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{{Short description|Hungarian mathematician}} | |||
{{Infobox scientist | {{Infobox scientist | ||
|name = Abraham Wald | | name = Abraham Wald | ||
|image = Abraham Wald in his youth.jpg | | image = Abraham Wald in his youth.jpg | ||
| caption = A young Wald | |||
|caption = A young Wald | | birth_date = {{birth date|1902|10|31}} | ||
|birth_date = {{birth date|1902|10|31}} | | birth_place = [[Cluj-Napoca|Kolozsvár]], [[Kingdom of Hungary]], [[Austria-Hungary]] | ||
|birth_place = [[Cluj-Napoca|Kolozsvár]], [[Kingdom of Hungary]] | | death_date = {{death date and age|1950|12|13|1902|10|31}} | ||
|death_date = {{death date and age|1950|12|13|1902|10|31}} | | death_place = [[Nilgiri Mountains]], [[India]] | ||
|death_place = [[Nilgiri Mountains]], [[India | | field = [[Mathematics]]<br />[[Statistics]]<br />[[Mathematical economics|Economics]] | ||
| work_institutions = [[Columbia University]]<br />[[Cowles Commission for Research in Economics]] | |||
| alma_mater = [[Babeș-Bolyai University|King Ferdinand I University]]<br/>[[University of Vienna]] | |||
| doctoral_advisor = [[Karl Menger]] | |||
|field = [[Mathematics]]<br /> [[Statistics]] <br /> [[Mathematical economics|Economics]] | | doctoral_students = [[Herman Chernoff]]<br />[[Meyer Abraham Girshick]]<br />[[Milton Sobel]]<br />[[Charles Stein (statistician)|Charles Stein]] | ||
|work_institutions = [[Columbia University]] <br /> [[Cowles Commission for Research in Economics]] | | known_for = [[Wald's equation]]<br />[[Wald test]]<br />[[Wald distribution]]<br />[[Wald–Wolfowitz runs test]]<br />[[Wald's martingale]]<br />[[Wald's maximin model]]<br />[[Mann–Wald theorem]]<br />[[Decision theory]]<br />[[Sequential analysis]]<br />[[Sequential probability ratio test]] | ||
|alma_mater = [[Babeș-Bolyai University|King Ferdinand I University]]<br/>[[University of Vienna]] | | author_abbrev_bot = | ||
|doctoral_advisor = [[Karl Menger]] | | author_abbrev_zoo = | ||
|doctoral_students = [[Herman Chernoff]]<br />[[Milton Sobel]] <br /> [[Charles Stein (statistician)|Charles Stein]] | | prizes = | ||
|known_for = [[Wald's equation]] <br /> [[Wald test]] <br /> [[Wald distribution]] <br /> [[Wald–Wolfowitz runs test]] <br /> [[Wald's martingale]] <br /> [[Wald's maximin model]] <br /> [[Mann–Wald theorem]] <br /> [[Decision theory]] <br> [[Sequential analysis]] <br /> [[Sequential probability ratio test]] | | footnotes = | ||
|author_abbrev_bot = | | children = [[Robert Wald]] | ||
|author_abbrev_zoo = | | signature = | ||
|prizes = | |||
|footnotes = | |||
|children = [[Robert Wald]] | |||
|signature = | |||
}} | }} | ||
'''Abraham Wald''' ({{IPAc-en|w|ɔː|l|d}}; {{lang | '''Abraham Wald''' ({{IPAc-en|w|ɔː|l|d}}; {{IPA|de|valt|lang}}; {{langx|hu|Wald Ábrahám}}, {{langx|yi|אברהם וואַלד}}; {{birth date|1902|10|31|df=y}} – {{death date|1950|12|13|df=y}}) was a Hungarian and American mathematician and statistician who contributed to [[decision theory]],<ref name=DuppeWeintraub2015>{{cite journal | ||
| author = Duppe, Till | |||
| author-link = Till Duppe | |||
| author2= Weintraub, E. Roy | |||
| author2-link = E. Roy Weintraub | |||
| journal = History of Political Economy | |||
| volume = 48 | |||
| issue = 191 | |||
| pages = 635–655 | |||
| year = 2015 | |||
| title = Losing Equilibrium: On the Existence of Abraham Wald's Fixed-Point Proof of 1935 | |||
| publisher = Econometrica, Vol. 19, No. 4 | |||
| doi = 10.1215/00182702-3687283 | |||
| hdl = 10419/149731 | |||
| hdl-access = free | |||
}}</ref> [[geometry]] and [[econometrics]], and founded the field of [[sequential analysis]].<ref name=Morgenstern1951>{{cite journal | |||
