亚伯拉罕·瓦尔德Abraham Wald

亚伯拉罕·瓦尔德出生于匈牙利的一个犹太家庭。这是一个知识分子家庭，但作为犹太人，他们被迫从事远低于其能力的行业谋生。当时匈牙利的初等和中等学校都要求学生在星期六上课，而瓦尔德一家不能让他们的儿子在犹太安息日上学。因此，马克斯·亚伯拉罕既没有上初等学校也没有上中等学校，而是在家中由家庭成员教育。从教育的角度来看，这当然没有使他处于不利地位，因为他的家人都是知识渊博、能力出众的教师。

第一次世界大战后，曾是匈牙利一部分的许多土地被割让给邻国，当时克卢日成为罗马尼亚的一部分。瓦尔德获准进入克卢日大学，但看来这对他来说并不容易，因为他是犹太人。然而，他在数学方面的杰出才能使他希望继续从事数学研究，并于1927年进入维也纳大学，师从Karl 卡尔·门格尔。他在卡尔·门格尔的指导下研究几何学，并于1931年获得博士学位。

20世纪30年代的维也纳，一个年轻的犹太人无论多么有才华，都无法获得学术职位。事实上，当时几乎没有什么学术职位空缺。瓦尔德能够维持生计以便继续研究的唯一途径就是受雇工作。他这样做了，担任了卡尔·施莱辛格的数学导师，后者是奥地利一位重要的银行家和经济学家。1931年至1937年间，瓦尔德发表了21篇关于几何学的论文，卡尔·门格尔在[5]中将其描述为：-

……深刻、优美且具有根本重要性。

然而，他与施莱辛格的工作不仅给了他经济保障，从而有机会从事几何学研究。这也使他对将自己的数学技能应用于施莱辛格感兴趣的经济学和计量经济学问题产生了兴趣。在此期间，瓦尔德发表了10篇关于经济学和计量经济学的论文，他还在1936年出版了一部关于时间序列季节性变动的重要专著。其主要目的是给出消除这种季节性变动的方法。在[8]中，Morgenstern写道，在这部专著中，瓦尔德：-

……发展出了优于所有其他方法的技术。

瓦尔德当时在维也纳因其犹太人身份而遭遇的问题，可以从卡尔·门格尔讨论班的下场看出端倪。瓦尔德是该讨论班的成员，讨论班既有成员讲座，也邀请客人演讲。演讲者提出未解决的问题，并报告近期出版物和研究。瓦尔德向讨论班报告了他在计量经济学方面的工作，特别是他为讨论班写了一篇关于竞争经济模型解的存在性的论文。然而，该讨论班在1936年因犹太贡献者受到批评而被迫停止工作，其中一位当然是瓦尔德。如果说1936年犹太人在维也纳的生活已经很艰难，那么很快会变得更糟。

1938年，纳粹军队入侵奥地利。对于像瓦尔德这样的犹太人来说，纳粹统治下的条件往好里说是极其艰难，往坏里说是非常危险。考尔斯委员会邀请他去美国进行计量经济学研究，他于1938年夏天离开奥地利。移居美国几乎肯定救了他的命，因为他留在身后的九名家庭成员中，除一人外，其余全部死于纳粹奥斯维辛集中营的毒气室。到1938年9月，瓦尔德已成为卡内基公司的研究员，在纽约哥伦比亚大学跟随哈罗德·霍特林学习统计学。

瓦尔德一直是卡内基公司的研究员，直到1941年，但那时他已经开始在哥伦比亚大学授课，始于1939-40学年。他于1941年被任命为哥伦比亚大学教员，并一直在那里工作直至去世。除了在哥伦比亚大学的教学和研究外，美国参加第二次世界大战后，他还承担了战争工作，与哥伦比亚大学的统计研究小组一起从事军事项目。他利用自己的统计专长开发了一种估算飞机易损性的方法。

正如我们上面提到的，在维也纳，瓦尔德从事纯数学研究，主要是几何学，以及计量经济学。他的第一篇纯数学工作是关于度量空间的，将施泰尼茨的工作推广到无穷维向量空间，以及关于微分几何的一些漂亮结果。

然而，瓦尔德最重要的工作是在统计学领域。在[14]中，伍佛维兹——起初是他的学生，后来成为他的同事和合作者——将瓦尔德于1939年发表在Annals of Mathematical Statistics上的一篇论文描述为：-

……可能是瓦尔德最重要的单篇论文。

在这篇1939年的论文中瓦尔德[13]：-

……指出当时统计理论的两大问题，即假设检验和估计，都可以视为一个更一般问题的简单特例——如今称为“统计决策问题”。……他定义了损失函数、风险函数、先验分布、托马斯·贝叶斯决策规则、容许决策规则和极小极大决策规则，并证明在某些正则条件下极小极大决策规则具有常风险。

