English+GermanWikipedia - /page/entry.mu

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Der Wald-Test ist in der <a href="%C3%96konometrie" title="Ökonometrie">Ökonometrie</a> ein <a href="Parametrische_Statistik" title="Parametrische Statistik">parametrischer</a> <a href="Statistischer_Test" title="Statistischer Test">statistischer Test</a>, der 1939 von <a href="Abraham_Wald" title="Abraham Wald">Abraham Wald</a> (1902–1950) entwickelt worden ist.

Mit dem Test kann die Verteilung einer geeigneten <a href="Teststatistik" title="Teststatistik">Teststatistik</a> unter Gültigkeit der <a href="Nullhypothese" class="mw-redirect" title="Nullhypothese">Nullhypothese</a> bestimmt werden. Eine allgemeine Teststatistik für verschiedenste ökonometrische Fragestellungen ist die Wald-Statistik, die asymptotisch einer <a href="Chi-Quadrat-Verteilung" title="Chi-Quadrat-Verteilung">Chi-Quadrat-Verteilung</a> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi ^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c0cc9237ec72a1da6d18bc8e7fb24cdda43a49a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:3.009ex;" alt="{\displaystyle \chi ^{2}}" loading="lazy">-Verteilung folgt. Der Wald-Test basiert auf der Tatsache, dass der <a href="Maximum-Likelihood-Methode" title="Maximum-Likelihood-Methode">Maximum-Likelihood-Schätzer</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\vartheta }}_{\text{ML}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\vartheta }}_{\text{ML}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d59e086a40cea5f8db9970a3cbb18538af01f608.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.228ex; height:3.176ex;" alt="{\displaystyle {\hat {\vartheta }}_{\text{ML}}}" loading="lazy"> für den unbekannten Parameter für große Beobachtungszahlen <a href="Konvergenz_in_Verteilung" title="Konvergenz in Verteilung">in Verteilung</a> gegen eine Normalverteilung strebt. Viele Tests lassen sich daher als Spezialfälle des Wald-Tests auffassen.

<div class="mw-heading mw-heading2"><h2 id="Eindimensionaler_Fall">Eindimensionaler Fall</h2></div>

Aus der Maximum-Likelihood-Theorie weiß man, dass der <a href="Maximum-Likelihood-Methode" title="Maximum-Likelihood-Methode">Maximum-Likelihood-Schätzer</a> des unbekannten Parameters in Verteilung für große Beobachtungszahlen gegen eine Normalverteilung strebt. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta }">

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<annotation encoding="application/x-tex">{\displaystyle \vartheta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d00eaf197c35bbfa391b9477490a4af955416837.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.374ex; height:2.176ex;" alt="{\displaystyle \vartheta }" loading="lazy"> ein unbekannter Parameter in der <a href="Grundgesamtheit" title="Grundgesamtheit">Grundgesamtheit</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \vartheta _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2d2fe479bcd30fad9d68cc45a1760ff38de03c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle \vartheta _{0}}" loading="lazy"> ein vorgegebener Wert. Um die folgende <a href="Nullhypothese" class="mw-redirect" title="Nullhypothese">Nullhypothese</a> gegen korrespondierende <a href="Alternativhypothese" class="mw-redirect" title="Alternativhypothese">Alternativhypothese</a> zu testen

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<annotation encoding="application/x-tex">{\displaystyle H_{0}\colon \vartheta =\vartheta _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef65bf0c10072f9fbbe45f846c84d69993d465e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.92ex; height:2.509ex;" alt="{\displaystyle H_{0}\colon \vartheta =\vartheta _{0}}" loading="lazy">  gegen  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}\colon \vartheta \neq \vartheta _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle H_{1}\colon \vartheta \neq \vartheta _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/777a887af25781165b5058bb7da71745df6bb516.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.92ex; height:2.676ex;" alt="{\displaystyle H_{1}\colon \vartheta \neq \vartheta _{0}}" loading="lazy">,</dd></dl>