| doi = 10.2307/1907462 | | doi = 10.2307/1907462 | ||
| author = Morgenstern, Oskar | | author = Morgenstern, Oskar | ||
| Line 40: | Line 49: | ||
| publisher = Econometrica, Vol. 19, No. 4 | | publisher = Econometrica, Vol. 19, No. 4 | ||
| jstor = 1907462 | | jstor = 1907462 | ||
}}</ref> One of | }}</ref> One of his well-known statistical works was written during [[World War II]] on how to minimize the damage to bomber aircraft and took into account the [[survivorship bias]] in his calculations.<ref name="Mangel1984">{{cite journal |last1=Mangel |first1=Marc |last2=Samaniego |first2=Francisco J. |title=Abraham Wald's Work on Aircraft Survivability |journal=Journal of the American Statistical Association |date=June 1984 |volume=79 |issue=386 |pages=259–267 |doi=10.1080/01621459.1984.10478038 |publisher=[[American Statistical Association]] |url=http://people.ucsc.edu/~msmangel/Wald.pdf |jstor=2288257}}</ref> He spent his research career at [[Columbia University]]. He was the grandson of Rabbi [[Moshe Shmuel Glasner]]. | ||
== Life and career == | == Life and career == | ||
[[File:Abraham Wald.jpg|thumb|left|upright|Photograph of Abraham Wald from the | [[File:Abraham Wald.jpg|thumb|left|upright|Photograph of Abraham Wald from the [[Oberwolfach Research Institute for Mathematics]]]] | ||
Wald was born on 31 October 1902 in [[Cluj-Napoca|Kolozsvár]], [[Transylvania]], in the [[Kingdom of Hungary]]. A religious [[Jew]], he did not attend school on Saturdays, as was required | Wald was born on 31 October 1902 in [[Cluj-Napoca|Kolozsvár]], [[Transylvania]], in the [[Kingdom of Hungary]]. A religious [[Jew]], he did not attend school on Saturdays, as was then required by the Hungarian school system, and so he was [[homeschooled]] by his parents until college.<ref name=Morgenstern1951/> His parents were quite knowledgeable and competent as teachers.<ref>{{MacTutor Biography|id=Wald}}</ref> | ||
In 1928, he graduated in mathematics from the [[Babeș-Bolyai University|King Ferdinand I University]].<ref>Anuarul Universității Regele Ferdinand I pe anul școlar 1927/28. p. 187. [http://documente.bcucluj.ro/web/bibdigit/periodice/anuaruluniversitatiidincluj/1927-1928/BCUCLUJ_FP_452283_1_1927_1928_007_0001.pdf Online access], University Library in Cluj, Romania.</ref> In 1927, he had entered [[Postgraduate education|graduate school]] at the [[University of Vienna]], from which he graduated in 1931 with a Ph.D. in mathematics. His advisor there was [[Karl Menger]].<ref name=Morgenstern1951/><ref name=DuppeWeintraub2015/> | |||
Despite Wald's brilliance, he could not obtain a university position because of Austrian discrimination against Jews. However, [[Oskar Morgenstern]] created a position for Wald in economics. When [[Nazi Germany]] [[Anschluss|annexed Austria]] in 1938, the discrimination against Jews intensified. In particular, Wald and his family were persecuted as Jews. Wald emigrated to the [[United States]] at the invitation of the [[Cowles Commission for Research in Economics]], to work on econometrics research.<ref name=Morgenstern1951/> By September 1938 Wald was a Fellow of the Carnegie Corporation at Columbia University learning about modern English language statistics from [[Harold Hotelling]]. He was appointed to the Columbia Faculty in 1941 and remained there until his death. | |||
In | |||
[[File:Survivorship-bias.svg|thumb|left|The damaged portions of returning planes show locations where they can sustain damage and still return home; those hit in other places presumedly do not survive. (Image shows hypothetical data.)]] | |||
