is given by ${\displaystyle {\frac {(n-1)S^{2}}{\sigma ^{2}}}\sim \chi _{n-1}^{2}}$

${\displaystyle P\left(\chi _{n-1;\alpha /2}^{2}{\frac {\sigma ^{2}}{n-1}}\leq S^{*^{2}}\leq \chi _{n-1;1-\alpha /2}^{2}{\frac {\sigma ^{2}}{n-1}}\right)=1-\alpha }$.

The values are the - quantiles of the chi-square distribution with degrees of freedom. ${\displaystyle \chi _{m;q}^{2}}$ ${\displaystyle q}$ ${\displaystyle m}$

In both cases the central fluctuation interval is not symmetrical . ${\displaystyle \sigma ^{2}}$

For the proportional value of the population ${\displaystyle \pi }$

A dichotomous random variable number of successes in draws with replacement is binomially distributed depending on the unknown probability of success . When the approximation conditions are met, the distribution is normal and so is the estimator . The central fluctuation interval is therefore given by ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle \pi }$ ${\displaystyle X}$ ${\displaystyle \Pi =X/n\approx N(\pi ,{\sqrt {\pi (1-\pi )/n}})}$

${\displaystyle P\left(\pi -z_{1-\alpha /2}{\sqrt {\pi (1-\pi )/n}}\leq \Pi \leq \pi +z_{1-\alpha /2}{\sqrt {\pi (1-\pi )/n}}\right)\approx 1-\alpha }$.

For the practical calculations you can either estimate with . Alternatively, you can replace with , and is the proportional value from the sample. ${\displaystyle \pi (1-\pi )}$ ${\displaystyle 1/4=\max _{0\leq \pi \leq 1}\pi (1-\pi )}$ ${\displaystyle \pi (1-\pi )}$ ${\displaystyle p(1-p)}$ ${\displaystyle p}$

Examples

Example 1 : If we want to estimate the mean length of study in semesters of students precisely with a certainty probability , then this means that the central fluctuation interval must not deviate from the true value by more than semesters. The length of the central fluctuation interval must therefore be semester. ${\displaystyle 0{,}1}$ ${\displaystyle 1-\alpha =95\,\%}$ ${\displaystyle \mu }$ ${\displaystyle \pm 0{,}05}$ ${\displaystyle 0{,}1}$

For the mean study duration, it is not known whether it is normally distributed, i.e. H. it follows

${\displaystyle 0{,}1=\left(\mu +z_{1-\alpha /2}{\frac {s}{\sqrt {n}}}\right)-\left(\mu -z_{1-\alpha /2}{\frac {s}{\sqrt {n}}}\right)}$,

d. H. Depending on ( ), a sample size can be determined in order to achieve this accuracy: ${\displaystyle s}$ ${\displaystyle z_{0{,}975}=1{,}96}$

${\displaystyle n=1536{,}64s^{2}}$.

With the semester, 1537 students have to be surveyed, if the semester is, then 6147 students would be needed. In this example only the position, but not the width of the central fluctuation interval, depends on the true parameter. ${\displaystyle s=1}$ ${\displaystyle s=2}$

Example 2 : In election surveys, around 1000 eligible voters are usually asked. With what accuracy, given a probability of certainty of, can an election researcher predict the outcome of a party? ${\displaystyle 1-\alpha =95\,\%}$

The length of the central fluctuation interval is

${\displaystyle \left(\pi +z_{1-\alpha /2}{\frac {1}{\sqrt {4n}}}\right)-\left(\pi +z_{1-\alpha /2}{\frac {1}{\sqrt {4n}}}\right)={\frac {2z_{1-\alpha /2}}{\sqrt {4n}}}={\frac {z_{1-\alpha /2}}{\sqrt {n}}}}$,

and with , results in a length of . I.e. there is a 95% probability that the proportional value from the random sample will deviate from the true proportional value at most . With a true proportional value of , the central fluctuation interval is thus ; This great inaccuracy is one of the reasons why the press / polling institutes rarely mention the accuracy of forecasts. ${\displaystyle z_{0{,}975}=1{,}96}$ ${\displaystyle n=1000}$ ${\displaystyle 0{,}062=6{,}2\,\%}$ ${\displaystyle \pm 3{,}1\,\%}$ ${\displaystyle \pi }$ ${\displaystyle \pi =50\,\%}$ ${\displaystyle [46{,}9\,\%;53{,}1\,\%]}$

Central fluctuation interval and confidence interval

Derivation

The confidence intervals are derived directly from the central fluctuation intervals :

${\displaystyle P\left(\vartheta -c_{u}^{*}{\sqrt {Var(\theta )}}\leq \theta \leq \vartheta +c_{o}^{*}{\sqrt {Var(\theta )}}\right)=1-\alpha }$

• Subtract from ${\displaystyle \vartheta }$

${\displaystyle P\left(-c_{u}^{*}{\sqrt {Var(\theta )}}\leq \theta -\vartheta \leq +c_{o}^{*}{\sqrt {Var(\theta )}}\right)=1-\alpha }$

• Subtract from ${\displaystyle \theta }$

${\displaystyle P\left(-\theta -c_{u}^{*}{\sqrt {Var(\theta )}}\leq -\vartheta \leq -\theta +c_{o}^{*}{\sqrt {Var(\theta )}}\right)=1-\alpha }$

• Multiplication of ${\displaystyle -1}$

${\displaystyle P\left(\theta -c_{u}^{*}{\sqrt {Var(\theta )}}\leq \vartheta \leq \theta +c_{o}^{*}{\sqrt {Var(\theta )}}\right)=1-\alpha }$

And that gives the confidence interval.

differences

The following table sums up some of the differences between the central fluctuation interval and the confidence interval.

	Central fluctuation interval	Confidence interval
Limits	Are the same for every sample, i.e. fixed values	Change with every sample, so they are random variables
location	Includes the unknown population parameter	Includes the estimated parameter of the sample
interpretation	Indicates the probability with which the parameter estimated from the sample is included in the interval	Indicates what proportion of the estimation intervals contain the true parameter