Harold Hotelling

Key takeaways

Harold Hotelling belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Harold Hotelling to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Harold Hotelling from memory before moving on to harder problems.

Reference excerpt

Harold Hotelling (; September 29, 1895 – December 26, 1973) was an American mathematical statistician and an influential economic theorist, known for Hotelling's law, Hotelling's lemma, and Hotelling's rule in economics, as well as Hotelling's T-squared distribution in statistics. He also developed and named the principal component analysis method widely used in finance, statistics and computer science. He was associate professor of mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a professor of Mathematical Statistics at the University of North Carolina at Chapel Hill from 1946 until his death. A street in Chapel Hill bears his name. In 1972, he received the North Carolina Award for contributions to science.

Statistics Hotelling is known to statisticians because of Hotelling's T-squared distribution which is a generalization of the Student's t-distribution in multivariate setting, and its use in statistical hypothesis testing and confidence regions. He also introduced canonical correlation analysis. At the beginning of his statistical career Hotelling came under the influence of R.A. Fisher, whose Statistical Methods for Research Workers had "revolutionary importance", according to Hotelling's review. Hotelling was able to maintain professional relations with Fisher, despite the latter's temper tantrums and polemics. Hotelling suggested that Fisher use the English word "cumulants" for Thiele's Danish "semi-invariants". Fisher's emphasis on the sampling distribution of a statistic was extended by Jerzy Neyman and Egon Pearson with greater precision and wider applications, which Hotelling recognized. Hotelling sponsored refugees from European anti-semitism and Nazism, welcoming Henry Mann and Abraham Wald to his research group at Columbia. While at Hotelling's group, Wald developed sequential analysis and statistical decision theory, which Hotelling described as "pragmatism in action". In the United States, Hotelling is known for his leadership of the statistics profession, in particular for his vision of a statistics department at a university, which convinced many universities to start statistics departments. Hotelling was known for his leadership of departments at Columbia University and the University of North Carolina.

Economics Hotelling has a crucial place in the growth of mathematical economics; several areas of active research were influenced by his economics papers. While at the University of Washington, he was encouraged to switch from pure mathematics toward mathematical economics by the famous mathematician Eric Temple Bell. Later, at Columbia University (where during 1933-34 he taught Milton Friedman statistics) in the '40s, Hotelling in turn encouraged young Kenneth Arrow to switch from mathematics and statistics applied to actuarial studies towards more general applications of mathematics in general economic theory. Hotelling is the eponym of Hotelling's law, Hotelling's lemma, and Hotelling's rule in economics. Hotelling was influenced by the writing of Henry George and was an editorial adviser for the Georgist journal AJES.

Spatial economics

One of Hotelling's most important contributions to economics was his conception of "spatial economics" in his 1929 article. Space was not just a barrier to moving goods around, but rather a field upon which competitors jostled to be nearest to their customers. Hotelling considers a situation in which there are two sellers at point A and B in a line segment of size l. The buyers are distributed uniformly in this line segment and carry the merchandise to their home at cost c. Let p1 and p2 be the prices charged by A and B, and let the line segment be divided in 3 parts of size a, x+y and b, where x+y is the size of the segment between A and B, a the portion of segment to the left of A and b the portion of segment to the right of B. Therefore, a+x+y+b=l. Since the product being sold is a commodity, the point of indifference to buying is given by p1+cx=p2+cy. Solving for x and y yields:

x = 1 2 ( l − a − b + p 2 − p 1 c ) {\displaystyle x={\frac {1}{2}}\left(l-a-b+{\frac {p_{2}-p_{1}}{c}}\right)}

y = 1 2 ( l − a − b + p 1 − p 2 c ) {\displaystyle y={\frac {1}{2}}\left(l-a-b+{\frac {p_{1}-p_{2}}{c}}\right)}

Let q1 and q2 indicate the quantities sold by A and B. The sellers profit are:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Harold Hotelling

Start with the simplest possible case. Write down what Harold Hotelling claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Harold Hotelling before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Harold Hotelling ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Harold Hotelling

In research: Harold Hotelling appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Harold Hotelling in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Harold Hotelling is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1895 births, 1973 deaths, 20th-century American economists, so understanding it makes those chapters shorter.
In everyday life: Look for Harold Hotelling outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Harold Hotelling in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Harold Hotelling means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Harold Hotelling out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Harold Hotelling in simple terms?

Harold Hotelling (; September 29, 1895 – December 26, 1973) was an American mathematical statistician and an influential economic theorist, known for Hotelling's law, Hotelling's lemma, and Hotelling's rule in economics, as well as Hotelling's T-squared distribution in statistics. He also developed…

Why does Harold Hotelling matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Harold Hotelling?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Harold Hotelling.