A pivot statistic , also known as a pivot size , or a pivot for short , is a special function in mathematical statistics . These are statistics with certain invariance properties that are used to construct confidence ranges . The name is derived from the French pivot (German anchor, here in the sense of pivot and pivot point) and is based on the invariance properties.
definition
A statistical model is given
A pivot is a random variable as a function of the sample variable and , the distribution of which is independent of .
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![\ vartheta](https://wikimedia.org/api/rest_v1/media/math/render/svg/d00eaf197c35bbfa391b9477490a4af955416837)
![\ vartheta](https://wikimedia.org/api/rest_v1/media/math/render/svg/d00eaf197c35bbfa391b9477490a4af955416837)
A pivot statistic is strictly formally defined as follows: A decision space and a function to be estimated are given
![{\ displaystyle (\ Gamma, {\ mathcal {\ mathcal {A}}} _ {\ Gamma})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00f7bcd2e4cd8a1e10ea49a3333519394da2bb59)
-
.
Mostly is . Then it is called a measurable figure
![{\ displaystyle T \ colon {\ mathcal {X}} \ times \ Gamma \ to \ Gamma}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89820a4b04316e93d4c5756d9d419aa60eb64da8)
a pivot statistic for if it fulfills the following properties:
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
- For all and all the amount is in included.
![{\ displaystyle B \ in {\ mathcal {A}} _ {\ Gamma}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f516ad6730749d84401bbbe9868e1b2d8abf3a31)
![\ vartheta \ in \ Theta](https://wikimedia.org/api/rest_v1/media/math/render/svg/bec0011b0631782a18328c0582af915b43a061ca)
![{\ displaystyle \ {x \ in {\ mathcal {X}} \ mid T (x, g (\ vartheta)) \ in B \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25f3dc75fbaca47331a2bb848974194dc8ba8daa)
![\ mathcal A](https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8)
- There is a probability distribution on , so for all always applies.
![Q](https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed)
![{\ displaystyle (\ Gamma, {\ mathcal {A}} _ {\ Gamma})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0c5cbcc5e21285223018a34b393222aac6f0af)
![\ vartheta \ in \ Theta](https://wikimedia.org/api/rest_v1/media/math/render/svg/bec0011b0631782a18328c0582af915b43a061ca)
![{\ displaystyle P _ {\ vartheta} \ circ T (\ cdot, g (\ vartheta)) ^ {- 1} = Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2be8da06c7b680404fdbea635b3c1eaa967a483)
example
A fixed one is given and the normal distribution with expected value and variance is assumed . Let be n times the product measure .
![{\ displaystyle \ sigma _ {0}> 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06919e9a09e767e4c8b9a7c41c86fa8e33c03109)
![{\ displaystyle \ sigma _ {0} ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48d1e886ce1a6572f6f726d3ac797d392a01d3ea)
![{\ displaystyle {\ mathcal {N}} ^ {n} (\ mu, \ sigma _ {0} ^ {2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c13b8622bf888d71ee466f099d99694c0e2d6bbd)
The product model with a fixed variance and an unknown expected value is considered as a statistical model .
![{\ displaystyle (\ mathbb {R} ^ {n}, {\ mathcal {B}} (\ mathbb {R} ^ {n}), ({\ mathcal {N}} ^ {n} (\ mu, \ sigma _ {0} ^ {2})) _ {\ mu \ in \ mathbb {R}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11cf15d051bc5f654a27373c35acafe4ff0713c7)
Then a pivot statistic is given by
-
.
Here is
![{\ displaystyle {\ overline {X}} = {\ frac {1} {n}} \ sum _ {i = 1} ^ {n} X_ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bba8c8160cf89fc97efe65f1c8c47126eded5c1)
the sample mean . The fact that it is a pivot follows directly from the calculation rules for normally distributed random variables (see invariance of the normal distribution versus convolution ), because it is . By normalizing with the standard deviation , one obtains that there is always standard normal distribution , i.e. for all .
