Package 'OwenQ'

First Owen Q-function

Description

Evaluates the first Owen Q-function (integral from $0$ to $R$) for an integer value of the degrees of freedom.

Usage

OwenQ1(nu, t, delta, R, algo = 2)

Arguments

nu	integer greater than $1$, the number of degrees of freedom
t	number, positive or negative, possibly infinite
delta	vector of finite numbers, with the same length as R
R	(upper bound of the integral) vector of finite positive numbers, with the same length as delta
algo	the algorithm, 1 or 2

Value

A vector of numbers between $0$ and $1$, the values of the integral from $0$ to $R$.

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.

Examples

# As R goes to Inf, OwenQ1(nu, t, delta, R) goes to pt(t, nu, delta):
OwenQ1(nu=5, t=3, delta=2, R=100)
pt(q=3, df=5, ncp=2)

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Second Owen Q-function

Description

Evaluates the second Owen Q-function (integral from $R$ to $\infty$) for an integer value of the degrees of freedom.

Usage

OwenQ2(nu, t, delta, R, algo = 2)

Arguments

nu	integer greater than $1$, the number of degrees of freedom
t	number, positive or negative, possibly infinite
delta	vector of finite numbers, with the same length as R
R	(lower bound of the integral) vector of finite positive numbers, with the same length as delta
algo	the algorirthm used, 1 or 2

Value

A vector of numbers between $0$ and $1$, the values of the integral from $R$ to $\infty$.

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.

Examples

# OwenQ1(nu, t, delta, R) + OwenQ2(nu, t, delta, R) equals pt(t, nu, delta):
OwenQ1(nu=5, t=3, delta=2, R=1) + OwenQ2(nu=5, t=3, delta=2, R=1)
pt(q=3, df=5, ncp=2)

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Owen T-function

Description

Evaluates the Owen T-function.

Usage

OwenT(h, a)

Arguments

h	numeric scalar
a	numeric scalar

Details

This is a port of the function owens_t of the boost collection of C++ libraries.

Value

A number between 0 and 0.25.

References

Owen, D. B. (1956). Tables for computing bivariate normal probabilities. Ann. Math. Statist. 27, 1075-1090.

Examples

integrate(function(x) pnorm(1+2*x)^2*dnorm(x), lower=-Inf, upper=Inf)
pnorm(1/sqrt(5)) - 2*OwenT(1/sqrt(5), 1/3)

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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

---

Owen distribution functions

Description

Evaluates the Owen cumulative distribution function for an integer number of degrees of freedom.

• psbt1 evaluates P(T₁ ≤ t₁, T₂ ≤ t₂)

• psbt2 evaluates P(T₁ ≤ t₁, T₂ > t₂)

• psbt3 evaluates P(T₁ > t₁, T₂ > t₂)

• psbt4 evaluates P(T₁ > t₁, T₂ ≤ t₂)

Usage

psbt1(nu, t1, t2, delta1, delta2, algo = 2)

psbt2(nu, t1, t2, delta1, delta2, algo = 2)

psbt3(nu, t1, t2, delta1, delta2, algo = 2)

psbt4(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, possibly infinite
delta1, delta2	two vectors of possibly infinite numbers with the same length, the noncentrality parameters
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

It is better to use powen if delta1>delta2.

Examples

nu=5; t1=1; t2=2; delta1=2; delta2=3
( p1 <- psbt1(nu, t1, t2, delta1, delta2) )
( p2 <- psbt2(nu, t1, t2, delta1, delta2) )
( p3 <- psbt3(nu, t1, t2, delta1, delta2) )
( p4 <- psbt4(nu, t1, t2, delta1, delta2) )
# the sum should be 1
p1+p2+p3+p4

---

Student CDF with integer number of degrees of freedom

Description

Cumulative distribution function of the noncentrel Student distribution with an integer number of degrees of freedom.

Usage

ptOwen(q, nu, delta = 0)

Arguments

q	quantile, a finite number
nu	integer greater than $1$, the number of degrees of freedom; possibly infinite
delta	numeric vector of noncentrality parameters; possibly infinite

Value

Numeric vector, the CDF evaluated at q.

Note

The results are theoretically exact when the number of degrees of freedom is even. When odd, the procedure resorts to the Owen T-function.

References

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

Examples

ptOwen(2, 3) - pt(2, 3)
ptOwen(2, 3, delta=1) - pt(2, 3, ncp=1)

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Special case of second Owen distribution function

Description

Evaluation of the second Owen distribution function in a special case (see details).

Usage

spowen2(nu, t, delta, algo = 2)

Arguments

nu	positive integer, possibly infinite
t	positive number
delta	vector of positive numbers
algo	the algorithm used, 1 or 2

Details

The value of spowen2(nu, t, delta) is the same as the value of powen2(nu, t, -t, delta, -delta), but it is evaluated more efficiently.

Value

A vector of numbers between 0 and 1.

See Also

powen2

Examples

spowen2(4, 1, 2) == powen2(4, 1, -1, 2, -2)

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