Why statistics requires high performance computing?

Besides numerous reasons, such as; efficiency of statistical algorithms, accuracy of results, replicability of studies and reproducibility of results (Wesonga et al. 2019), below I will briefly describe three:

1. An almost parallel development of statistics theory and computing tools
2. Definitional reasons
3. Sampling and Central Limit Theory

Development of Computing and Statistics

The developers of mechanical calculating machines have been in a way or the other closely associated to statistics (Sint 1984; Grossmann, Schimek, and Sint 2004). Brief outline: (Wilson 2015, 2001; Rojas-Sola et al. 2021; Kistermann 1998; Coale, Demeny, and Vaughan 2013; Coale and Trussell 1996)

• Wilhelm Schickard (1592-1635) built the first mechanical calculator, which performed multiplication and divisions using an algorithmic idea of Kepler in 1623.
• Blaise Pascal (1623-1662) constructed a machine for addition and subtraction, which was used in accounting. He also studied the theory of games of chance, giving birth to the modern theory of probability .
• Gottfried Wilhelm Leibnitz (1646-1716) constructed a four species machine that incorporated the Leibnitz wheel. He, thus created a link between mathematical computational aspect “old” statistics and political arithmetic leading to the “modern” statistics, known as official statistics. Brought data on registration figures of christenings and funerals in Breslau to the Royal Society, which Halleys used for his famous paper on mortality containing real life tables in 1693 (John 1884, Pearson 1978).

Result 1

If \(X_1, X_2, \cdots,X_n\) are independent identically distributed random variables (Data), and \[Statistics \propto f(Data, Computing)\] Computing seems to be lagged statistics. That is, the growth of computing power is highly correlated with the growth of statistics.

Definition of Statistics and Computing

Statistics is derived from the Italian/Latin to describe the knowledge of statesmanship or state affairs. “Statist” in Shakespears Hamlet and Cymbeline was used for that purpose. However, Gabriel Naude (1639) was the first to use the adjective statisticus in print (Pearson 1978 p.4). Subsequently, Leibnitz, Hermann Conring (1606-1681) founded “statistics” as the art of collecting relevant data for decision making by government. Gottfried Anchenwell (1719-1772) though widely recognised as the founder of statistics, himself knew he just extended Conring’s ideas, but appreciated mathematical methods and information derived from the collected data. Therefore, plainly speaking statistics is the process of planning, collecting, processing, analysing and interpretation of data for knowledge discovery. Similarly, computing is the art of processing input data to generate output information (Sint 1984; Grossmann, Schimek, and Sint 2004).

Result 2

If \(X_1, X_2, \cdots,X_n\) are independent identically distributed random variables, and \[Statistics = Theory + Data + Computing\] Thus, Computing is one of the Attributes of Statistics

Sampling Theory and Central Limit Theorem

CLT is a popular statistical concept: (Le Cam 1986; Johnson 2004; Fischer 2011)

Definition of CLT

Let \(X_1, X_2, \cdots,X_n\) be independent identically distributed random variables with mean \(\mu\) and variance \(\sigma^2\) both finite. Then, for any constant \(z\) \[lim_{n\rightarrow\infty}P\left(\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}\le z\right)=\Phi(z)\] where \(\Phi\) is the cdf of the standard normal distribution.

Demonstrating the large sample

• The \(n \longrightarrow \infty\) concept

Result 3

For a given set of parameters, belonging to some distribution \(X_1, X_2, \cdots,X_p \sim D(\mu, \Sigma)\) then \[D(\mu, \Sigma)_{n\rightarrow \infty} \longrightarrow N(\mu, \Sigma)\] Implication Computing Power is Necessary to Achieve Normality.