During [[World War II]], Wald was a member of the [[The Statistical Research Group of World War II|Statistical Research Group]] (SRG) at [[Columbia University]], where he applied his statistical skills to various wartime problems.<ref name="Wallis1980">{{cite journal|doi=10.1080/01621459.1980.10477469 | volume=75 | issue=370 | title=The Statistical Research Group, 1942–1945 | journal=Journal of the American Statistical Association | pages=320–330| year=1980 | last1=Wallis | first1=W. Allen |jstor=2287451}}</ref> There was a difficulty: the work was secret and Wald was officially an enemy alien and, as such, barred from access to restricted matter. Hotelling recounts, “This impasse led to a facetious suggestion that each page he wrote should immediately be snatched away and never shown to him again, but was resolved when a federal court granted him a hearing long before his turn on the docket and promptly naturalized him.” <ref>{{cite journal |last=Hotelling|first= Harold|title= Abraham Wald |journal= American Statistician|date=1951|volume=5 |issue=1 |pages=18–19|doi= 10.1080/00031305.1951.10481914}}</ref> | |||
The problems that the SRG worked on included methods of sequential analysis and sampling inspection.<ref name="Wallis1980" /> Another was to examine the distribution of [[Survivorship bias#Military|damage to aircraft returning after flying missions]] to provide advice on how to minimize bomber losses to enemy fire. Wald derived a useful means of estimating the damage distribution for all aircraft that flew from the data on the damage distribution of all aircraft that returned.<ref name="Mangel1984" /><ref name=Wald1943>{{cite tech report |first=Wald |last=Abraham |title=A Reprint of 'A Method of Estimating Plane Vulnerability Based on Damage of Survivors' |number=ADA091073 |institution=Center for Naval Analyses; [[Statistical Research Group]], [[National Defense Research Committee]] |via=[[Defense Technical Information Center]] |date=1980 |orig-date=1943 |url=https://apps.dtic.mil/sti/pdfs/ADA091073.pdf|archive-url=https://web.archive.org/web/20210512125414/https://apps.dtic.mil/sti/pdfs/ADA091073.pdf|url-status=live|archive-date=May 12, 2021}}</ref> His work is considered seminal in the discipline of [[operational research]], which was then fledgling. | |||
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Wald and his wife died in 1950 when the Air India plane (VT-CFK, a DC-3 aircraft<ref> | Wald and his wife died in 1950 when the [[Air India]] plane (VT-CFK, a DC-3 aircraft<ref>{{cite web |title=Aircraft accident Douglas C-47B-5-DK (DC-3) VT-CFK Kotagiri |url=https://aviation-safety.net/database/record.php?id=19501213-1 |website=[[Aviation Safety Network]] |publisher=[[Flight Safety Foundation]] |date=26 September 2021}}</ref>) in which they were travelling crashed near the [[Rangaswamy Peak and Pillar|Rangaswamy Pillar]] in the northern part of the [[Nilgiri Mountains]], in southern [[India]], on an extensive lecture tour at the invitation of the Indian government.<ref name=Morgenstern1951/> He had visited the [[Indian Statistical Institute]] at Calcutta and was to attend the Indian Science Congress at Bangalore in January. Their two children were back at home in the United States.<ref>{{cite news|url=http://spectatorarchive.library.columbia.edu/cgi-bin/columbia?a=d&d=cs19501215-01|title=Prof. Wald Reported Among Victims of India Plane Crash|date=15 December 1950|newspaper=Columbia Daily Spectator|access-date=2018-02-08|issue=52|volume=XCV|page=1}}</ref> | ||
After his death, Wald was criticized by Sir [[Ronald A. Fisher]] [[Fellow of the Royal Society|FRS]]. Fisher attacked Wald for being a mathematician without scientific experience who had written an incompetent book on statistics. Fisher particularly criticized Wald's work on the design of experiments and alleged ignorance of the basic ideas of the subject, as set out by Fisher and [[Frank Yates]].<ref name=Fisher1955>{{cite journal | |||