![{\ displaystyle {\ overline {X}} - \ mu \ sim {\ mathcal {N}} (0, \ sigma _ {0} ^ {2} / n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c43d20ea440dfd8d10fc37929cc106da471abd)
![{\ displaystyle \ sigma _ {0} / {\ sqrt {n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb0b26975fc49df109d70291776cf59a02a805e)
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
![{\ displaystyle T (\ cdot, \ mu) \ sim {\ mathcal {N}} (0,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd04cb38b75c8b0ae2d35feaa6efcd137fae161f)
![{\ displaystyle \ mu \ in \ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a48f0e84328dc53dec2ad301bb321c00dcf422)
Construction of confidence areas using pivots
If a pivot statistic exists and a set is given, then through
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
![{\ displaystyle B \ in {\ mathcal {A}} _ {\ Gamma}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f516ad6730749d84401bbbe9868e1b2d8abf3a31)
![{\ displaystyle C_ {B} (x) = \ {\ gamma \ in \ Gamma \ mid T (x, \ gamma) \ in B \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eac373e8b5daa337e2112da0b9d9eeaae4875833)
an area estimator is defined. Based on the definition of the range estimator, then
![{\ displaystyle \ {x \ in {\ mathcal {X}} \ mid g (\ vartheta) \ in C_ {B} (x) \} = \ {x \ in {\ mathcal {X}} \ mid T ( x, g (\ vartheta)) \ in B \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec0ebaf7e7758e6c0026748f38b2d286ae3ca391)
and thus
![{\ displaystyle P _ {\ vartheta} (\ {x \ in {\ mathcal {X}} \ mid g (\ vartheta) \ in C_ {B} (x) \}) = P _ {\ vartheta} (\ {x \ in {\ mathcal {X}} \ mid T (x, g (\ vartheta)) \ in B \}) = Q (B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f53a156df42f1374b6f3e38deb5915fea1e29c)
for everyone due to the pivot property of . The range estimator thus provides a confidence range for the confidence level . The choice of the set thus determines the confidence level and geometry of the confidence range.
![\ vartheta \ in \ Theta](https://wikimedia.org/api/rest_v1/media/math/render/svg/bec0011b0631782a18328c0582af915b43a061ca)
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
![{\ displaystyle C_ {B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82e66cdda0ceb560bfab11747176e76650cf35d6)
![{\ displaystyle Q (B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9368557c9be8c8cc5573541bd80832643592621c)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
Example for the construction of confidence ranges
Under the same conditions as in the example above, a confidence range is to be determined for the mean value for the confidence level . There is a set must first be chosen so that
![{\ displaystyle 1- \ alpha}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9afa7876fb8b4fb8c4d8039ebed6cd1cbc4781cd)
![{\ displaystyle Q = {\ mathcal {N}} (0,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79c40abc66bbd6a708b8d8a0ca7d92b9a66580eb)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
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.
The choice of depends largely on the application. One-sided confidence intervals are common
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
-
or
or two-sided confidence intervals
-
.
You must now choose that for is. For this one selects the appropriate - quantiles of the standard normal distribution and obtains as well as and .
![a_1, a_2, a_3](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd8e11b0b055a7c3471e39b9742a8e7df9883e99)
![{\ displaystyle {\ mathcal {N}} (0.1) (B_ {i}) = 1- \ alpha}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee6aba9e7e85fddb670a0edfa5e965c387fe976)
![{\ displaystyle i = 1,2,3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a87b7e5d81c62b040a35078417fb9294b1ec7c7)
![{\ displaystyle u_ {p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a113d37c1d0f165030579e70c73c7608eb9c1e4d)
![{\ displaystyle a_ {1} = u _ {\ alpha}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62c90f88d74122e9339c00e2eab1ef6ebfa05615)
![{\ displaystyle a_ {2} = u_ {1- \ alpha}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b29c23406d5a4b09d60671a13588079d2b87f6)
![{\ displaystyle a_ {3} = u_ {1- \ alpha / 2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5339a3b988f9bc75cddceef7304a965f5561a58)
This results for the range estimator with the quantity
-
,
because due to the symmetry of the standard normal distribution applies.
![{\ displaystyle u_ {1- \ alpha} = - u _ {\ alpha}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88f39ee104e4c7630f196d2cf79b3735b2ad7f48)
The one-sided confidence interval for the confidence level for the expected value is thus obtained
![{\ displaystyle 1- \ alpha}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9afa7876fb8b4fb8c4d8039ebed6cd1cbc4781cd)
-
.
By proceeding in the same way with the quantities and , the second one-sided confidence interval is obtained
![{\ displaystyle B_ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/199944d59dcc18842dfd1deab6000a1d1dadcbae)
![{\ displaystyle B_ {3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db17e43bc3f2416503df52a301a7d2779ab3a8ac)
![{\ displaystyle C_ {B_ {2}} (X) = \ left (- \ infty, {\ overline {X}} + {\ frac {\ sigma _ {0} u_ {1- \ alpha}} {\ sqrt {n}}} \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12b6485b2811bc3073465bf365cfdc98e892b5f1)
and as a two-sided confidence interval
-
.
Related concepts
The approximate pivot statistics are closely related to the pivot statistics . They serve to construct approximate confidence ranges and are based on limit value considerations.
Individual evidence
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^ Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 234 , doi : 10.1515 / 9783110215274 .
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↑ Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 142 , doi : 10.1007 / 978-3-642-17261-8 .
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^ A b Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 231 , doi : 10.1007 / 978-3-642-41997-3 .