kann man eine der folgenden Test-Statistiken benutzen:<a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {I({\hat {\vartheta }}_{\text{ML}})}}({\hat {\vartheta }}_{\text{ML}}-\vartheta _{0})\;{\stackrel {a}{\sim }}\;{\mathcal {N}}(0,1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88c08dc6278a43b7e814276bf7b74ddfe9d0a545.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:31.38ex; height:4.843ex;" alt="{\displaystyle {\sqrt {I({\hat {\vartheta }}_{\text{ML}})}}({\hat {\vartheta }}_{\text{ML}}-\vartheta _{0})\;{\stackrel {a}{\sim }}\;{\mathcal {N}}(0,1)}" loading="lazy"></dd></dl>

oder

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {J({\hat {\vartheta }}_{\text{ML}})}}({\hat {\vartheta }}_{\text{ML}}-\vartheta _{0})\;{\stackrel {a}{\sim }}\;{\mathcal {N}}(0,1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/901dad280d262e148cf494df561ddf8cd0ff1df3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:31.68ex; height:4.843ex;" alt="{\displaystyle {\sqrt {J({\hat {\vartheta }}_{\text{ML}})}}({\hat {\vartheta }}_{\text{ML}}-\vartheta _{0})\;{\stackrel {a}{\sim }}\;{\mathcal {N}}(0,1)}" loading="lazy">,</dd></dl>

die beide unter der Nullhypothese <a href="Asymptotische_Normalit%C3%A4t" title="Asymptotische Normalität">asymptotisch normalverteilt</a> sind. Hierbei bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\cdot )}">

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<annotation encoding="application/x-tex">{\displaystyle I(\cdot )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bbfc685a51919256883dacd3ad86eec3dc7a258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.628ex; height:2.843ex;" alt="{\displaystyle I(\cdot )}" loading="lazy"> die <a href="Fisher-Information" title="Fisher-Information">Fisher-Information</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(\cdot )}">

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<annotation encoding="application/x-tex">{\displaystyle J(\cdot )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6126a8ab97bab23dd0afd7f11698496b2ff2789.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle J(\cdot )}" loading="lazy"> die erwartete Fisher-Information. Beide Teststatistiken sind approximative <a href="Pivotgr%C3%B6%C3%9Fe" class="mw-redirect" title="Pivotgröße">Pivotgrößen</a> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta }">

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<annotation encoding="application/x-tex">{\displaystyle \vartheta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d00eaf197c35bbfa391b9477490a4af955416837.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.374ex; height:2.176ex;" alt="{\displaystyle \vartheta }" loading="lazy"> und werden Wald-Statistiken genannt.

Betrachtet man die quadrierte Teststatistik, so gilt:

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<annotation encoding="application/x-tex">{\displaystyle W:=F({\hat {\vartheta }}_{\text{ML}})({\hat {\vartheta }}_{\text{ML}}-\vartheta _{0})^{2}\;{\stackrel {a}{\sim }}\;\chi ^{2}(1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c0b9e32a039158d9043f09cd7b22d78796ecba9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.898ex; height:3.343ex;" alt="{\displaystyle W:=F({\hat {\vartheta }}_{\text{ML}})({\hat {\vartheta }}_{\text{ML}}-\vartheta _{0})^{2}\;{\stackrel {a}{\sim }}\;\chi ^{2}(1)}" loading="lazy">,</dd></dl>

d. h., sie ist bei großen Stichproben asymptotisch <a href="Chi-Quadrat-Verteilung" title="Chi-Quadrat-Verteilung">Chi-Quadrat-verteilt</a>. Dies gilt, da eine quadrierte standardnormalverteilte Zufallsgröße einer Chi-Quadrat-Verteilung mit einem Freiheitsgrad folgt.

<div class="mw-heading mw-heading3"><h3 id="Wald-Vertrauensintervall">Wald-Vertrauensintervall</h3></div>

Bezeichne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\vartheta }}_{\text{ML}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\vartheta }}_{\text{ML}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d59e086a40cea5f8db9970a3cbb18538af01f608.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.228ex; height:3.176ex;" alt="{\displaystyle {\hat {\vartheta }}_{\text{ML}}}" loading="lazy"> den <a href="Maximum-Likelihood-Sch%C3%A4tzer" class="mw-redirect" title="Maximum-Likelihood-Schätzer">Maximum-Likelihood-Schätzer</a> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta }">