|title=Statistical methods and scientific induction | |title=Statistical methods and scientific induction | ||
|author-link=Ronald A. Fisher |first=Ronald |last=Fisher | |author-link=Ronald A. Fisher |first=Ronald |last=Fisher | ||
| Line 77: | Line 75: | ||
|jstor=2983785 | |jstor=2983785 | ||
|issue=1 | |issue=1 | ||
}} (criticism of statistical theories of [[Jerzy Neyman]] and Abraham Wald)</ref> Wald's work was defended by [[Jerzy Neyman]] | |doi=10.1111/j.2517-6161.1955.tb00180.x }} (criticism of statistical theories of [[Jerzy Neyman]] and Abraham Wald)</ref> Wald's work was defended by [[Jerzy Neyman]] the next year. Neyman explained Wald's work, particularly with respect to the design of experiments.<ref name=Neyman1956>{{cite journal | ||
|title=Note on an Article by Sir Ronald Fisher | |title=Note on an Article by Sir Ronald Fisher | ||
|author-link=Jerzy Neyman |first=Jerzy |last=Neyman | |author-link=Jerzy Neyman |first=Jerzy |last=Neyman | ||
| Line 86: | Line 84: | ||
|jstor=2983716 | |jstor=2983716 | ||
|issue=2 | |issue=2 | ||
}} (reply to Fisher 1955)</ref> | |doi=10.1111/j.2517-6161.1956.tb00236.x }} (reply to Fisher 1955)</ref> [[Lucien Le Cam]] credits him in his own book, ''Asymptotic Methods in Statistical Decision Theory'': "The ideas and techniques used reflect first and foremost the influence of Abraham Wald's writings."<ref name=LeCam1986>{{cite book | ||
[[Lucien Le Cam]] credits him in his own book, ''Asymptotic Methods in Statistical Decision Theory'': "The ideas and techniques used reflect first and foremost the influence of Abraham Wald's writings" | |||
|title=Asymptotic Methods in Statistical Decision Theory | |title=Asymptotic Methods in Statistical Decision Theory | ||
|author-link=Lucien Le Cam |first=Lucien |last=Le Cam | |author-link=Lucien Le Cam |first=Lucien |last=Le Cam | ||
|year=1986 | |year=1986 | ||
|pages=xiii | |pages=xiii | ||
}} | }}</ref> | ||
He is the father of the noted American physicist [[Robert Wald]]. | |||
== Notable publications == | == Notable publications == | ||
| Line 134: | Line 131: | ||
| title=Contributions to the Theory of Statistical Estimation and Testing Hypotheses | | title=Contributions to the Theory of Statistical Estimation and Testing Hypotheses | ||
| url=https://projecteuclid.org/download/pdf_1/euclid.aoms/1177732144 | | url=https://projecteuclid.org/download/pdf_1/euclid.aoms/1177732144 | ||
| doi-access=free}} | | doi-access=free | ||
| bibcode=1939AnnMS..10..299W | |||
}} | |||
* {{citation | * {{citation | ||
| doi = 10.1214/aoms/1177731868 | | doi = 10.1214/aoms/1177731868 | ||
| Line 148: | Line 147: | ||
| title = The Fitting of Straight Lines if Both Variables Are Subject to Error | | title = The Fitting of Straight Lines if Both Variables Are Subject to Error | ||
| doi-access = free | | doi-access = free | ||
| bibcode = 1940AnnMS..11..284W | |||
}} | }} | ||
* {{citation | * {{citation | ||
| Line 172: | Line 172: | ||
| publisher = [[John Wiley and Sons]] | | publisher = [[John Wiley and Sons]] | ||
| location = [[New York City|New York]] | | location = [[New York City|New York]] | ||
}} Reprinted, Dover publications, 1973, {{isbn|0-486-43912-7}}. | |||
* {{cite book | * {{cite book | ||
| first=Abraham | | first=Abraham | ||
| Line 202: | Line 200: | ||
| doi-access = free | | doi-access = free | ||
}} | }} | ||
