Pao-Lu Hsu
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Pao-Lu Hsu or Xu Baolu (; September 1, 1910 – December 18, 1970) was a Chinese mathematician noted for his work in
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
.


Life and career

Pao-Lu Hsu was born in Beijing on September 1, 1910, with his ancestral home in
Hangzhou Hangzhou, , Standard Mandarin pronunciation: ; formerly romanized as Hangchow is a sub-provincial city in East China and the capital of Zhejiang province. With a population of 13 million, the municipality comprises ten districts, two counti ...
,
Zhejiang ) , translit_lang1_type2 = , translit_lang1_info2 = ( Hangzhounese) ( Ningbonese) (Wenzhounese) , image_skyline = 玉甑峰全貌 - panoramio.jpg , image_caption = View of the Yandang Mountains , image_map = Zhejiang i ...
Province. He was from a prominent intellectual family. In his childhood, he received solid training in both traditional Chinese and modern western cultures. He graduated from
Tsinghua University Tsinghua University (THU) is a public university in Haidian, Beijing, China. It is affiliated with and funded by the Ministry of Education of China. The university is part of Project 211, Project 985, and the Double First-Class Constructio ...
in 1933, majoring in mathematics. After his graduation, he worked at Peking University as a teacher. In the meantime, he published a joint paper with Tsai-han Kiang ( Jiang Zehan) on the numbers of nondegenerate critical points, which showed his solid mathematical foundation and research capability. In 1936, he went to University College London and spend four years studying mathematical statistics. During this period, with his strong mathematical skill combining with advanced statistical ideas, he wrote a series remarkable papers. He earned his Ph.D. in 1938 and Sc.D. in 1940. From London, he returned to China, accepting a professorship in the Department of Mathematics, Peking University. Nevertheless, the ensuing years were a period of great difficulty and hardship for him. During the years 1943–44, he corresponded with
Jerzy Neyman Jerzy Spława-Neyman (April 16, 1894 – August 5, 1981; ) was a Polish mathematician and statistician who first introduced the modern concept of a confidence interval into statistical hypothesis testing and, with Egon Pearson, revised Ronald Fis ...
(who was by this time at Berkeley) about statistics-related matters, but also mentioning in these letters the great hardship he was suffering, particularly starvation. In 1945, he went to the United States, visiting the
University of California at Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California), is a public land-grant research university in Berkeley, California, United States. Founded in 1868 and named after the Anglo-Irish philosopher George Berkele ...
,
Columbia University Columbia University in the City of New York, commonly referred to as Columbia University, is a Private university, private Ivy League research university in New York City. Established in 1754 as King's College on the grounds of Trinity Churc ...
, and the
University of North Carolina at Chapel Hill The University of North Carolina at Chapel Hill (UNC, UNC–Chapel Hill, or simply Carolina) is a public university, public research university in Chapel Hill, North Carolina, United States. Chartered in 1789, the university first began enrolli ...
. In 1947, he returned to Beijing and thereafter he was engaged in teaching mathematics at Peking University for more than 20 years. Unfortunately, poor health had dogged Hsu from 1950 onward. Although many concerned about his health repeatedly suggested he go abroad to recuperate, he politely refused, insisting on teaching and continuing his research work. In 1956, Dr. Hsu was made Director of the first research institute for probability and statistics to be established in China. But his health continued to worsen, forcing him to work at home, which included giving lectures at his home to graduate students and young teachers using a blackboard hanging on a wall of his room. A month before his death in 1970 his manuscript on the relationship between experimental design and algebraic coding theory was completed, with such being his final legacy. Hsu died in his home on the campus of Peking University on December 18, 1970, from chronic tuberculosis. Found beside his bed the day after his death were piles of manuscripts that "serve as a testimony to the super-human fortitude with which he exerted himself for a period of more than 20 years..." In total, he had published more than 40 papers. There was a memorial meeting and Pao-Lu Hsu Award in his honor. Hsu is a Fellow of the Institute of Mathematical Statistics (IMS) and Academician of the Academy of Science of China. Memorial meetings were held in 1980, 1990, 2000 & 2010 at
Peking University Peking University (PKU) is a Public university, public Types of universities and colleges in China#By designated academic emphasis, university in Haidian, Beijing, China. It is affiliated with and funded by the Ministry of Education of the Peop ...
. Professor Hsu's main research areas were mathematical statistics and probability theory. He was an expert in operating matrix, manipulating characteristic functions and integral transformation. Hsu was a world-class statistician and is considered to be the founder of probability and statistics in China. He influenced many students such as
Theodore Wilbur Anderson Theodore Wilbur Anderson (June 5, 1918 – September 17, 2016) was an American mathematician and statistician who specialized in the analysis of multivariate data. Life and career He was born in Minneapolis, Minnesota. He was on the facu ...
,
Erich Leo Lehmann Erich Leo Lehmann (20 November 1917 – 12 September 2009) was a German-born American statistician, who made a major contribution to nonparametric hypothesis testing. He is one of the eponyms of the Lehmann–Scheffé theorem and of the Hodges ...
, and Chung Kai-lai.