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<annotation encoding="application/x-tex">{\displaystyle \vartheta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d00eaf197c35bbfa391b9477490a4af955416837.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.374ex; height:2.176ex;" alt="{\displaystyle \vartheta }" loading="lazy">, dann gilt für die Wahrscheinlichkeit, dass die Wald-Statistik innerhalb der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\alpha /2}">

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<annotation encoding="application/x-tex">{\displaystyle 1-\alpha /2}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d89cd111892baef3eb29d9b8943859a18ed4b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.815ex; height:2.843ex;" alt="{\displaystyle 1-\alpha /2}" loading="lazy">-Quantile der Standardnormalverteilung liegt

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<annotation encoding="application/x-tex">{\displaystyle P\left({-z_{\left(1-{\tfrac {\alpha }{2}}\right)}\leq {\sqrt {I({\hat {\vartheta }}_{\text{ML}})}}({\hat {\vartheta }}_{\text{ML}}-\vartheta )\leq z_{\left(1-{\tfrac {\alpha }{2}}\right)}}\right)\approx 1-\alpha }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7a95c9ce76f7d6b3e83f8ae83b38f6a3b274388.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.461ex; height:6.176ex;" alt="{\displaystyle P\left({-z_{\left(1-{\tfrac {\alpha }{2}}\right)}\leq {\sqrt {I({\hat {\vartheta }}_{\text{ML}})}}({\hat {\vartheta }}_{\text{ML}}-\vartheta )\leq z_{\left(1-{\tfrac {\alpha }{2}}\right)}}\right)\approx 1-\alpha }" loading="lazy"></dd></dl>

und damit ergibt sich das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\alpha )}">

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<annotation encoding="application/x-tex">{\displaystyle (1-\alpha )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1dcc25f05dca60e358d4d22e8342fad5ad7affbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.3ex; height:2.843ex;" alt="{\displaystyle (1-\alpha )}" loading="lazy">-Wald-<a href="Vertrauensintervall" class="mw-redirect" title="Vertrauensintervall">Vertrauensintervall</a> zu<a href="#cite_note-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle KI_{1-\alpha }(\vartheta )=\left[{\hat {\vartheta }}_{\text{ML}}-z_{\left(1-{\tfrac {\alpha }{2}}\right)}{\frac {1}{\sqrt {I({\hat {\vartheta }}_{\text{ML}})}}};{\hat {\vartheta }}_{\text{ML}}+z_{\left(1-{\tfrac {\alpha }{2}}\right)}{\frac {1}{\sqrt {I({\hat {\vartheta }}_{\text{ML}})}}}\right]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/767ba3913e2d4b38fc5b26e7d6fb55a4bc2e59d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:66.294ex; height:10.509ex;" alt="{\displaystyle KI_{1-\alpha }(\vartheta )=\left[{\hat {\vartheta }}_{\text{ML}}-z_{\left(1-{\tfrac {\alpha }{2}}\right)}{\frac {1}{\sqrt {I({\hat {\vartheta }}_{\text{ML}})}}};{\hat {\vartheta }}_{\text{ML}}+z_{\left(1-{\tfrac {\alpha }{2}}\right)}{\frac {1}{\sqrt {I({\hat {\vartheta }}_{\text{ML}})}}}\right]}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Mehrdimensionaler_Fall">Mehrdimensionaler Fall</h2></div>