* {{ | * {{cite web |last1=Casselman |first1=Bill |author-link=Bill Casselman |title=The Legend of Abraham Wald |url=http://www.ams.org/publicoutreach/feature-column/fc-2016-06 |website=Feature Column |publisher=[[American Mathematical Society]] |date=June 2016 |access-date=29 May 2020}} | ||
== External links == | == External links == | ||
* {{MathGenealogy|id=36887}} | * {{MathGenealogy|id=36887}} | ||
*[https://apps.dtic.mil/sti/citations/ADA091073 ''A Method of Estimating Plane Vulnerability Based on Damage of Survivors'' by Abraham Wald, Defense Technical Information Center 1943] | |||
{{Authority control}} | {{Authority control}} | ||
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[[Category:1950 deaths]] | [[Category:1950 deaths]] | ||
[[Category:Scientists from Cluj-Napoca]] | [[Category:Scientists from Cluj-Napoca]] | ||
[[Category: | [[Category:Jews from Austria-Hungary]] | ||
[[Category:Jewish emigrants from Austria to the United States after the Anschluss]] | [[Category:Jewish emigrants from Austria after the Anschluss to the United States]] | ||
[[Category:Emigrants from Austria after the Anschluss to the United States]] | |||
[[Category:Presidents of the Institute of Mathematical Statistics]] | [[Category:Presidents of the Institute of Mathematical Statistics]] | ||
[[Category:Fellows of the American Statistical Association]] | [[Category:Fellows of the American Statistical Association]] | ||
[[Category:Hungarian statisticians]] | [[Category:Hungarian statisticians]] | ||
[[Category:Econometricians]] | [[Category:Econometricians]] | ||
<!-- [[Category:Mathematical analysts]] --> | <!-- [[Category:Mathematical analysts]] --> | ||
[[Category:Columbia University faculty]] | [[Category:Columbia University faculty]] | ||
[[Category:Jewish American scientists]] | [[Category:Jewish American scientists]] | ||
[[Category:Victims of aviation accidents or incidents in India]] | [[Category:Victims of aviation accidents or incidents in India]] | ||
[[Category:Hungarian mathematicians]] | [[Category:20th-century Hungarian mathematicians]] | ||
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[[Category:American operations researchers]] | [[Category:American operations researchers]] | ||
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[[Category:Fellows of the Econometric Society]] | [[Category:Fellows of the Econometric Society]] | ||
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Latest revision as of 19:22, 25 January 2026
Abraham Wald | |
|---|---|
 A young Wald | |
| Born | October 31, 1902 |
| Died | December 13, 1950 (aged 48) |
| Alma mater | King Ferdinand I University University of Vienna |
| Known for | Wald's equation Wald test Wald distribution Wald–Wolfowitz runs test Wald's martingale Wald's maximin model Mann–Wald theorem Decision theory Sequential analysis Sequential probability ratio test |
| Children | Robert Wald |
| Scientific career | |
| Fields | Mathematics Statistics Economics |
| Institutions | Columbia University Cowles Commission for Research in Economics |
| Doctoral advisor | Karl Menger |
| Doctoral students | Herman Chernoff Meyer Abraham Girshick Milton Sobel Charles Stein |
Abraham Wald (/wɔːld/; de; Script error: The function "langx" does not exist., Script error: The function "langx" does not exist.; 31 October 1902 – 13 December 1950) was a Hungarian and American mathematician and statistician who contributed to decision theory,[1] geometry and econometrics, and founded the field of sequential analysis.[2] One of his well-known statistical works was written during World War II on how to minimize the damage to bomber aircraft and took into account the survivorship bias in his calculations.[3] He spent his research career at Columbia University. He was the grandson of Rabbi Moshe Shmuel Glasner.