瓦尔德还发展了赌徒破产问题的推广，这些推广在统计序贯分析中起着重要作用。他发明了序贯分析这一课题，以回应二战期间对更高效的工业质量控制方法的需求。这里的想法很简单，但瓦尔德是第一个将其构建为统计理论的人。最好是对序贯产生的数据进行分析，而不是收集所有数据后再进行分析。在这种方法中，人们不选择固定的样本量，但如果结果证明合理，可以在任何时候结束抽样。

瓦尔德是第一个解决统计假设序贯检验一般问题的人。序贯概率比检验的最优性质由瓦尔德于1943年猜想，并在1948年与伍佛维兹合作的一篇论文中证明了这一性质。这项工作及相关工作非常注重实际应用，他关于所需观测数分布以及误差相关概率的定理立即得到了应用。他在序贯分析和决策函数理论（另一个由他创立的课题）方面的主要结果，汇集在他的专著Sequential Analysis（1947）中。

瓦尔德在与Karl Schlesinger共事期间持续关注的兴趣之一是经济学。他证明了重要的结果，也许最重要的是竞争经济模型解的存在性，正如我们上面所指出的，这是为卡尔·门格尔的讨论班写的。他在这一领域的其他工作涉及[5]：-

……时间序列的季节性校正、经济指数的近似公式、无差异曲面、生产瓦尔拉斯方程组扩展形式的解的存在性和唯一性、安托万·奥古斯丁·古诺双头垄断问题，最后，在他与Mann合著的被广泛使用的工作（1943）中，随机差分方程。

伍佛维兹在[1]中写道：-

他对统计学的伟大贡献之一是在问题的表述中引入数学精确性，在论证中引入数学严谨性。这些品质在他1938年开始统计学生涯时往往是缺乏的，它们已经改变了这门学科——尽管不一定让每个人都满意。

瓦尔德不仅在研究中对统计学产生了显著影响，而且尽管他只教了大约十年书，作为教师他也产生了显著影响。他在研究中表现出的精确和严谨的品质也被带到了他的教学中，但这并不意味着他的讲座很复杂。相反，他的讲座以清晰和[13]而闻名：-

他擅长以极其简单的方法推导出复杂的结果。

他的学生在哥伦比亚大学听课时所做的笔记被传抄，由于其出色的清晰性，这些笔记传到了美国许多不同大学学习统计学的学生手中。

瓦尔德移居美国后遇到了Lucille Lang，两人结了婚。1950年，瓦尔德收到印度政府的邀请，去该国讲授统计学。他与妻子一同前往印度，不幸的是两人都在一次飞机失事中遇难。

Freeman曾在哥伦比亚大学听过瓦尔德的课，他在5中写到瓦尔德的性格：-

瓦尔德是一个安静温和的人，深深沉浸于自己的工作。他相当不喜闲聊，几乎没有爱好。但他并非不在意认可……

Abraham Wald was born into a Jewish family in Hungary. It was a family of intellectuals but, being Jewish, they were forced to earn their living in trades well below their abilities. At this time both primary and secondary schools in Hungary required pupils to attend on Saturdays and the Wald family could not allow their son to attend school on the Jewish Sabbath. As a result Abraham did not attend either primary of secondary school and was educated at home by members of his family. This certainly did not put him at a disadvantage from an educational point of view for his family were very knowledgeable and competent teachers.

After World War I much of the land that had been part of Hungary was given to neighbouring countries and at that time Cluj became part of Romania. Wald was allowed to attend the University of Cluj but it appears that this was not made easy for him because he was Jewish. However his outstanding abilities in mathematics led him to wish to continue to undertake mathematical research and in 1927 he entered the University of Vienna to study with Karl Menger. He worked under Menger's supervision on geometry and was awarded his doctorate in 1931.

Vienna in the 1930s was no place for a young Jewish man to obtain an academic position, no matter how talented. In fact there were few academic positions open at all. The only way that Wald could support himself so as to be able to continue with his research was to take employment. This he did taking the position of mathematics tutor to Karl Schlesinger, a leading Austrian banker and economist. Between 1931 and 1937 Wald published 21 papers on geometry which Menger describes in [5] as:-

... deep, beautiful and of fundamental importance.

However his work with Schlesinger did not only give him financial security and hence the opportunity to undertake research in geometry. It also gave him an interest in applying his mathematical skills to the problems in economics and econometrics which interested Schlesinger. During this period Wald published 10 papers on economics and econometrics, and he also published an important monograph in 1936 on seasonal movements in time series. The main aim was to give methods to eliminate this seasonal variation. In [8] Morgenstern writes that in this monograph Wald:-

... developed techniques superior to all others.