Hsu's work in statistics

In 1938, Hsu's first two statistical papers were published, concerning the
Behrens–Fisher problem In statistics, the Behrens–Fisher problem, named after Walter-Ulrich Behrens and Ronald Fisher, is the problem of interval estimation and hypothesis testing concerning the difference between the means of two normally distributed populations ...
and an optimal estimate of the
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
''σ''2 in Gauss-Markov processes. In another important paper published around this time, Hsu established the first optimal property for
hypotheses A hypothesis (: hypotheses) is a proposed explanation for a phenomenon. A scientific method, scientific hypothesis must be based on observations and make a testable and reproducible prediction about reality, in a process beginning with an educ ...
in the
likelihood-ratio test In statistics, the likelihood-ratio test is a hypothesis test that involves comparing the goodness of fit of two competing statistical models, typically one found by maximization over the entire parameter space and another found after imposing ...
for univariate
linear model In statistics, the term linear model refers to any model which assumes linearity in the system. The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model. However, t ...
s, which was also the first nonlocal optimal property for any hypothesis specifying the value of more than one parameter. From 1938 to 1945, Hsu published several papers at the forefront of the development of the theory of multivariate analysis. During this time, he obtained several exact or asymptotic estimates of important statistical distributions in the field of
multivariate analysis Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable, i.e., '' multivariate random variables''. Multivariate statistics concerns understanding the differ ...
.


Hsu's work in probability

Hsu was an expert in manipulating
characteristic functions In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function \mathbf_A\colon X \to \, which for a given subset ''A'' of ''X'', has value 1 at points o ...
, which he used as a tool in obtaining the distributions of certain
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s and in determining the limiting distribution of series of random variables (as in the
Hsu–Robbins–Erdős theorem In the mathematical theory of probability, the Hsu–Robbins–Erdős theorem states that if X_1, \ldots ,X_n is a sequence of i.i.d. random variables with zero mean and finite variance and : S_n = X_1 + \cdots + X_n, \, then : \sum\limits_ P ...
for which he is partially named). Another example of Hsu' work in this field, published around 1940, concerned a solution to the most general form of the
Central Limit Theorem In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which drew the attention of many famed mathematicians, such as Paul Lévy,
William Feller William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian–American mathematician specializing in probability theory. Early life and education Feller was born in Zagreb to Ida Oemichen-Perc, a Cro ...
,
Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Soviet ...
, and Boris Gnedenko. Hsu made contributions to this problem in his paper, "A general weak limit theorem for independent distributions", a manuscript mailed to Kai Lai Chung in 1947. In this paper, Hsu independently obtained a
necessary and sufficient In logic and mathematics, necessity and sufficiency are terms used to describe a material conditional, conditional or implicational relationship between two Statement (logic), statements. For example, in the Conditional sentence, conditional stat ...
condition under which the row sums of a
triangular matrix In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are z ...
of infinitesimal random variables,
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...
in each row, will converge in distribution to a given
infinitely divisible Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
distribution. Despite the fact that Gnedenko obtained the same result in 1944, Hsu's method is direct and has its own trait.Chung, K. L. (1979). Hsu's work in probability. The Annals of Statistics, 479–483.


Publications

* ''Collected Papers by P. L. Hsu''; edited by K.L. Chung, Springer-VerLag. (1983) .New York; . * ''许宝騄. (1982). 抽样论. 北京, 北京大学出版杜.''


References

{{DEFAULTSORT:Hsu, Pao-Lu 1910 births 1970 deaths 20th-century Chinese mathematicians Educators from Beijing Mathematicians from Beijing Members of Academia Sinica Members of the Chinese Academy of Sciences Academic staff of the National Southwestern Associated University Peking University alumni Academic staff of Peking University Mathematical statisticians