Life and career[edit | edit source]
Wald was born on 31 October 1902 in Kolozsvár, Transylvania, in the Kingdom of Hungary. A religious Jew, he did not attend school on Saturdays, as was then required by the Hungarian school system, and so he was homeschooled by his parents until college.[2] His parents were quite knowledgeable and competent as teachers.[4]
In 1928, he graduated in mathematics from the King Ferdinand I University.[5] In 1927, he had entered graduate school at the University of Vienna, from which he graduated in 1931 with a Ph.D. in mathematics. His advisor there was Karl Menger.[2][1]
Despite Wald's brilliance, he could not obtain a university position because of Austrian discrimination against Jews. However, Oskar Morgenstern created a position for Wald in economics. When Nazi Germany annexed Austria in 1938, the discrimination against Jews intensified. In particular, Wald and his family were persecuted as Jews. Wald emigrated to the United States at the invitation of the Cowles Commission for Research in Economics, to work on econometrics research.[2] By September 1938 Wald was a Fellow of the Carnegie Corporation at Columbia University learning about modern English language statistics from Harold Hotelling. He was appointed to the Columbia Faculty in 1941 and remained there until his death.
During World War II, Wald was a member of the Statistical Research Group (SRG) at Columbia University, where he applied his statistical skills to various wartime problems.[6] There was a difficulty: the work was secret and Wald was officially an enemy alien and, as such, barred from access to restricted matter. Hotelling recounts, “This impasse led to a facetious suggestion that each page he wrote should immediately be snatched away and never shown to him again, but was resolved when a federal court granted him a hearing long before his turn on the docket and promptly naturalized him.” [7]
The problems that the SRG worked on included methods of sequential analysis and sampling inspection.[6] Another was to examine the distribution of damage to aircraft returning after flying missions to provide advice on how to minimize bomber losses to enemy fire. Wald derived a useful means of estimating the damage distribution for all aircraft that flew from the data on the damage distribution of all aircraft that returned.[3][8] His work is considered seminal in the discipline of operational research, which was then fledgling.
Wald and his wife died in 1950 when the Air India plane (VT-CFK, a DC-3 aircraft[9]) in which they were travelling crashed near the Rangaswamy Pillar in the northern part of the Nilgiri Mountains, in southern India, on an extensive lecture tour at the invitation of the Indian government.[2] He had visited the Indian Statistical Institute at Calcutta and was to attend the Indian Science Congress at Bangalore in January. Their two children were back at home in the United States.[10]
After his death, Wald was criticized by Sir Ronald A. Fisher FRS. Fisher attacked Wald for being a mathematician without scientific experience who had written an incompetent book on statistics. Fisher particularly criticized Wald's work on the design of experiments and alleged ignorance of the basic ideas of the subject, as set out by Fisher and Frank Yates.[11] Wald's work was defended by Jerzy Neyman the next year. Neyman explained Wald's work, particularly with respect to the design of experiments.[12] Lucien Le Cam credits him in his own book, Asymptotic Methods in Statistical Decision Theory: "The ideas and techniques used reflect first and foremost the influence of Abraham Wald's writings."[13]
He is the father of the noted American physicist Robert Wald.
Notable publications[edit | edit source]
For a complete list, see "The Publications of Abraham Wald", Annals of Mathematical Statistics, 23 (1): 29–33, 1952, doi:10.1214/aoms/1177729483
- — (1939), "A New Formula for the Index of Cost of Living", Econometrica, 7 (4), Econometrica, Vol. 7, No. 4: 319–331, doi:10.2307/1906982, JSTOR 1906982
- — (1939), "Contributions to the Theory of Statistical Estimation and Testing Hypotheses", Annals of Mathematical Statistics, 10 (4): 299–326, Bibcode:1939AnnMS..10..299W, doi:10.1214/aoms/1177732144
- — (1940), "The Fitting of Straight Lines if Both Variables Are Subject to Error", Annals of Mathematical Statistics, 11 (3): 284–300, Bibcode:1940AnnMS..11..284W, doi:10.1214/aoms/1177731868
- — (June 1945), "Sequential Tests of Statistical Hypotheses", The Annals of Mathematical Statistics, 16 (2): 117–186, doi:10.1214/aoms/1177731118
- — (1947). Sequential Analysis. New York: John Wiley and Sons. Reprinted, Dover publications, 1973, ISBN 0-486-43912-7.