An indication of the problems that Wald had in Vienna at this time because he was Jewish is indicated by the fate of Menger's seminar. Wald was a member of this seminar which included lectures by both members of the seminar and also invited guests. Speakers raised open problems, and reported on recent publications and research. Wald reported to the seminar on his work in econometrics, in particular he wrote a paper for the seminar on the existence of a solution to the competitive economic model. However, the seminar was forced to stop work in 1936 after it was criticised for its Jewish contributors, one of whom of course was Wald. If life was difficult for a Jew in Vienna in 1936, they would soon become much worse.

In 1938 the Nazi forces invaded Austria. For a Jewish person like Wald conditions under the Nazis were at best extremely difficult and at worst very dangerous. The Cowles Commission invited him to the United States to do econometric research in the United States and he left Austria in the summer of 1938. The move to the United States almost certainly saved his life for all but one of the nine members of his family left behind died in the gas chambers of the Nazi concentration camp at Auschwitz. By September 1938 Wald was a Fellow of the Carnegie Corporation studying statistics at Columbia University in New York under Hotelling.

Wald remained a Fellow of the Carnegie Corporation until 1941 but by that time he had already begun lecturing at Columbia University which he began in academic year 1939-40. He was appointed to the Faculty of Columbia University in 1941 and he remained on the staff there until his death. In addition to his teaching and research at Columbia, he undertook war work after the United States entered World War II, working on military projects with the Statistics Research Group at Columbia. He used his statistical expertise to develop a method to estimate aircraft vulnerability.

As we mentioned above, in Vienna Wald worked on pure mathematics, mostly geometry, and on econometrics. His first pure mathematical work was on metric spaces, an extension of Steinitz's work to infinite dimensional vector spaces, and some beautiful results on differential geometry.

Wald's most important work, however, was in statistics. In [14] Wolfowitz, who was first his student, then his colleague and collaborator, described a paper Wald published in the Annals of Mathematical Statistics in 1939 as:-

... probably Wald's most important single paper.

In this 1939 paper Wald [13]:-

... points out that the two major problems of statistical theory at that time, testing hypotheses and estimation, can both be regarded as simple special cases of a more general problem - known nowadays as a "statistical decision problem". ... He defines loss functions, risk functions, a priori distributions, Bayes decision rules, admissible decision rules, and minimax decision rules, and proves that a minimax decision rule has a constant risk under certain regularity conditions.

Wald also developed generalizations of the problem of gambler's ruin which play an important role in statistical sequential analysis. He invented the topic of sequential analysis in response to the demand for more efficient methods of industrial quality control during World War II. The idea here is a simple one yet Wald was the first to build it into a statistical theory. It is better to analyse data produced sequentially rather than collect all the data and then analyse it. In this approach one does not choose a fixed sample size but can end the sampling at any time if the results justify it.

Wald was the first to solve the general problem of sequential tests of statistical hypotheses. The optimum property of the sequential probability ratio test was conjectured by Wald in 1943 and, in a joint paper with Wolfowitz in 1948, he proved this property. This and related work was very much aimed at practical applications and his theorems on the distribution of the required number of observations, and on the probabilities associated with errors, found immediate applications. His main results on sequential analysis and the theory of decision functions, another topic which was founded by him, were gathered together in his monograph Sequential Analysis (1947).

One of Wald's continuing interests from his time working with Karl Schlesinger was economics. He proved important results, perhaps the most significant being the existence of a solution to the competitive economic model which, as we noted above was written for Menger's seminar. His other work in this area related to [5]:-

... seasonal corrections to time series, approximate formulas for economic index numbers, indifference surfaces, the existence and uniqueness of solutions of extended forms of the Walrasian system of equations of production, the Cournot duopoly problem, and finally, in his much used work written with Mann (1943), stochastic difference equations.

Wolfowitz writes in [1] that:-

One of his great contributions to statistics was to bring to it mathematical precision in the formulation of problems and mathematical rigour in argument. These qualities, which were often lacking when he began his statistical career in 1938, have transformed the subject - although not necessarily to the satisfaction of everyone.

It was not only in research that Wald had a remarkable influence on statistics, but although he only taught for about ten years, he also had a marked influence as a teacher. The same qualities of precision and rigour he showed in research were brought to his teaching but this did not mean that his lectures were complicated. On the contrary his lectures were renowned for their clarity and [13]:-

He was a master at deriving complicated results in amazingly simple ways.

The notes which his students took during his lectures in Columbia were circulated and because of their outstanding clarity they reached students studying statistics at many different universities in the United States.

After Wald emigrated to the United States he met Lucille Lang and the two were married. In 1950 Wald received an invitation from the Indian government to lecture on statistics in that country. He went to India with his wife and tragically they were both killed in a plane crash.

Freeman, who attended Wald's lectures at Columbia, writes in [5] about Wald's character:-

Wald was a quiet and gentle man, deeply immersed in his work. He was fairly aloof from small talk, and he had few hobbies. But he was not indifferent to recognition...