Pao-Lu Hsu (Xu, Bao-lu): The Grandparent of Probability and Statistics in China
aa r X i v : . [ s t a t . M E ] O c t Statistical Science (cid:13)
Institute of Mathematical Statistics, 2012
Pao-Lu Hsu (Xu, Bao-lu):The Grandparent of Probability andStatistics in China
Dayue Chen and Ingram Olkin
Abstract.
The years 1910–1911 are auspicious years in Chinese math-ematics with the births of Pao-Lu Hsu, Luo-Keng Hua and Shiing-ShenChern. These three began the development of modern mathematics inChina: Hsu in probability and statistics, Hua in number theory, andChern in differential geometry. We here review some facts about thelife of P.-L. Hsu which have been uncovered recently, and then discusssome of his contributions. We have drawn heavily on three papers inthe 1979
Annals of Statistics (volume 7, pages 467–483) by T. W. An-derson, K. L. Chung and E. L. Lehmann, as well as an article by JiangZe-Han and Duan Xue-Fu in Hsu’s collected papers.
Key words and phrases:
Multivariate analysis, Wishart distribution,Student’s t , Hotelling’s T , determinantal equation, eigenvalues, designof experiments, mathematics in China.
1. HSU’S LIFE
Pao-Lu Hsu was born in Beijing on September 1,1910, “into a Mandarin family from the famed lakecity of Hangchow” in the Zhejiang Province in East-ern China. His family was well educated: not only hisfather, but also his grandfather, his great-grandfatherand the father of his great-grandfather, as well astheir brothers and brothers-in-law. This reflectedthe tradition in old China of excelling in local exams,provincial exams and finally in national exams. Thetradition was abandoned after China lost the warwith Japan in 1895 and turned to Western meth-ods.
Dayue Chen is Professor of Mathematics, School ofMathematical Sciences, Peking University, Beijing,China 100871 e-mail: [email protected]. Ingram Olkinis Professor Emeritus of Statistics and Education,Department of Statistics, Stanford University, Stanford,California 94305, USA e-mail: [email protected].
This is an electronic reprint of the original articlepublished by the Institute of Mathematical Statistics in
Statistical Science , 2012, Vol. 27, No. 3, 434–445. Thisreprint differs from the original in pagination andtypographic detail.
Hsu was the youngest of seven children, with twobrothers and four sisters. His father died when hewas 14. During his childhood, Hsu moved from Bei-jing to Tianjin to Hangzhou and back to Beijing.His early education was with private tutors, a lux-ury that few people could afford at that time. He be-gan attending school at the age of 15, and enrolledat Yenching University when he was 18. He firststudied chemistry, then decided to study mathemat-ics and transferred to Tsinghua University in 1929.Both Yenching and Tsinghua universities had con-nections with universities in the United States. Forexample, the Yenching–Harvard Institute, foundedin 1928, was designed to foster education in the hu-manities and social sciences in Asia.Hsu received a bachelor of science degree from Ts-inghua University, and then went to Peking Univer-sity, where he was an assistant in the Departmentof Mathematics. He passed the examination in 1936,after which he went to the University of London tocontinue his studies. He served as a lecturer, ob-taining a Ph.D. in 1938 and a Sc.D. in 1940. Thus,he remained in England for the four years 1936–1940. However, it is known that he spent some timein Paris (perhaps during the academic year 1939–1940) to study with Jacques Hadamard. Hsu then D. CHEN AND I. OLKIN
Fig. 1.
Abstract of Pao-Lu Hsu’s 1938 thesis, part I: “Contribution to the theory of the t -test as applied to the problem oftwo samples.” returned to China, which was again at war withJapan. (The Second Sino–Japanese War lasted from1937 to 1945.) After his return, Hsu was appointedas professor at Peking University, which was relo-cated to Kunming during World War II.Constance Reid writes in her biography of JerzyNeyman: “The most outstanding of Neyman’s stu-dents at that time (eds. 1937–1938) was a Chinese,. . . P. L. Hsu. (Neyman expresses to me his admira-tion for Hsu with a Polish phrase which he trans-lates—with a little bow and a gracious wave of thehand—as Please sit down!)” (Reid (1982), page 153).The Neyman biography further notes that, “Hsuwas, in Neyman’s opinion, absolutely on a level withWald—they were the two outstanding statisticiansin the generation coming up!” Hsu was invited to lecture at Berkeley for six months with an appoint-ment for the following year. Harold Hotelling was atColumbia at this time and suggested that Columbiaand Berkeley join together to bring Hsu to each uni-versity for a semester. (Hsu was also sought after byChicago and Yale.) Hsu accepted the joint offer andindicated that he preferred to first visit the WestCoast.Erich Lehmann in his autobiography lists Hsu asone of his three Ph.D. godfathers. At that time (1945)Lehmann was a doctoral student at Berkeley, andNeyman asked Hsu to give him a thesis topic:Within a few days, Hsu presented me witha new possible topic: applying methods ofNeyman, Scheff´e and himself to some sit-uations for which they had not been tried
AO-LU HSU Fig. 2.
