powen {OwenQ}

Owen distribution functions when δ₁>δ₂

Description

Evaluates the Owen distribution functions when the noncentrality parameters satisfy δ₁>δ₂ and the number of degrees of freedom is integer.

• powen1 evaluates P(T₁ ≤ t₁, T₂ ≤ t₂) (Owen's equality 8)

• powen2 evaluates P(T₁ ≤ t₁, T₂ > t₂) (Owen's equality 9)

• powen3 evaluates P(T₁ > t₁, T₂ > t₂) (Owen's equality 10)

• powen4 evaluates P(T₁ > t₁, T₂ ≤ t₂) (Owen's equality 11)

Usage

powen1(nu, t1, t2, delta1, delta2, algo = 2)

powen2(nu, t1, t2, delta1, delta2, algo = 2)

powen3(nu, t1, t2, delta1, delta2, algo = 2)

powen4(nu, t1, t2, delta1, delta2, algo = 2)

Arguments

nu

integer greater than 1, the number of degrees of freedom; infinite allowed

t1, t2

two numbers, positive or negative, possible infinite

delta1, delta2

two vectors of possibly infinite numbers with the same length, the noncentrality parameters; must satisfy delta1>delta2

algo

the algorithm used, 1 or 2

Value

A vector of numbers between 0 and 1, possibly containing some NaN.

Note

When the number of degrees of freedom is odd, the procedure resorts to the Owen T-function (OwenT).

References

Owen, D. B. (1965). A special case of a bivariate noncentral t-distribution. Biometrika 52, 437-446.

See Also

Use psbt for general values of delta1 and delta2.

Examples

nu=5; t1=2; t2=1; delta1=3; delta2=2
# Wolfram integration gives 0.1394458271284726
( p1 <- powen1(nu, t1, t2, delta1, delta2) )
# Wolfram integration gives 0.0353568969628651
( p2 <- powen2(nu, t1, t2, delta1, delta2) )
# Wolfram integration gives 0.806507459306199
( p3 <- powen3(nu, t1, t2, delta1, delta2) )
# Wolfram integration gives 0.018689824158
( p4 <- powen4(nu, t1, t2, delta1, delta2) )
# the sum should be 1
p1+p2+p3+p4