Jackknife-Methode

Die Jackknife-Methode (englisch für ‚Taschenmesser‘) ist in der Statistik eine Methode des Resampling. Jackknife dient dazu, den zufälligen Fehler einer Schätzmethode und eine etwaige Verzerrung (engl. bias) zu schätzen. Aus Überlegungen zur Verbesserung der Jackknife-Methode entstand das Bootstrapping-Verfahren. Die Jackknife-Methode wurde 1956 bzw. 1958 zuerst von M. H. Quenouille und John W. Tukey veröffentlicht^[1]^[2]. Der Name soll die allgemeine Einsetzbarkeit der Methode für statistische Zwecke betonen.

Methode

Häufig wird Jackknife mit delete-1 Jackknife gleichgesetzt. Dabei wird aus der ursprünglichen Stichprobe $x_{1}\dots x_{N}$ jeweils ein Wert weggelassen und der Schätzer für diese reduzierte Stichprobe berechnet. Wird aus der ursprünglichen Stichprobe nicht nur ein Wert weggelassen, sondern d viele, so spricht man von delete-d Jackknife. Durch das Weglassen von $d$ von insgesamt $N$ Werten, können ${\binom {N}{d}}$ unterschiedliche reduzierte Stichproben erzeugt werden, die $N-d$ viele Werte haben.

Der Stichprobenmittelwert der ursprünglichen Stichprobe sei ${\hat {A}}={\frac {1}{N}}\sum _{k=1}^{N}x_{k}$ . Im Folgenden wird die Delete-1-Jackknife-Methode beschrieben. Der Mittelwert der reduzierten Jackknife-Stichprobe $i$ , welche durch Streichen des Wertes $X_{i}$ entsteht, sei:

A_{(i)}:={\frac {1}{N-1}}\sum _{k\neq i}x_{k}.

Dann ist der Mittelwert über alle Jackknife-Stichproben gegeben durch:

A_{(*)}:={\frac {1}{N}}\sum _{i=1}^{N}A_{(i)}

Die Varianz des Stichprobenmittelwertes kann durch folgende Formel abgeschätzt werden:^[3]

\operatorname {Var} ({\overline {x}})={\frac {N-1}{N}}\sum _{i=1}^{N}\left(A_{(i)}-{A}_{(*)}\right)^{2}

Die Jackknife-Methode liefert für die Verzerrung der Schätzfunktion ${\hat {A}}$ den geschätzten Wert:

{\widehat {\text{bias}}}_{\mathrm {(A)} }:=(N-1)({A}_{(*)}-{\hat {A}})

und somit ist der um die Verzerrung korrigierte Wert

{\hat {A}}_{\text{jack}}={\hat {A}}-{\widehat {\text{bias}}}_{\mathrm {A} }=N{\hat {A}}-(N-1){\hat {A}}_{\mathrm {(*)} }

Literatur

Joseph Lee Rodgers: The Bootstrap, the Jackknife, and the Randomization Test: A Sampling Taxonomy. Multivariate Behavioral Research, 34, Nr. 4 S. 441ff (1999)

Einzelnachweise

↑ M. H. Quenouille: Notes on bias in estimation. Biometrika, 43, S. 353ff (1956)
↑ J. W. Tukey: Bias and confidence in not quite large samples. Annls. Math. Stat. 29, S. 614 (1958)
↑ Bradley Efron, Charles Stein: The Jackknife Estimate of Variance. The Annals of Statistics, 9(3), S. 586–596 (1981)

Read other articles:

NewJeans discography

NewJeans discographyNewJeans in 2022EPs2Singles8Single albums1Remix albums1Promotional singles5 South Korean girl group NewJeans has released two extended plays, one single album, eight singles, and three promotional singles. The group's first release, their self-titled extended play (2022), topped the South Korean Circle Album Chart and became the best-selling debut album by a K-pop female act, with over one million copies sold.[1] It was supported by three singles: Attention, Hype ...

List of countries ranked by ethnic and cultural diversity level

List of countries ranked by ethnic and cultural diversity level. List based on Fearon's analysis This article contains lists of countries ranked by ethnic and cultural diversity level. Methodology The lists are commonly used in economics literature to compare the levels of ethnic, cultural, linguistic and religious fractionalization in different countries.[1][2] Fractionalization is the probability that two individuals drawn randomly from the country's groups are not from the...

Beaver Valley, Delaware and Pennsylvania

This article possibly contains original research. Please improve it by verifying the claims made and adding inline citations. Statements consisting only of original research should be removed. (January 2014) (Learn how and when to remove this template message) Unincorporated community in Delaware and Pennsylvania, United StatesBeaver Valley, Delaware and PennsylvaniaUnincorporated communityA vista in Beaver Valley, PennsylvaniaBeaver ValleyShow map of DelawareBeaver ValleyShow map of Pennsylv...

