|---| | [ View source on GitHub](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/math/special.py#L2589-L2659) | Computes Owen's T function of `h` and `a` element-wise. tfp.math.owens_t( h, a, name=None ) ### Used in the notebooks | Used in the tutorials | |---| | - [TFP Release Notes notebook (0.13.0)](https://www.tensorflow.org/probability/examples/TFP_Release_Notebook_0_13_0) | 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 (\[Patefield and Tandy, 2000\]\[1\]). The Owen's T function has several notable properties which we list here for convenience. (\[Owen, 1980\]\[2\], 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(-(x**2 + y**2)/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 (\[Owen, 1980\]\[2\], 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


| ## Args || |---|---| | `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). |



| ## Returns || |---|---| | `owens_t` | A `Tensor` with the same type as `h` and `a`, |
#### References \[1\]: 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. \[2\]: Owen, Donald Bruce. "A table of normal integrals: A table." Communications in Statistics-Simulation and Computation 9.4 (1980): 389-419.