Tolerance interval

Tolerant interval is a term used in mathematical statistics when determining on the basis of sample data an interval that at a given confidence level contains a given probability measure of an unknown distribution function.

The concepts of tolerance and confidence intervals are close to each other.

The tolerance interval is an interval in the sample space of observed random variables. It is determined by sufficient statistics based on the requirement that for a given confidence level it contains a probability measure of a statistical distribution that is not less than a given level. ^[one]

The confidence interval is determined for a certain parameter of the distribution function and is an interval in the parametric space. It is determined by sufficient statistics based on the requirement that the probability that it contains the true value of the unknown parameter is not less than the confidence level. ^[one]

Content

Definition

Let the random variable $Y$ ${\ displaystyle Y}$ does not depend on $X$ ${\ displaystyle X}$ and has a distribution function $F_{\theta }$ ${\ displaystyle F _ {\ theta}}$ . Tolerant spacing $(\beta ,\gamma )$ ${\ displaystyle (\ beta, \ gamma)}$ with measure $\beta$ ${\ displaystyle \ beta}$ and level of trust $\gamma$ ${\ displaystyle \ gamma}$ called interval $\left[L_{1,\beta ,\gamma }(X),L_{2,\beta ,\gamma }(X)\right]$ ${\ displaystyle \ left [L_ {1, \ beta, \ gamma} (X), L_ {2, \ beta, \ gamma} (X) \ right]}$ for which the condition is met $P_{\theta }\left\{P_{\theta }\left\{L_{1,\beta ,\gamma }(X)\leqslant Y\leqslant L_{2,\beta ,\gamma }(X)|X\right\}\geqslant \beta \right\}\geqslant \gamma$ ${\ displaystyle P _ {\ theta} \ left \ {P _ {\ theta} \ left \ {L_ {1, \ beta, \ gamma} (X) \ leqslant Y \ leqslant L_ {2, \ beta, \ gamma} ( X) | X \ right \} \ geqslant \ beta \ right \} \ geqslant \ gamma}$ for all parameter values $\theta$ ${\ displaystyle \ theta}$ . ^[2]

Explanation

Let be $\xi$ ${\ displaystyle \ xi}$ - quantile distribution function $F_{\theta }$ ${\ displaystyle F _ {\ theta}}$ denoted by $F^{-1}(\xi ;\theta )$ ${\ displaystyle F ^ {- 1} (\ xi; \ theta)}$ . By definition, we have $P_{\theta }\left\{F^{-1}(\xi _{1};\theta )\leqslant X\leqslant F^{-1}(\xi _{2};\theta )\right\}\geqslant \xi _{2}-\xi _{1}$ ${\ displaystyle P _ {\ theta} \ left \ {F ^ {- 1} (\ xi _ {1}; \ theta) \ leqslant X \ leqslant F ^ {- 1} (\ xi _ {2}; \ theta ) \ right \} \ geqslant \ xi _ {2} - \ xi _ {1}}$ . Interval measures $\beta$ ${\ displaystyle \ beta}$ distribution functions $F_{\theta }$ ${\ displaystyle F _ {\ theta}}$ called interval $\left[F^{-1}(\xi _{1};\theta ),F^{-1}(\xi _{2};\theta )\right]$ ${\ displaystyle \ left [F ^ {- 1} (\ xi _ {1}; \ theta), F ^ {- 1} (\ xi _ {2}; \ theta) \ right]}$ , if a $\beta =\xi _{2}-\xi _{1}$ ${\ displaystyle \ beta = \ xi _ {2} - \ xi _ {1}}$ . ^[3]

Notes

↑ ¹ ² Zacks, 1975 , p. 42
↑ Zacks, 1975 , p. 658.
↑ Zacks, 1975 , p. 657.

Literature

Sh. Zaks. The theory of statistical conclusions. - M .: Mir, 1975. - 776 p.

[_551502e7353879e4-1] ¹ ² Zacks, 1975 , p. 42

[_cb26cddf6ef0f707-2] Zacks, 1975 , p. 658.

[_cb26cddf6ef0f708-3] Zacks, 1975 , p. 657.