Abstract of Pao-Lu Hsu’s 1938 thesis, part II: “Onthe best unbiased quadratic estimate of the variance.” before. Hsu then got me started on thisline of work. In a letter of January 24to Neyman, about which I learned onlymuch later, he wrote: “I have passed theproblem of testing for independence be-tween successive observations to Erich forhis doctoral thesis. Will do all I had doneindependently,and then add a new partwhich I have not done. I hope this schemewill meet with your approval, so that Erichcan look forward to the degree with cer-tainty.”This was an act of greatest generosity.Hsu made me a present of work he hadplanned to do himself and on which hehad already obtained some results. I hadhoped to see him on his return to Berkeleyafter the term at Columbia. However, thiswas not to be; in fact I never saw himagain. (Lehmann (2008), page 39)When Hotelling moved to the University of NorthCarolina in 1946 to found a department of mathe-matical statistics, he offered Hsu an associate pro-fessorship. Hsu accepted the offer and spent the pe-riod 1946–1947 at Chapel Hill, but the pull to returnto China was too strong, and he returned in thesummer of 1947. He was committed to China, and wanted to participate in “the emerging new societyin his homeland.” On a trip across country (1946–1947) Neyman visited Hotelling in Chapel Hill andsaw Hsu again, whom he hoped to entice to Berkeley.“He found the Chinese scholar miserably unhappy,disappointed in love, and desiring only to return tohis native land.” (Reid (1982), page 214)After his return to China, Hsu’s research was un-known in the West. “Apart from his published pa-pers and a few remarks given us by an old friend,we are unable to obtain further information aboutHsu’s life and work in the twenty-some years he livedin Peking” (Anderson, Chung and Lehmann (1979)).His colleagues at Peking University did not see himeasily either. As reported by Boju Jiang, who joinedthe department as a faculty member in 1957 (but didnot meet Hsu until 1968), “Mr. Hsu was essentiallya legendary hero, somewhat mysterious to us.”Hsu was the first teacher to offer courses in prob-ability and statistics in China, from the early 1940sin Kunming. Kai-Lai Chung was a teaching assis-tant at that time, and became interested in proba-bility by taking courses and discussing research withHsu. For this reason Chung always regarded himselfas a student of Hsu. Other students included Shou-Jen Wang, L.C. Hsu and Chin-long Chiang. UnderHsu’s supervision, Zhong-Zhe Zhao completed grad-uate study in 1951 and was the first graduate stu-dent to major in probability in China. After threelectures given in the fall of 1955, Hsu could no longerteach in a classroom due to his poor health.In 1956, probability and statistics (together withcomputational mathematics and differential equa-tions) were identified as key subjects of mathemat-ics to be developed with high priority in China.Only a few Chinese researchers knew probabilityand statistics at that time, and in order to producequalified teachers at an accelerated pace, a specialprogram was created at Peking University, with 34juniors from PKU, 10 juniors from Nankai Univer-sity in Tianjin and 10 juniors from Sun Yat-sen Uni-versity in Guangzhou (Canton). In addition, some10 teachers came from all over the country to auditthe courses. Instructors were brought from the Chi-nese Academy of Science and Sun Yat-sen Univer-sity. Hsu was a great teacher, and served as leaderof the program. The curriculum he created therelater became the national standard. Textbooks werecompiled quickly, some based on the notes from hislectures and some translated from Russian. Aftera two-year training period, students were dispatchedto other universities to teach probability and statis-
D. CHEN AND I. OLKIN
Fig. 3.