Im mehrdimensionalen Fall, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\vartheta }}}=({\hat {\vartheta _{1}}},{\hat {\vartheta _{2}}},\dotsc ,{\hat {\vartheta _{k}}})^{\top }}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\vartheta }}}=({\hat {\vartheta _{1}}},{\hat {\vartheta _{2}}},\dotsc ,{\hat {\vartheta _{k}}})^{\top }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/019185c5dbb680f92057ab1f82986176c4243280.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.558ex; height:3.343ex;" alt="{\displaystyle {\hat {\boldsymbol {\vartheta }}}=({\hat {\vartheta _{1}}},{\hat {\vartheta _{2}}},\dotsc ,{\hat {\vartheta _{k}}})^{\top }}" loading="lazy"> der Vektor der <a href="Sch%C3%A4tzfunktion" title="Schätzfunktion">Schätzfunktionen</a> ist und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Sigma }}_{\hat {\boldsymbol {\vartheta }}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Sigma }}_{\hat {\boldsymbol {\vartheta }}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77053695cf82b2d0a6768d6f09fa4fe02734e4a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.366ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {\Sigma }}_{\hat {\boldsymbol {\vartheta }}}}" loading="lazy"> die asymptotische <a href="Regul%C3%A4re_Matrix" title="Reguläre Matrix">nichtsinguläre</a> <a href="Kovarianzmatrix" title="Kovarianzmatrix">Kovarianzmatrix</a> des Maximum-Likelihood-Schätzers ist, kann die Nullhypothese <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}:{\boldsymbol {\vartheta }}={\boldsymbol {\vartheta }}_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle H_{0}:{\boldsymbol {\vartheta }}={\boldsymbol {\vartheta }}_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11690b57a07663dba3ec3f2d05b13eca6de404e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.292ex; height:2.509ex;" alt="{\displaystyle H_{0}:{\boldsymbol {\vartheta }}={\boldsymbol {\vartheta }}_{0}}" loading="lazy"> mit folgender Teststatistik getestet werden<a href="#cite_note-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle W=({\hat {\boldsymbol {\vartheta }}}-{\boldsymbol {\vartheta }}_{0})^{\top }{\boldsymbol {\Sigma }}_{\hat {\boldsymbol {\vartheta }}}^{-1}({\hat {\boldsymbol {\vartheta }}}-{\boldsymbol {\vartheta }}_{0})\ \;{\stackrel {a,H_{0}}{\sim }}\;\chi ^{2}(k)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c20854490964e814e6c7ed16c6bb80bd960b2dd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:40.075ex; height:4.509ex;" alt="{\displaystyle W=({\hat {\boldsymbol {\vartheta }}}-{\boldsymbol {\vartheta }}_{0})^{\top }{\boldsymbol {\Sigma }}_{\hat {\boldsymbol {\vartheta }}}^{-1}({\hat {\boldsymbol {\vartheta }}}-{\boldsymbol {\vartheta }}_{0})\ \;{\stackrel {a,H_{0}}{\sim }}\;\chi ^{2}(k)}" loading="lazy"></dd></dl>

ist dann asymptotisch <a href="Chi-Quadrat-Verteilung" title="Chi-Quadrat-Verteilung">Chi-Quadrat-verteilt</a> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> <a href="Anzahl_der_Freiheitsgrade_(Statistik)" title="Anzahl der Freiheitsgrade (Statistik)">Freiheitsgraden</a>.

Die Restriktionsfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r({\hat {\vartheta }})=({\hat {\vartheta }}-\vartheta _{0})}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r({\hat {\vartheta }})=({\hat {\vartheta }}-\vartheta _{0})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90676fde275cca8a948d1c2e2f2177ccd9fcc1fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.957ex; height:3.343ex;" alt="{\displaystyle r({\hat {\vartheta }})=({\hat {\vartheta }}-\vartheta _{0})}" loading="lazy"> muss hierzu unter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43910602a221b7a4c373791f94793e3008622070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{0}}" loading="lazy"> vollständig differenzierbar sein und vollen Rang haben.

<div class="mw-heading mw-heading2"><h2 id="Wald-Statistiken_für_allgemeine_lineare_Hypothesen">Wald-Statistiken für allgemeine lineare Hypothesen</h2></div>