مقاطعة كاسيا (أيداهو)

هذه المقالة تحتاج للمزيد من الوصلات للمقالات الأخرى للمساعدة في ترابط مقالات الموسوعة. فضلًا ساعد في تحسين هذه المقالة بإضافة وصلات إلى المقالات المتعلقة بها الموجودة في النص الحالي. (مارس 2018) مقاطعة كاسيا الإحداثيات 42°16′N 113°37′W / 42.27°N 113.61°W / 42.27; -113.61...

Révolte spartakiste de Berlin

Révolte spartakiste de Berlin Barricade à Berlin durant le soulèvement, en janvier 1919. Informations générales Date 5 janvier - 12 janvier 1919 Lieu Berlin Issue Victoire gouvernementale Belligérants Conseil des commissaires du peuple Parti communiste d'AllemagneLigue spartakisteParti social-démocrate indépendant d'Allemagne Commandants Gustav Noske Karl LiebknechtRosa Luxemburg Forces en présence 3 000 hommes Pertes 17 morts20 blessés 139-180 morts Données clés modifi...

Sindrom Crouzon

Sindrom Crouzon Sindrom Crouzon merupakan penyakit autosomal dominan dengan gejala yang bervariasi yang disebabkan oleh mutasi gen pertumbuhan FGFR2 (Fibroblast Growth Factor Receptor 2) pada kromosom 10.[1][2] Penyakit ini dikarakteristikkan dengan tulang calvaria yang terlalu cepat menutup dan sutura basis kranial dan juga seperti halnya orbital dan maksila secara kompleks (craniosynostosis).[1] Kranium tersusun atas beberapa tulang yang dipisahkan oleh sutura.[1...

Valdir Bigode

Valdir Bigode Nazionalità Brasile Altezza 180[1] cm Peso 79[1] kg Calcio Ruolo Allenatore (ex attaccante) Termine carriera 2007 - giocatore Carriera Giovanili ?-1992 Vasco da Gama Squadre di club1 1992-1994 Vasco da Gama36 (12)1995-1996 San Paolo16 (11)1996 Atlético Mineiro11 (5)1996 Benfica10 (3)1997 San Paolo9 (0)1997-1998 Atlético Mineiro50 (34)1999 Botafogo20 (5)1999 Santos21 (0)2000 Atlético Mineiro16 (2)2001-20...

Gropiusstadt

Quarter of Berlin in GermanyGropiusstadt Quarter of Berlin Overview of the Gropiusstadt ComplexLocation of Gropiusstadt in Neukölln district and Berlin Gropiusstadt Show map of GermanyGropiusstadt Show map of BerlinCoordinates: 52°25′33″N 13°27′41″E / 52.42583°N 13.46139°E / 52.42583; 13.46139CountryGermanyStateBerlinCityBerlin BoroughNeukölln Founded1960Area • Total2.66 km2 (1.03 sq mi)Elevation52 m (171 ft)Population...

June 1927 Irish general election

Election to the 5th Dáil June 1927 Irish general election ← 1923 9 June 1927 Sep 1927 → ← outgoing memberselected members →153 seats in Dáil Éireann[a]77 seats needed for a majorityTurnout68.1% 6.8pp First party Second party Third party Leader W. T. Cosgrave Éamon de Valera Thomas Johnson Party Cumann na nGaedheal Fianna Fáil Labour Leader since April 1923 26 March 1926 1922 Leader's seat Carlow–Kilkenny C...

Secret Combination (song)

Song by Greek American singer Kalomira Secret CombinationSingle by Kalomirafrom the album Secret Combination Released26 February 2008 (2008-02-26)Genre Pop R&B Length3:02LabelHeavenComposer(s)Konstantinos PantzisLyricist(s)Poseidonas GiannopoulosKalomira singles chronology Gine Mazi Mou Paidi (2006) Secret Combination (2008) Please Don't Break My Heart (2010) Music videoSecret Combination on YouTubeEurovision Song Contest 2008 entryCountryGreeceArtist(s)KalomiraLanguageEngl...

羅德島州

本條目存在以下問題，請協助改善本條目或在討論頁針對議題發表看法。此條目需要擴充。 (2013年1月1日)请協助改善这篇條目，更進一步的信息可能會在討論頁或扩充请求中找到。请在擴充條目後將此模板移除。此條目需要补充更多来源。 (2013年1月1日)请协助補充多方面可靠来源以改善这篇条目，无法查证的内容可能會因為异议提出而被移除。致使用者：请搜索一下条目的...