Pao-Lu Hsu’s handwritten letter to Kai-Lai Chung in 1947. K.-L. Chung (1917–2009) was Professor Hsu’s studentduring the Second World War and later became a well-known probabilist. tics. In some sense, all Chinese probabilists and statis-ticians are students or grand-students of Hsu. Onteaching, he once said: “One can feel proud to bethe advisor of a Nobel laureate. It means nothingjust to be a student of a Nobel laureate.”Hsu continued his teaching by running seminarsat his home. He began the practice informally inthe early 1950s, and continued on a regular basisfor eight consecutive years until 1964. This was verymuch in the Chinese tradition of private educationin which students were required to learn by them-selves and to present their findings each week forevaluation by the advisor. Quite often the seminarbecame a small class taught by Hsu himself. Only a few people were fortunate to serve as his appren-tices. During the peak period, he ran three seminarsa week in the living room of his one-bedroom apart-ment on campus, which was about 140 square feet insize. Seminar participants were selected by Hsu him-self; in contrast, his graduate students were assignedto him. Among the few photographs available todayare two taken with students of his seminars in 1959(Figures 6 and 7). The topics of the seminars cov-ered a wide range: mathematical statistics, limit the-ory, Markov processes, stationary processes, exper-imental designs, sampling techniques, order statis-tics, and topology. Research conducted by membersof the seminars represented the first coordinated ef-
AO-LU HSU Fig. 4.
Pao-Lu Hsu’s handwritten letter to Herbert Robbins in 1948 (page 1 of 2). forts in probability and statistics in China. Hsu cre-ated pen names such as Ban-cheng, Ban-guo andBan-ji for the students to write and submit joint pa-pers. (The Chinese word “Ban” means class.) From1958 to 1962 he also supervised six graduate stu-dents, including Yongquan Yin.The reader is reminded that China and the SovietUnion were in a “honeymoon” period in the 1950s,and China learned much from the Soviet Union. Fol-lowing the Soviet pattern, a subdivision within themathematics department was set up at Peking Uni-versity in 1956, named
Teaching and Research Unit of Probability Theory and Mathematical Statistics .This was the very first of its kind in China, and itevolved into an independent department in 1985. Asthe founding director of the unit, Hsu watched overthe career development of young colleagues, becausemost members only had undergraduate training. Healso organized scientific exchanges with foreign col-leagues, for example, Marek Fisz and Kazimierz Ur-banik from Poland in 1957 and Eugene Dynkin fromthe USSR in 1958. Careful preparation was madebefore each visit in order to better understand theforthcoming lectures. Several months before these
D. CHEN AND I. OLKIN
Fig. 5.
Hsu’s 1948 letter to Robbins (page 2 of 2). Herbert Robbins (1915–2002) was Professor Hsu’s colleague at ColumbiaUniversity. He was an eminent statistician and a member of the US National Academy of Sciences. Robbins had kept this letterfor a long time, and after his death the letter was sent to Peking University by Professor T. L. Lai of Stanford University. visits, Hsu would assign related papers for youngfaculty members and students to study.However, although Hsu himself was immune frompolitics, Hsu’s efforts were discounted. Seminars wereended unexpectedly because students were requiredto devote themselves fully to a political movement.Graduate students were selected based on politi-cal criteria and were assigned to professors, with-out much consideration of the academic interest andability of the student. Some were not well prepared for graduate study. Hsu must have been annoyedby the requirement to submit a research plan eachyear, simply because everything had to be part ofa planned economy. His solution was to propose hisnew papers as the research plan for the next year.Shortly before the Communist victory in 1949,Hsu and most other professors declined the offer ofChiang Kai-shek to airlift them to southern China.He even sent a telegram to a foreign friend saying“. . . am happy after liberation.” However, when Hsu