Um <a href="Testen_allgemeiner_linearer_Hypothesen" title="Testen allgemeiner linearer Hypothesen">allgemeine lineare Hypothesen zu testen</a>, spielt die asymptotische Verteilung der Wald-Statistik eine große Rolle. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {R}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {R}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7945da372695d49cd4229e2a84ac6dc8ae6c99b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.047ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {R}}}" loading="lazy"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\times (k+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\times (k+1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2d7b243c8f803da5a0ce57f885638007414e005.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.933ex; height:2.843ex;" alt="{\displaystyle q\times (k+1)}" loading="lazy"> Restriktionsmatrix, mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\leq (k+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\leq (k+1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee26d18053f4fd9d1a3ec5a3b9f0a8f51d83b975.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.191ex; height:2.843ex;" alt="{\displaystyle q\leq (k+1)}" loading="lazy"> Sei weiterhin angenommen, dass die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> Restriktionen an den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k+1)\times 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k+1)\times 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb0324bd0379f701eeac993029b3df27555519ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.026ex; height:2.843ex;" alt="{\displaystyle (k+1)\times 1}" loading="lazy"> Parametervektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/702cafc420cc00c54896f6d125112820956aaf6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.534ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}}" loading="lazy"> ausgedrückt werden können als :<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}:{\boldsymbol {R}}{\boldsymbol {\beta }}={\boldsymbol {r}}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{0}:{\boldsymbol {R}}{\boldsymbol {\beta }}={\boldsymbol {r}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55380d6ba7d3a629a5053cc338d561d4985273f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.832ex; height:2.509ex;" alt="{\displaystyle H_{0}:{\boldsymbol {R}}{\boldsymbol {\beta }}={\boldsymbol {r}}}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {r}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {r}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6a5e169814762d75ef0dd3a3d0bc99b4a5a06e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {r}}}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\times 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\times 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34eb51bf72eabafab73183262f081e14390100e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.072ex; height:2.509ex;" alt="{\displaystyle q\times 1}" loading="lazy">-Vektor bestehend aus bekannten Konstanten darstellt. Unter bestimmten Voraussetzungen folgt unter der Nullhypothese die gewichtete <a href="Summe_der_Abweichungsquadrate#Hypothesenquadratsumme" title="Summe der Abweichungsquadrate">Hypothesenquadratsumme</a>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W={\frac {1}{\sigma ^{2}}}({\boldsymbol {R}}{\boldsymbol {\hat {\beta }}}-{\boldsymbol {r}})^{\top }({\boldsymbol {R}}(\mathbf {X} ^{\top }\mathbf {X} )^{-1}{\boldsymbol {R}}^{\top })^{-1}({\boldsymbol {R}}{\boldsymbol {\hat {\beta }}}-{\boldsymbol {r}})\sim \chi ^{2}(q)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dfe55ad216ce83a745531f6d76193c84723ea7db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:57.367ex; height:5.509ex;" alt="{\displaystyle W={\frac {1}{\sigma ^{2}}}({\boldsymbol {R}}{\boldsymbol {\hat {\beta }}}-{\boldsymbol {r}})^{\top }({\boldsymbol {R}}(\mathbf {X} ^{\top }\mathbf {X} )^{-1}{\boldsymbol {R}}^{\top })^{-1}({\boldsymbol {R}}{\boldsymbol {\hat {\beta }}}-{\boldsymbol {r}})\sim \chi ^{2}(q)}" loading="lazy"></dd></dl>

einer <a href="Chi-Quadrat-Verteilung" title="Chi-Quadrat-Verteilung">Chi-Quadrat-Verteilung</a> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> (Anzahl der Restriktionen) <a href="Anzahl_der_Freiheitsgrade_(Statistik)" title="Anzahl der Freiheitsgrade (Statistik)">Freiheitsgraden</a>. Hierbei misst <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {R}}{\boldsymbol {\hat {\beta }}}-{\boldsymbol {r}}}">

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {R}}{\boldsymbol {\hat {\beta }}}-{\boldsymbol {r}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b280ba6cbac488009481fb61894d14acc24acdd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.775ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {R}}{\boldsymbol {\hat {\beta }}}-{\boldsymbol {r}}}" loading="lazy"> wie weit der geschätzte Wert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\hat {\beta }}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\hat {\beta }}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db74a112c44daacba85b2a69f8daabdf4208e155.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.658ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {\hat {\beta }}}}" loading="lazy"> von der Nullhypothese <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {R}}{\boldsymbol {\beta }}-{\boldsymbol {r}}=\mathbf {0} }">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {R}}{\boldsymbol {\beta }}-{\boldsymbol {r}}=\mathbf {0} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6a1c892fca510c2a87380d5b9b95f25cda1b10a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.087ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {R}}{\boldsymbol {\beta }}-{\boldsymbol {r}}=\mathbf {0} }" loading="lazy"> abweicht. Weiterhin ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\boldsymbol {R}}{\boldsymbol {\beta }}-{\boldsymbol {r}})^{\top }({\boldsymbol {R}}{\boldsymbol {\beta }}-{\boldsymbol {r}})}">

