# tfp.math.owens_t

Computes Owen's T function of `h` and `a` element-wise.

### Used in the notebooks

Used in the tutorials

Owen's T function is defined as the combined probability of the event `X > h` and `0 < Y < a * X`, where `X` and `Y` are independent standard normal random variables.

In integral form this is defined as `1 / (2 * pi)` times the integral of `exp(-0.5 * h ** 2 * (1 + x ** 2)) / (1 + x ** 2)` from `0` to `a`. `h` and `a` can be any real number

The Owen's T implementation below is based on ([Patefield and Tandy, 2000]).

The Owen's T function has several notable properties which we list here for convenience. ([Owen, 1980], page 414)

• P2.1 `T( h, 0) = 0`
• P2.2 `T( 0, a) = arctan(a) / (2 pi)`
• P2.3 `T( h, 1) = Phi(h) (1 - Phi(h)) / 2`
• P2.4 `T( h, inf) = (1 - Phi(|h|)) / 2`
• P2.5 `T(-h, a) = T(h, a)`
• P2.6 `T( h,-a) = -T(h, a)`
• P2.7 `T( h, a) + T(a h, 1 / a) = Phi(h)/2 + Phi(ah)/2 - Phi(h) Phi(ah) - [a<0]/2`
• P2.8 `T( h, a) = arctan(a)/(2 pi) - 1/(2 pi) int_0^h int_0^{ax}` exp(-(x2 + y2)/2) dy dx`
• P2.9 `T( h, a) = arctan(a)/(2 pi) - int_0**h phi(x) Phi(a x) dx + Phi(h)/2 - 1/4`

`[a<0]` uses Iverson bracket notation, i.e., `[a<0] = {1 if a<0 and 0 otherwise`.

Let us also define P2.10 as:

• P2.10 `T(inf, a) = 0`
• Proof

Note that result #10,010.6 ([Owen, 1980], pg 403) states that: `int_0^inf phi(x) Phi(a+bx) dx = Phi(a/rho)/2 + T(a/rho,b) where rho = sqrt(1+b**2).` Using `a=0`, this result is: `int_0^inf phi(x) Phi(bx) dx = 1/4 + T(0,b) = 1/4 + arctan(b) / (2 pi)` Combining this with P2.9 implies

``````T(inf, a)
=  arctan(a)/(2 pi) - [ 1/4 + arctan(a) / (2 pi)]  + Phi(inf)/2 - 1/4
= -1/4 + 1/2 -1/4 = 0.
``````

QED

`h` A `float` `Tensor` defined as in `P({X > h, 0 < Y < a X})`. Must be broadcastable with `a`.
`a` A `float` `Tensor` defined as in `P({X > h, 0 < Y < a X})`. Must be broadcastable with `h`.
`name` A name for the operation (optional).

`owens_t` A `Tensor` with the same type as `h` and `a`,

: Patefield, Mike, and D. A. V. I. D. Tandy. "Fast and accurate calculation of Owen’s T function." Journal of Statistical Software 5.5 (2000): 1-25. http://www.jstatsoft.org/v05/i05/paper : Owen, Donald Bruce. "A table of normal integrals: A table." Communications in Statistics-Simulation and Computation 9.4 (1980): 389-419.

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