Statistical Approaches to Length of Reign Boyko Zlatev University of Alberta Canadian Young Researchers Conference in Mathematics and Statistics. Edmonton, May 2010

The Date of Founding of Rome Last Roman King deposed c. 503 BC. Traditional date of founding of Rome – c. 760 BC. Reigns of 7 Roman Kings cover 257 years in total. Isn’t it a very large value?

Sir Isaac Newton ( ) and “The Chronology of Ancient Kingdoms Amended” (1728) Newton was not much interested in probability (Sheynin, 1971). But he was interested in chronology! He reconstructed ancient chronology and made it “fit better with the course of Nature”. For that he analyzed some data, available in his time.

Newton’s Data and Results Average lengths of reign for different states concentrated in the interval 18÷20 years. In fact, this is roughly a 65% confidence interval, as shown by Stigler (JASA, 1977).

François-Marie Arouet de Voltaire ( ) Confirmed Newton’s results on the reigns on the German Emperors (elected!) – average is 18.4 years. Applied Newton’s rule to Chinese Kings. Noted that in states where revolutions are frequent, Newton’s rule shows too large a value.

Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet ( ) Determined the chance that 7 kings shall reigned exactly 257 years. Applied “la probabilité propre” – “appropriate probability” (not clear what is it) to find the solution. Contributed to shortening the reign of Louis XVI – King of France.

Condorcet’s Solution Following De Moivre’s Law of Mortality with 90 years as limit of life, and the ages of (s)elected kings between 30 and 60 years, the probability seven kings reigning s years is the coefficient c s of x s in the expansion of:

Condorcet’s Solution (continued) Condorcet found P = c 257 = (may be it is an approximation only) Computation with R gives P = c 257 = , but it is also not very accurate: Mathematica gives P = c 257 =

Condorcet’s Solution (continued) “Appropriate probability” for 414 events possible: If P from R computation is used, the result is To find P(s≥257), one has to compute:

Karl Pearson “Biometry and Chronology” (Biometrika, 1928) “The frequency distribution of the reigns of souvereigns is given. What is the chance that a sample of seven selected at random will give a total length of reigns of 257 or more years?”

Karl Pearson “Biometry and Chronology” (Biometrika, 1928) (continued) The frequency distribution we have is not for the same country at the same time (further discussions are possible about which is more important – time or space). In reality averages for different countries and times are (according to Pearson) in surprising agreement with Newton’s results, i.e. concentrated between 18 and 20 years.

Karl Pearson “Biometry and Chronology” (Biometrika, 1928) (continued) Histogram for 250 European Monarchs

Karl Pearson “Biometry and Chronology” (Biometrika, 1928) (continued) Distribution of Means in Samples of Seven Reigns According to this distribution, Pearson found the probability for seven kings reigning at least 257 years to be This is about ½ of the result based on de Moivre’s Law of Mortality

Karl Pearson “Biometry and Chronology” (Biometrika, 1928) (continued) “Seven men are chosen at random between the ages of 30 and 60. Find the chance that a sample of seven selected at random will give a total length of reigns of 257 or more years”. The life-table chosen will not be from same time and same country. The choice of age interval is questionable.

Solution by Charles F. Trustam (Biometrika, 1928) H M life table is used (healthy males from England and Scotland, mid. XIX century). Linear approximation for λ between 0 and 45. P =

1943 E.Rubin. The Place of Statistical Methods in Modern Historiography. American Journal of Economics and Sociology. S.Ya.Lurie. Newton – Historian of the Antiquity. In Proceedings of Symposium (USSR), dedicated to 300-th anniversary from Newton’s birth In both papers Newton’s approach to the date of founding of Rome and similar problems is examined. As pointed by Lurie, in some cases Newton’s method gives surprisingly accurate results, confirmed by evidence found later. According to Rubin: “Statistical methods may only be applied if the historian knows something as to the reliability of the data. (…) There must be a clear comprehension of these techniques, which implies an understanding of their possibilities and limitations”.

Thomas R. Trautmann. Length of Generation and Reign in Ancient India. Journal of the American Oriental Society (1969) Hypotheses concerning ancient Indian chronology must be tested by comparing with a sample of kings from medieval India whose dates are better known. The problems of interregna and missing data are considered. Estimations based on the number of generations are more accurate then those based on the number of reigns.

E. Khmaladze, R. Browning, J. Haywood Brittle power: On Roman Emperors and exponential lengths of rule. Statistics & Probability Letters (2007) Hypothesis tested: a sample T 1, …, T n of n independent r.vs. follows exponential distribution with some a priori unspecified λ>0. One could consider an empirical distribution function of the sample and compare it to, with estimated by from the same data. Instead of taking the difference there is an advantage to take where K is called the “compensator” in the martingale theory of point processes. For the family of exponential distributions its form is simple. If the hypothesis of exponentiality is true, the version of the empirical process converges in distribution to the standard Brownian motion. So, the classical Kolmogorov-Smirnov statistic can be used.

E. Khmaladze et al. (continued) Results For the three chronological tables of Roman Emperors (those of Kienast, Parkin and Gibbon) the p-values are 0.26, 0.17 and 0.20, respectively. Hence, the hypothesis of exponentiality is not rejected. Exponentiality rejected for Chinese Emperors (p=0.028). Age of ascent dependence for British and Spanish monarchs.

Why? “…their [Roman Emperors’] lives were constantly exposed to turmoil and life threatening challenges, no matter in peace or war” (Khmaladze at al.) But the same must hold for other monarchs also! There are differences in the inheritance rules!

Topology of Data: Family tree (graph!) mapped on a time interval

Idea: to apply FDA Considering non-homogeneous Poisson process with intensity µ(t) = exp(W(t)), representing W(t) in terms of a set of basis functions (B-splines, Fourier, etc.) and minimizing the following: (Ramsay & Silverman, Functional Data Analysis, 2006).

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