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {R}}{\boldsymbol {\beta }}-{\boldsymbol {r}})^{\top }({\boldsymbol {R}}{\boldsymbol {\beta }}-{\boldsymbol {r}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/787d0d845527e1b67eea02b5945666f81e181c63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.433ex; height:3.176ex;" alt="{\displaystyle ({\boldsymbol {R}}{\boldsymbol {\beta }}-{\boldsymbol {r}})^{\top }({\boldsymbol {R}}{\boldsymbol {\beta }}-{\boldsymbol {r}})}" loading="lazy"> die dazugehörige <a href="Summe_der_Abweichungsquadrate" title="Summe der Abweichungsquadrate">Summe der Abweichungsquadrate</a> (Analog zur <a href="Residuenquadratsumme" title="Residuenquadratsumme">Residuenquadratsumme</a>).

Diese Summe der Abweichungsquadrate wird mit der inversen Kovarianzmatrix der Nullhypothese <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\boldsymbol {R}}(\mathbf {X} ^{\top }\mathbf {X} )^{-1}{\boldsymbol {R}}^{\top })^{-1}/\sigma ^{2}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {R}}(\mathbf {X} ^{\top }\mathbf {X} )^{-1}{\boldsymbol {R}}^{\top })^{-1}/\sigma ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3349912f693dc215c340c26cf7a94f6c88e8e226.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.993ex; height:3.176ex;" alt="{\displaystyle ({\boldsymbol {R}}(\mathbf {X} ^{\top }\mathbf {X} )^{-1}{\boldsymbol {R}}^{\top })^{-1}/\sigma ^{2}}" loading="lazy"> gewichtet, weil für eine große Kovarianz ebenso so große Abweichungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {R}}{\boldsymbol {\hat {\beta }}}-{\boldsymbol {r}}}">

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {R}}{\boldsymbol {\hat {\beta }}}-{\boldsymbol {r}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b280ba6cbac488009481fb61894d14acc24acdd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.775ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {R}}{\boldsymbol {\hat {\beta }}}-{\boldsymbol {r}}}" loading="lazy"> nicht notwendigerweise ein Indikator für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43910602a221b7a4c373791f94793e3008622070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{0}}" loading="lazy"> sind. Falls der <a href="Erwartungstreue_Sch%C3%A4tzung_der_Varianz_der_St%C3%B6rgr%C3%B6%C3%9Fen" title="Erwartungstreue Schätzung der Varianz der Störgrößen">erwartungstreue Schätzer für die Störgrößenvarianz</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\sigma }}^{2}={\tfrac {1}{n-k-1}}\sum \nolimits _{i=1}^{n}{\hat {\varepsilon }}_{i}^{2}}">

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}^{2}={\tfrac {1}{n-k-1}}\sum \nolimits _{i=1}^{n}{\hat {\varepsilon }}_{i}^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d0b6a2b92da4c5bb2ddf2c313572f30acc1011a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.98ex; height:3.843ex;" alt="{\displaystyle {\hat {\sigma }}^{2}={\tfrac {1}{n-k-1}}\sum \nolimits _{i=1}^{n}{\hat {\varepsilon }}_{i}^{2}}" loading="lazy"> benutzt wird, kann man zeigen, dass die Wald-Statistik <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> dividiert durch die Anzahl der Restriktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> genau der <a href="Testen_allgemeiner_linearer_Hypothesen#F-Test_für_das_multiple_Regressionsmodell" title="Testen allgemeiner linearer Hypothesen">F-Statistik des multiplen linearen Testproblems</a> entspricht.<a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Einstichproben-Gauß-Test_als_Spezialfall_des_Wald-Tests">Einstichproben-Gauß-Test als Spezialfall des Wald-Tests</h3></div>

Wenn eine Variable in einer Grundgesamtheit normalverteilt ist mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim {\mathcal {N}}(\mu ;\sigma ^{2})}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\sim {\mathcal {N}}(\mu ;\sigma ^{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21e3f4903fec1d7ced803a0b287401c93a383990.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.983ex; height:3.176ex;" alt="{\displaystyle X\sim {\mathcal {N}}(\mu ;\sigma ^{2})}" loading="lazy"> mit unbekanntem Parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> und bekanntem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">, dann ist der <a href="Stichprobenmittelwert" class="mw-redirect" title="Stichprobenmittelwert">Stichprobenmittelwert</a>