- — (1950). Statistical Decision Functions. John Wiley and Sons, New York; Chapman and Hall, London. p. ix+179.[14]
References[edit | edit source]
- ↑ 1.0 1.1 Duppe, Till; Weintraub, E. Roy (2015). "Losing Equilibrium: On the Existence of Abraham Wald's Fixed-Point Proof of 1935". History of Political Economy. 48 (191). Econometrica, Vol. 19, No. 4: 635–655. doi:10.1215/00182702-3687283. hdl:10419/149731.
- ↑ 2.0 2.1 2.2 2.3 2.4 Morgenstern, Oskar (1951). "Abraham Wald, 1902–1950". Econometrica. 19 (4). Econometrica, Vol. 19, No. 4: 361–367. doi:10.2307/1907462. JSTOR 1907462.
- ↑ 3.0 3.1 Mangel, Marc; Samaniego, Francisco J. (June 1984). "Abraham Wald's Work on Aircraft Survivability" (PDF). Journal of the American Statistical Association. 79 (386). American Statistical Association: 259–267. doi:10.1080/01621459.1984.10478038. JSTOR 2288257.
- ↑ O'Connor, John J.; Robertson, Edmund F., "Abraham Wald", MacTutor History of Mathematics archive, University of St Andrews.
- ↑ Anuarul Universității Regele Ferdinand I pe anul școlar 1927/28. p. 187. Online access, University Library in Cluj, Romania.
- ↑ 6.0 6.1 Wallis, W. Allen (1980). "The Statistical Research Group, 1942–1945". Journal of the American Statistical Association. 75 (370): 320–330. doi:10.1080/01621459.1980.10477469. JSTOR 2287451.
- ↑ Hotelling, Harold (1951). "Abraham Wald". American Statistician. 5 (1): 18–19. doi:10.1080/00031305.1951.10481914.
- ↑ Template:Cite tech report
- ↑ "Aircraft accident Douglas C-47B-5-DK (DC-3) VT-CFK Kotagiri". Aviation Safety Network. Flight Safety Foundation. 26 September 2021.
- ↑ "Prof. Wald Reported Among Victims of India Plane Crash". Columbia Daily Spectator. Vol. XCV, no. 52. 15 December 1950. p. 1. Retrieved 2018-02-08.
- ↑ Fisher, Ronald (1955). "Statistical methods and scientific induction". Journal of the Royal Statistical Society, Series B. 17 (1): 69–78. doi:10.1111/j.2517-6161.1955.tb00180.x. JSTOR 2983785. (criticism of statistical theories of Jerzy Neyman and Abraham Wald)
- ↑ Neyman, Jerzy (1956). "Note on an Article by Sir Ronald Fisher". Journal of the Royal Statistical Society, Series B. 18 (2): 288–294. doi:10.1111/j.2517-6161.1956.tb00236.x. JSTOR 2983716. (reply to Fisher 1955)
- ↑ Le Cam, Lucien (1986). Asymptotic Methods in Statistical Decision Theory. pp. xiii.
- ↑ Robbins, Herbert (1951), "Review: A. Wald, Statistical decision functions", Bull. Amer. Math. Soc., 57 (5): 383–384, doi:10.1090/S0002-9904-1951-09520-8
Further reading[edit | edit source]
- Wolfowitz, Jacob (1952). "Abraham Wald, 1902–1950". Annals of Mathematical Statistics. 23 (1): 1–13. doi:10.1214/aoms/1177729480.
- Casselman, Bill (June 2016). "The Legend of Abraham Wald". Feature Column. American Mathematical Society. Retrieved 29 May 2020.