AO-LU HSU returned from the US in 1947, China was in themiddle of their civil war. His return from the UK toChina in 1940 had been even worse. At that time,China was involved in WWII and living conditionsin Kunming were miserable. On May 4th of 1919,students of Peking University demonstrated in Tian-an-men Square, crying out for the adoption of princi-ples of democracy and science. (This was the first ofa number of Tian-an-men Square protests, the mostrecent occurring in 1989.) The May 4th Movementwas a turning point in the modern history of China.It is reasonable to assume that Hsu, like many Chi-nese intellectuals of his generation, wished to builda strong country by introducing science to China.This may also explain in part why Hsu submittedpapers to Chinese journals.In the 1950s Hsu wanted to create a Chinese jour-nal in probability and statistics, as a launch pad foryoung researchers to publish their papers. He wasprepared to subsidize the journal even with his ownmoney. Hsu had a tremendous linguistic talent, witha command of English, German, French and Rus-sian. He learned Russian by himself, and helped cor-rect several textbooks by Alexander Khintchine andVyacheslav Stepanov, translated from their originaleditions. Indeed, the last accomplishment of his lifewas to proofread a set of manuscripts scheduled forcompletion in one month. He looked at the task andsaid he could do it in ten days; he finished the jobin a little over nine days.Hsu’s health had been fragile since he was young.He was 5 feet, 9 inches tall, but only weighed 88pounds at his maximum. Because of his light weight,he was disqualified for a government fellowship tostudy abroad in 1933. According to medical records,he was hospitalized in 1948 and in the early 1950s,and recuperated in hospital from illnesses in 1933and 1957. In a letter to Herbert Robbins in 1948,Hsu wrote, “It appears that my system has gonewrong, a stomach ulcer and a lung TB are just se-rious enough to force upon me a complete rest forone year at least.” In his final decade or so he wasessentially confined to bed, where he continued toread and write. He never married and lived alone.For his important contributions, Pao-Lu Hsu wasone of five mathematicians elected as Academiciansin 1948. He was elected as an Academician againin 1955, along with eight other mathematicians. Hewas a Fellow of the Institute of Mathematical Statis-tics. After his death, memorial meetings were heldevery ten years at Peking University. In 2010, a me-morial collection of papers was published, a bronze statue was dedicated, an international conference onprobability and statistics was held in July, and anofficial commemoration was held on his centennialbirthday. The Pao-Lu Hsu Lecture Series was launch-ed at Peking University in 2009 with a roster of dis-tinguished speakers. Tsinghua University recentlyalso inaugurated the Pao-Lu Hsu Distinguished Lec-ture in Statistics and Probability, with Brad Efronas the first speaker. The P.-L. Hsu Conference onStatistical Machine Learning is now in its third year.The International Chinese Statistical Society (ICSA)has announced its intention to set up the P.-L. HsuAward. A memorial webpage will be built as partof PKU’s Department of Probability and Statisticsweb site, .
2. HSU’S RESEARCH
Some insight into Hsu’s views about research canbe gleaned from comments made to his students,such as the following: • “The merit of a paper is not just to get published,but is realized when it is cited repeatedly by oth-ers.” • “A good author should show the simplicity.” • “I do not want to become famous because my pa-per appears in a well-established journal. I wisha journal to be well established because my paperappears in that journal.”Pao-Lu Hsu authored 41 papers and three books,on a wide range of topics: limit theorems, randommatrices, Markov chains, experimental designs, char-acteristic functions. Almost all his papers were singlyauthored. He published with only three coauthors:Kai Lai Chung, Tsai-han Kiang and Herbert Rob-bins. Hsu used the pen name Ban-cheng to pub-lish papers with his students. As demonstrated inTable 1, his peak performance was from 1938 to1947. In a ten-year period he wrote 22 papers, allin English. For the next 15 years (1949–1964), hewrote 13 papers of which 11 articles were publishedin Chinese journals. Some of his manuscripts werepublished posthumously, with the assistance of hisstudents.
3. HSU’S RESEARCH IN MULTIVARIATEANALYSIS
In addition to probability and statistics, Hsu lec-tured on topology, matrix theory and analysis. Hedid not have access to outside literature, and pro-vided new proofs of some known results. He alsoobtained new results which still remain hidden.
D. CHEN AND I. OLKIN
Table 1
Hsu’s publications
JournalsPeriod ResidenceChineseInt’lTopics