世界政區索引

此條目可能包含不适用或被曲解的引用资料，部分内容的准确性无法被证實。 (2023年1月5日)请协助校核其中的错误以改善这篇条目。详情请参见条目的讨论页。各国相关主題列表索引国内生产总值石油储量国防预算武装部队（军事）官方语言人口統計人口密度生育率出生率死亡率自杀率谋杀率失业率储蓄率识字率出口额进口额煤产量发电量监禁率死刑国债 ...

لغات ميزاب-ورقلة

هذه مقالة غير مراجعة. ينبغي أن يزال هذا القالب بعد أن يراجعها محرر؛ إذا لزم الأمر فيجب أن توسم المقالة بقوالب الصيانة المناسبة. يمكن أيضاً تقديم طلب لمراجعة المقالة في الصفحة المخصصة لذلك. (أبريل 2021) مزاب-ورغلةالتوزيعالجغرافي:شمال إفريقياتصنيفات اللغوية:أفريقية آسيويةأما�...

Lori Grimes

Fictional character Lori GrimesThe Walking Dead characterLori Grimes, as she appears in the comic book series (left) and as portrayed by Sarah Wayne Callies in the television series (right).First appearance Comic: Issue #2 (2003) Rick Grimes 2000: Issue #75 (2010) Television: Days Gone Bye (2010) Literature: The Fall of the Governor - Part Two (2014) Last appearance Comic: Issue #55 (2008) Rick Grimes 2000: Skybound X (2021) Television: Say the Word (2012) Literature: The Fall of the Governor...

Spitsbergen

Bài này viết về một hòn đảo. Đối với quần đảo mà đảo này là một bộ phận, xem Svalbard. Spitsbergen Bản đồ Svalbard với Spitsbergen được tô đậm ở phía tâyĐịa lýVị tríBắc Băng DươngTọa độ78°45′B 16°00′Đ / 78,75°B 16°Đ / 78.750; 16.000Quần đảoSvalbardDiện tích39.044 km2 (15.075 mi2)Hạng diện tích36Độ cao tương đối lớn nhất1.717 m (5.633 ft)Đỉnh c...

Sarah Baartman

Khoikhoi woman (c. 1789–1815) For the South African patrol vessel, see Sarah Baartman (ship). Sarah BaartmanBornSsehura?[1]c. 1789[2]Camtoos Valley, Eastern CapeDied29 December 1815 (aged 26)Paris, Kingdom of France(now France)Resting placeVergaderingskop, Hankey, Eastern Cape, South Africa33°50′14″S 24°53′05″E / 33.8372°S 24.8848°E / -33.8372; 24.8848Other namesHottentot Venus, Saartjie Baartman Sarah Baartman (Afrikaans: ...

Краснобродский городской округ

Городской округКраснобродский городской округ Флаг 54°09′29″ с. ш. 86°26′34″ в. д.HGЯO Страна Россия Субъект федерации Кемеровская область Включает 3 населённых пункта Адм. центр Краснобродский Глава территориального отдела Тамашева Татьяна Ивановна История и ...

جيوف هورست

جيوف هورست معلومات شخصية الميلاد 8 ديسمبر 1941 (العمر 82 سنة)[1]أشتون أندر لاين [لغات أخرى]‏ الطول 1.82 م (5 قدم 11 1⁄2 بوصة)[2][2] مركز اللعب مهاجم الجنسية المملكة المتحدة[3] مسيرة الشباب سنوات فريق 1957–1959 West Ham United F.C. Under-21s and Academy [الإنجليز�...

Leopold Infeld

Polish physicist Leopold InfeldLeopold Infeld in 1960Born20 August 1898 (1898-08-20)Kraków, then Austria-Hungary, now PolandDied15 January 1968 (1968-01-16) (aged 69)Warsaw, PolandCitizenshipAustrian (1898–1918)Polish (1918–1968)Canadian (1939–50)Alma materJagiellonian UniversityKnown forInfeld–Van der Waerden symbolsBorn–Infeld theory Einstein–Infeld–Hoffmann equations Infeld–Hull factorization methodChildrenEryk Infeld [pl]Scientific ca...

Luapula River

Tributary of the Congo River This article is about the river. For the genus of crickets, see Laupala. The Luapula River is the middle stretch in red. The Luapula (bottom) forms a swampy delta where it enters Lake Mweru (center, in black), as shown in this false-colour NASA satellite image. Blue lines mark the swamp and its islands (9, 10); a dotted line marks the river's floodplain (11). Red dots (1–8) are towns. See note for full legend.[note 1] The Luapula River is a north-flowing...

Tochi Raina

/profillengkap.com/index.php/article/Funk B

Honolulu Challenger 2011 – Tunggal

Klumpit, Gebog, Kudus

Aindrita Ray

Screams Without Words

Reaktor Nuklir Ignalina

مجلس صيانة الدستور

جائحة فيروس كورونا في كرواتيا

WAGE-LP

Billy Brick

List of Manchester City F.C. seasons