# Owen's Q-function

### Description

Calculates Owen's Q function.

### Usage

1 | ```
OwensQ(nu, t, delta, a=0, b)
``` |

### Arguments

`nu` |
degree of Owen's Q |

`t` |
parameter t |

`delta` |
parameter delta |

`a` |
lower integration limit, only a=0 implemented |

`b` |
upper integration limit |

### Details

Uses the relationship to non-central t-distribution (see Chou YM)

OwensQ = pt(t, df=nu, ncp=delta) - Integal_b_Inf(Q_integrand)

The definite integral is numerically evaluated using `integrate()`

from
package `stats`

after a variables transformation resulting in the
integration range from 0 to 1 instead of the semi-infinite original range.
This may result in higher precision and better numerical stability.

The arguments to the function must be scalars. No vectors allowed.

### Value

Numeric value of Owen's Q-function at given input arguments.

### Note

This function is intended for internal use in the power calculations.

But may be useful for others.

### Author(s)

D. Labes

### References

Owen, DB (1965)

"A Special Case of a Bivariate Non-central t-Distribution"

Biometrika, 52, 437-446.

\Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.2307/2333696")}

Chou YM (1992)

"A bivariate noncentral T-distibution with applications"

Communications in Statistics - Theory and Methods, 21:12, 3427-3462

\Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1080/03610929208830988")}

### See Also

`OwensQOwen`

### Examples

1 2 3 4 5 | ```
# This function is mainly intended for internal use.
OwensQ(10, 2.5, 5, 0, 2)
#should give [1] 9.388137e-06
OwensQ(10, -2.5, -5, 0, 2)
#should give [1] 0.05264363
``` |