The Wishart distribution was a focal point in Hsu’searly work. He generated new derivations and ob-tained the distribution of the eigenvalues of the sam-ple covariance matix and the canonical correlations.The joint distribution of the elements of the co-variance matrix obtained from a sample of p -variatestandard normal variates was obtained by Fisher in1915 for the special case p = 2, and by Wishart in1928 for general p . (Although stated for standardnormal variables, by a simple transformation thevariates can have a covariance matrix Σ.) What istantalizing about this distribution is that the start-ing point is a set of pn ( p ≤ n ) random variables inthe p × n sample matrix X , and the ending pointis a set of p ( p + 1) / S = XX ′ . There are a number of routes andmethods that might be used to make this transi-tion. Wishart’s original derivation was a geometricargument and later in 1933, together with Bartlett,he gave a derivation using characteristic functions.In 1937, Mahalanobis, Bose and Roy gave a geomet-ric derivation of rectangular coordinates (describedbelow), which is a stepping stone to deriving theWishart distribution.In 1939, Hsu gave a new derivation of the Wishartdensity using a clever inductive argument. The caseof p = 1 reduces to the chi-square distribution. Theessence of the induction is to go from p − p vari-ables. Here Hsu uses a multivariate tranformation,an area that he later developed.In 1940 Hsu gave an algebraic derivation of rect-angular coordinates which led to a general result. Suppose the joint density p ( X ) of the qm elementsof the random q × m matrix X is of the functionalform p ( X ) = f ( XX ′ ) = f ( S ), where S = XX ′ . Let S = T T ′ , where T is lower triangular. The elementsof T are called rectangular coordinates . Hsu givesa detailed proof that such a factorization exists; thisis an example in which Hsu gives a new proof ofa known theorem, in this case by Toeplitz in 1907.Hsu next obtains the Jacobian of the transforma-tion, which leads to the joint density of the elementsof T : c ( q, m ) p Y t m − iii f ( T T ′ ) , < t ii , −∞ < t ij < ∞ , i = j, where c ( q, m ) is a normalizing constant explicitlydetermined. In two papers of 1933 and 1936, Hotelling openedthe door to consideration of the distribution of theroots of a determinantal equation | A − θI | = 0 or | A − θ ( A + B ) | = 0, where p -dimensional randommatrices A and B are independently distributed,each having a standard Wishart distribution.In the first case suppose that the random posi-tive definite matrix A has a density f ( A ) that isorthogonally invariant, that is, f ( A ) = g ( θ , . . . , θ p ),where the θ ’s are the eigenvalues of A . Make thetransformation A = ∆ D θ ∆ ′ , where ∆ is orthogonaland D θ = diag( θ , . . . , θ p ) to yield the joint densityof ∆ and θ . Integration over the orthogonal groupyields the joint distribution of the eigenvalues. Atthat time (1953), this integration was not very wellknown. Here again, Hsu found a clever way to carryout this integration. He showed that every orthogo-nal matrix Γ has a representation in terms of a skew-symmetric matrix Y , namely, Γ = 2( I + Y ) − − I .The beauty of this transformation is that whereasthe p ( p − / Y . Hsu then shows how to carry outthe integration to yield a general result. Theorem 1. If S is a random p -dimensionalpositive definite matrix with an orthogonally invari-ant density f ( S ) = g ( θ , . . . , θ p ) , where θ ≥ · · · ≥ θ p > are the eigenvalues of S , then the joint dis-tribution of the eigenvalues is c ( p ) Y i Pao-Lu Hsu and his group of statistics students in the Department of Mathematics and Mechanics at Peking Univer-sity in 1954. In the front row are Professor Hsu (center) and his assistants Yao-Ting Zhang (second from left) and Chong-FeiLu (second from right). To obtain the distribution of the roots of | A − θ ( A + B ) | = 0, Hsu transforms ( A, B ) to ( W, ϕ ) by A = W D ϕ W ′ , B = W W ′ , where D ϕ = diag( ϕ , . . . ,ϕ p ). Note that ϕ i = θ i / (1 − θ i ). The key problem isto evaluate the Jacobian of the transformation. HereHsu gave an explicit derivation for p = 3 but statedthe general result, later given in the exposition byDeemer and Olkin (1951). Hsu obtained the resultfor p < n , n and n < p < n , where n and n arethe sample sizes that lead to A and B . In the description above, both A and B have a cen-tral Wishart distribution. Hsu (1941) tackles thecase that B has a central Wishart distribution but A has a noncentral distribution. This distribution iscomplicated and Hsu obtains asymptotic results. SeeAnderson (1979) for a more detailed description. t and Hotelling’s T Distributions Hsu’s first statistical paper was in 1938, and in ithe obtained the distribution of the square of the Stu- D. CHEN AND I. OLKIN Fig. 7. Professor Pao-Lu Hsu (left), Professor Deehe Hu (right) and the Queueing Theory Discussion Group (1954 EnteringClass, Department of Mathematics and Mechanics, Peking University), circa July, 1959. dent’s t -statistic in which the denominator is a linearcombination, as + bs , of the variances in the twounderlying samples. This is now called the Behrens–Fisher problem.Hotelling obtained the null distribution of the mul-tivariate version of the Student’s t -statistic in 1931,and in 1938 Hsu obtained its noncentral distribu-tion. This may be the earliest noncentral distribu-tion in multivariate analysis, and was very much inthe spirit of the Neyman–Pearson view. (Recall thatNeyman invited Hsu to the first Berkeley Sympo- sium in 1945. His paper was titled “The limitingdistribution of functions of sample means and ap-plication to testing hypotheses.”)Hsu also wrote about power functions of severalmultivariate tests, and provided a canonical form forthe multivariate analysis of variance. Here the meanof the random p × m matrix Y is EY = Θ, the meanof the random p × n matrix Z is EZ = 0; the rowsof Y and Z are multivariate normal with a commoncovariance matrix Σ. The hypothesis in question is H : Θ = 0. AO-LU HSU Fig. 8. First page of handwritten notes, kept at the Department of Probability and Statistics of the School of MathematicalSciences, of Professor Pao-Lu Hsu’s lectures on point-set topology in 1963 at the Department of Mathematics and Mechanics,Peking University. Hsu entered into another realm, that of combi-natorial analysis. Here he defines a balanced incom-plete block (BIB) design in terms of a (0, 1) matrix T of dimension v × b with the properties e ′ T = ke ′ , T e = re and T T ′ = rI + ( λ + r ) J, where e is the vector of ones, and J is the matrixwith all elements equal to 1. Such a BIB design isdenoted by the five constants ( v, b, r, k, λ ). The fol-lowing is an example of his results. Suppose thata BIB( v, ∗ , r, k, λ ) exists, Hsu shows how to obtaina BIB( v + 1 , ∗ , r, k, λ ). He then defines a code as a new n × w matrix M with elements (1 , − M (without repetition). Let M M ′ = Q =( q ij ). Hsu obtains a number of results concerningthe matrix Q , such as the inequalitymax i = j q ij ≥ n ( n − X i = j q ij ≥ (cid:26) − w/ ( n − , if n is even, − w/n, if n is odd.Hsu defines a simple code to be a matrix M forwhich equality is achieved, and then obtains neces-sary and sufficient conditions for M to be a simplecode. An orthogonal code is one for which M M ′ = D. CHEN AND I. OLKIN wI , where I is the identity matrix. Hsu providesan example of a 12 × 20 orthogonal code. It is to benoted that a 4 w × w orthogonal code is a Hadamardmatrix, a subject of interest for a long time. Hsu isaware of this interest (recall that he studied withHadamard) so raises the following question. Sup-pose, by his construction, that we can generate an n × t orthogonal code: is it possible to extend thisto a 4 t × t orthogonal code? He provides two coun-terexamples, one of dimension 4 × 12 and the other12 × 20. Hsu’s interest in codes may stem from hisstudy with Jacques Hademard, who connected theconstruction of error-correcting codes with matriceswhose elements are +1 or − 4. EPILOGUE Hsu’s last paper, “BIB matrices, simple codes andorthogonal codes,” was published in 1970. ZhangYao-ting provided this introduction for its appear-ance in Hsu’s collected papers (page 566):This paper was Professor Pao-Lu Hsu’slast article, completed in October 1970.He died in December of the same year.The Cultural Revolution lasting alreadyfour years at the time had not yet ended.He had suffered tremendously in this cala-mity, and physically he was paralyzed. Thispaper was completed when he was bedrid-den. The only journal he could have ac-cess to was the Annals of MathematicalStatistics . It was said that when he gavethis paper to Mr. H. F. Tuan, being nolonger able to speak clearly, he was usinghis hands to express himself. The material discussed in this article isclosely related to the contents covered inhis early 1966 seminars on combinatorialanalysis. During the later years of his life,he was devoted to using matrices to de-scribe and prove results in combinatorialanalysis. This article is a typical represen-tative of this idea. ACKNOWLEDGMENT We thank Professor Tom Fearn of University Col-lege, London, for his persistence in uncovering Hsu’shistory as a doctoral candidate.REFERENCES Anderson, T. W. (1979). Hsu’s work in multivariate analy-sis. Ann. Statist. Anderson, T. W. , Chung, K. L. and Lehmann, E. L. (1979). Pao Lu Hsu: 1909–1970. Ann. Statist. Chung, K. L. (1979). Hsu’s work in probability. Ann. Statist. Deemer, W. L. and Olkin, I. (1951). The Jacobians of cer-tain matrix transformations useful in multivariate analysis. Biometrika Hsu, P. L. (1941). On the limiting distribution of roots ofa determinantal equation. J. London Math. Soc. Hsu, P. L. (1983). Collected Papers . Springer, New York.Edited by Kai Lai Chung, with the cooperation of Ching-Shui Cheng and Tse-Pei Chiang. Lehmann, E. L. (1979). Hsu’s work on inference. Ann.Statist. Lehmann, E. L. (2008). Reminiscences of a Statistician. TheCompany I Kept . Springer, New York. MR2367933 Reid, C. (1982).