Helmert’s Distribution


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Helmert’s distribution is another name for the chi-square distribution. It is named after F.R. Helmert, who proved the general reproductive property of chi-square distributions [1].

Helmert’s most noted contribution was establishing that if a set of independent, normally distributed random variables X1, X2, …, Xn, then

  1. Is distributed as a chi-square variable with n – 1 degrees of freedom
  2. This chi-square variable is statistically independent of “X-bar” (the sample mean).

Helmert also proved that the sample mean, and variance are independent in 1785 [2].  Helmert’s transformation allows us to obtain a set of k – 1 new independent and identically distributed normally distributed samples with μ = 0 and the same variance of the original distribution. Given a set of xk normally distributed i.i.d. variables, the transformation to yj random variables can be defined as [3]

Because of this important contribution to sampling theory, Kruskal [4] recommended naming the joint distribution of the two random variables “Helmert’s Distribution”. However, the recommendation did not take, and we still refer to Helmert’s result as the chi-square.

Helmert’s distribution is important in sampling theory because it leads to [4]

  1. Statistical independence of sample mean (μ) and sample standard deviation (σ).
  2. Separate single distributions of μ and σ.
  3. Distribution of Student’s t.

Helmert obtained his distribution by setting up a pair of linear transformations. These transformed the joint distribution of individual observation errors into a joint distribution of errors for μ and σ; any dummy variables could be integrated out.

References

[1] Helmert, F. R. (1876). Die Genauigkeit der Formel von Peters zue Berechnung des wahrscheinlichen Beobachtungsfehlers directer Beobachtungen gleicher Genauigkeit, Astronomische Nachrichten, 88, columns 113-120;  Helmert, F. R. (1876). ~ber die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und iiber einige damit in Zusammenhange stehende Fragen, Zeitschrift fir angewandte Mathematik und Physik, 21, 192-218.

[2] Helmert, F. R. (1875). ~ber die Berechnung der wahrscheinlichen Fehlers aus einer endlichen Anzahl wahrer Beobachtungsfehler, Zeitschrift fur angewandte Mathe[1]matik und Physik, 20, 300-303.

[3] Wu, M. et al. (2019). Differentiable Antithetic Sampling for Variance Reduction in Stochastic Variational Inference. Online: https://arxiv.org/pdf/1810.02555.pdf

[4]  Kruskal, W. Helmert’s Distribution. The American Mathematical Monthly. Vol. 53, No. 8 (Oct., 1946), pp. 435-438 (4 pages). Published By: Taylor & Francis, Ltd.