<mover>

</mover>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\sim {\mathcal {N}}(\mu ,\sigma ^{2}/n)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0440f9372800f870022f0143d991b85a107e2a24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.874ex; height:6.843ex;" alt="{\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\sim {\mathcal {N}}(\mu ,\sigma ^{2}/n)}" loading="lazy"></dd></dl>

auch der <a href="Maximum-Likelihood-Methode" title="Maximum-Likelihood-Methode">Maximum-Likelihood-Schätzer</a> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">. Eine der Hypothesen für den <a href="Gau%C3%9F-Test#Einstichproben-Gauß-Test" title="Gauß-Test">Einstichproben-Gauß-Test</a> lautet:

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<annotation encoding="application/x-tex">{\displaystyle H_{0}\colon \mu =\mu _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f322c9ebd24b67502fba0562fb24f7af61efd7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.975ex; height:2.676ex;" alt="{\displaystyle H_{0}\colon \mu =\mu _{0}}" loading="lazy">  gegen  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}\colon \mu \neq \mu _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle H_{1}\colon \mu \neq \mu _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a9e81f92efb64aa071993ed12aad18d66271804.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.975ex; height:2.676ex;" alt="{\displaystyle H_{1}\colon \mu \neq \mu _{0}}" loading="lazy"></dd></dl>

und die Teststatistik nach Wald wäre

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<annotation encoding="application/x-tex">{\displaystyle T={\frac {{\overline {X}}-\mu _{0}}{\sigma /{\sqrt {n}}}}\sim {\mathcal {N}}(0,1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d96db64990412fa32b13541ada0399ddb578dca4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.54ex; height:7.176ex;" alt="{\displaystyle T={\frac {{\overline {X}}-\mu _{0}}{\sigma /{\sqrt {n}}}}\sim {\mathcal {N}}(0,1)}" loading="lazy">.</dd></dl>

Somit kann der Einstichproben-Gauß-Test als Spezialfall des Wald-Tests aufgefasst werden.

<div class="mw-heading mw-heading3"><h3 id="Globaler_F-Test_als_Spezialfall_des_Wald-Tests">Globaler F-Test als Spezialfall des Wald-Tests</h3></div>

Einen weiteren Spezialfall des Wald-Tests stellt der <a href="Globaler_F-Test" title="Globaler F-Test">globale F-Test</a> dar. Bei diesem wird geprüft, ob mindestens eine erklärende Variable einen Erklärungsgehalt für das Modell liefert. Falls diese Hypothese verworfen wird, ist somit das Modell nutzlos. Die <a href="Nullhypothese" class="mw-redirect" title="Nullhypothese">Nullhypothese</a> des F-Tests auf Gesamtsignifikanz des Modells sagt aus, dass alle erklärenden Variablen keinen Einfluss auf die abhängige Variable haben, und die <a href="Alternativhypothese" class="mw-redirect" title="Alternativhypothese">Alternativhypothese</a>, dass mindestens eine erklärende Variable Einfluss auf sie hat. Sowohl die erklärenden Variablen als auch die unabhängigen Variablen können binär (kategoriell) oder metrisch sein. Der Wald-Test kann dann die <a href="Hypothesentest" class="mw-redirect" title="Hypothesentest">Hypothesen testen</a> (ohne Einbezug des <a href="Regressionsparameter" title="Regressionsparameter">Achsenabschnitts</a>):<a href="#cite_note-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle H_{0}\colon \beta _{1}=\beta _{2}=\ldots =\beta _{k}\;=\;0\Rightarrow \rho ^{2}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2ff000550e2d2e5236dd5b3b499f6d622d9d8ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.865ex; height:3.176ex;" alt="{\displaystyle H_{0}\colon \beta _{1}=\beta _{2}=\ldots =\beta _{k}\;=\;0\Rightarrow \rho ^{2}=0}" loading="lazy">  gegen  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}:\beta _{j}\;\neq \;0\;\mathrm {f{\ddot {u}}r\;mindestens\;ein} \;j\in \{1,\ldots ,k\}\Rightarrow \rho ^{2}>0}">

