Tine Distribution - Probability How To

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What is the Tine Distribution?

The little known Tine distribution (sometimes called the symmetric triangular distribution) made an entry in the Index to the Distributions of Mathematical Statistics [1] as:

The Annals of Mathematical Statistics [2] contains an expanded definition:

Schmidt’s notes on the tine distribution — *Schmidt’s notes on the “Tine Curve” from Annals of Mathematical Statistics [2].*

Rinne [3] defines Tine’s distribution as the distribution of two independent and identically distributed (i.i.d.) uniform variables (i.e., the convolution of two uniform distributions):

X₁, X₂ iid∼ UN(a, b) ⇒ X = X₁ + X₂ ∼ TS(2 a + b, b).

A convolution is an operation on two function (f and g) that produces a third function ( $f*g$ ), which expresses how the shape of one function is modified by the other.

The Tine distribution is also called Simpson’s distribution, after Thomas Simpson (1710-1761) who is thought to be the first to suggest the distribution [3].

The distribution isn’t widely known. In fact, if you try and Google “Tine Distribution” you’ll be redirected (at the time of writing) to pages on “time distribution” instead. It’s also not often used (most likely because it isn’t well known!), but there are a few specific use cases. For example, this Google patent for an “Authentication device and authentication method” includes the Tine distribution as a threshold measure.

The threshold value determination part 22 is Mahalanobis prescribed | regulated by the Mahalanobis distance prescribed | regulated by the mean value and the standard deviation of a person distribution, and the average value and standard deviation of a tine distribution. To match the distance, the threshold value Xth is determined.
Google patent for an authentication device

Note that the patent makes reference to a “person distribution” which is most likely the author’s name for the distribution of biometric authentication data.

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References

[1] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[2] Schmidt, R. Statistical Analysis of One-Dimensional Distributions. Annals of Mathematical Statistics. 5:33.

[3] Rinne, H. Location–Scale Distributions Linear Estimation and Probability Plotting Using MATLAB. Online: http://geb.uni-giessen.de/geb/volltexte/2010/7607/pdf/RinneHorst_LocationScale_2010.pdf