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<annotation encoding="application/x-tex">{\displaystyle H_{1}:\beta _{j}\;\neq \;0\;\mathrm {f{\ddot {u}}r\;mindestens\;ein} \;j\in \{1,\ldots ,k\}\Rightarrow \rho ^{2}>0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/992c60633b7085223120201e38cfc4cf6b302eb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:56.39ex; height:3.343ex;" alt="{\displaystyle H_{1}:\beta _{j}\;\neq \;0\;\mathrm {f{\ddot {u}}r\;mindestens\;ein} \;j\in \{1,\ldots ,k\}\Rightarrow \rho ^{2}>0}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Alternativen">Alternativen</h2></div>

Eine Alternative zum Wald-Test bietet der <a href="Likelihood-Quotienten-Test" title="Likelihood-Quotienten-Test">Likelihood-Quotienten-Test</a>. Dieser ist zwar rechenaufwändiger, dafür zeigt er in kleinen Stichproben jedoch auch bessere Eigenschaften. Eine weitere Alternative ist der sogenannte Lagrange-Multiplikator-Tests (LM-Tests, siehe auch <a href="Lagrange-Multiplikator" title="Lagrange-Multiplikator">Lagrange-Multiplikator</a>). Asymptotisch sind diese drei Tests jedoch identisch.

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Wald's W-Statistics. In: Encyclopedia of Statistical Sciences. Wiley, Hoboken 2006, S. 9028–9029.</li>

<li>Abraham Wald: Tests of Statistical Hypotheses Concerning Several Parameters When the Number of Observations is Large. In: Transactions of the American Mathematical Society. Vol. 54, No. 3, Nov 1943, S. 426–482, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090/S0002-9947-1943-0012401-3">10.1090/S0002-9947-1943-0012401-3</a>, <a href="JSTOR" title="JSTOR">JSTOR</a>:<a rel="nofollow" class="external text" href="http://www.jstor.org/stable/1990256">1990256</a>.</li>

<li>Tim F. Liao: Comparing Social Groups: Wald Statistics for Testing Equality Among Multiple Logit Models. In: International Journal of Comparative Sociology. Vol. 45, No. 1–2, 2004, S. 3–16, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177/0020715204048308">10.1177/0020715204048308</a>.</li>

<li>Robert F. Engle: Wald, Likelihood Ratio and Lagrange Multiplier Tests in Econometrics. In: Zvi Griliches, Michael D. Intriligator (Hrsg.): Handbook of Econometrics. Vol. 2, Elsevier, Amsterdam u. a. 1984, S. 775–826.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Leonhard Held und Daniel Sabanés Bové: <a rel="nofollow" class="external text" href="https://www.springer.com/de/book/9783642378867">Applied Statistical Inference: Likelihood and Bayes.</a> Springer Heidelberg New York Dordrecht London (2014). ISBN 978-3-642-37886-7, S. 99.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Leonhard Held und Daniel Sabanés Bové: <a rel="nofollow" class="external text" href="https://www.springer.com/de/book/9783642378867">Applied Statistical Inference: Likelihood and Bayes.</a> Springer Heidelberg New York Dordrecht London (2014). ISBN 978-3-642-37886-7, S. 100.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> George G. Judge, R. Carter Hill, W. Griffiths, <a href="Helmut_L%C3%BCtkepohl" title="Helmut Lütkepohl">Helmut Lütkepohl</a>, T. C. Lee. Introduction to the Theory and Practice of Econometrics. 2. Auflage. John Wiley & Sons, New York / Chichester / Brisbane / Toronto / Singapore 1988, ISBN 0-471-62414-4, S. 109.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Jeffrey Marc Wooldridge: Introductory econometrics: A modern approach. 4. Auflage. Nelson Education, 2015, S. 810

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> <a href="Ludwig_Fahrmeir" title="Ludwig Fahrmeir">Ludwig Fahrmeir</a>, Rita Künstler, <a href="Iris_Pigeot" title="Iris Pigeot">Iris Pigeot</a>, <a href="Gerhard_Tutz" title="Gerhard Tutz">Gerhard Tutz</a>: Statistik. Der Weg zur Datenanalyse. 8., überarb. und erg. Auflage. Springer Spektrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3, S. 458.

</li>

</ol></div><div><div class="zim-footer">

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