Let \(Z \sim {\cal N}(0,1)\) and \(X \sim \chi^2_\nu\) be two independent random variables. For real numbers \(\delta_1\) and \(\delta_2\), define the two random variables \[ T_1 = \frac{Z+\delta_1}{\sqrt{X/\nu}} \quad \text{and} \quad\; T_2 = \frac{Z+\delta_2}{\sqrt{X/\nu}}. \]

Both \(T_1\) and \(T_2\) follow a non-central Student distribution. The number of degrees of freedom is \(\nu\) for each of them, and their respective non-centrality parameters are \(\delta_1\) and \(\delta_2\) respectively.

Owen (1965) studied the distribution of the pair \((T_1, T_2)\).

The four Owen cumulative functions are \[ \begin{align} O_1(\nu, t_1, t_2, \delta_1, \delta_2) & = \Pr(T_1 \leq t_1, T_2 \leq t_2), \\ O_2(\nu, t_1, t_2, \delta_1, \delta_2) & = \Pr(T_1 \leq t_1, T_2 \geq t_2), \\ O_3(\nu, t_1, t_2, \delta_1, \delta_2) & = \Pr(T_1 \geq t_1, T_2 \geq t_2), \\ O_4(\nu, t_1, t_2, \delta_1, \delta_2) & = \Pr(T_1 \geq t_1, T_2 \leq t_2). \end{align} \]

Owen provided an efficient way to evaluate these functions *when \(\nu\) is an integer number*. Owen’s algorithms are implemented in the `OwenQ`

package.

For \(\delta_1 > \delta_2\), these four functions are implemented in the `OwenQ`

package under the respective names `powen1`

, `powen2`

, `powen3`

and `powen4`

. For general values of \(\delta_1\) and \(\delta_2\), they are implemented under the respective names `psbt1`

, `psbt2`

, `psbt3`

and `psbt4`

.

Owen (1965) also provided an algorithm to evaluate the cumulative distribution function of a univariate non-central Student distribution with an integer number of degrees of freedom. This evaluation is performed by the function `ptOwen`

of the `OwenQ`

package.

```
ptOwen(q=1, nu=3, delta=2)
## [1] 0.1573494
pt(q=1, df=3, ncp=2)
## [1] 0.1573494
```

It is known that the `pt`

function is not reliable when the non-centrality parameter `ncp`

is large. Below we compare the values given by `ptOwen`

and `pt`

to the value given by Wolfram|Alpha (returned by the command `N[CDF[NoncentralStudentTDistribution[4,70],80]]`

):

```
<- pt(q=80, df=4, ncp=70)
p1 <- ptOwen(q=80, nu=4, delta=70)
p2 <- 0.54742763380700947685
wolfram - wolfram
p1 ## [1] 0.02268711
- wolfram
p2 ## [1] 3.330669e-16
```

When `q`

\(=\)`delta`

, the value of `ptOwen(q, nu, delta)`

should go to `0.5`

as `nu`

increases to infinity. The examples below show the failure of this expectation when `nu`

is too large.

```
ptOwen(q=50, nu=3500, delta=50)
## [1] 0.4986697
ptOwen(q=50, nu=3600, delta=50)
## [1] 0.4989289
ptOwen(q=50, nu=3650, delta=50)
## [1] 0.4522949
ptOwen(q=50, nu=3660, delta=50)
## [1] 0.3349762
ptOwen(q=50, nu=3670, delta=50)
## [1] 0.7607795
ptOwen(q=50, nu=3680, delta=50)
## [1] 0
```

Since all Owen’s algorithms are somehow similar to the algorithm evaluating `ptOwen`

, we can suspect that the other ones also suffer from certain limitations. In the other vignette, we investigate the reliability and the limitations of `OwenQ`

.

In order to do some comparisons, and thanks to the `BH`

package, we have ported the `boost`

implementation of the cumulative Student distribution to `OwenQ`

. It is evaluated by the internal function `pt_boost`

. We concluded that this function is highly reliable. In particular it does not suffer from the failures of `ptOwen`

we have just shown:

```
:::pt_boost(q=50, nu=3500, delta=50)
OwenQ## [1] 0.4986697
:::pt_boost(q=50, nu=3600, delta=50)
OwenQ## [1] 0.4987041
:::pt_boost(q=50, nu=3650, delta=50)
OwenQ## [1] 0.4987206
:::pt_boost(q=50, nu=3660, delta=50)
OwenQ## [1] 0.4987238
:::pt_boost(q=50, nu=3670, delta=50)
OwenQ## [1] 0.4987271
:::pt_boost(q=50, nu=3680, delta=50)
OwenQ## [1] 0.4987303
```

The Owen distribution intervenes in the calculation of the power of equivalence tests.

Assume a statistical model given by a sample \(y_i \sim_{\text{iid}} {\cal N}(\mu, \sigma^2)\) for \(i=1, \ldots, n\). We want to demonstrate that, up to a given confidence level, the mean \(\mu\) belongs to a certain interval \([\Delta_1, \Delta_2]\). In other words, we are interested in the alternative hypothesis \(H_1\colon\{\Delta_1 \leq \mu \leq \Delta_2\}\).

Consider the usual \(100(1-2\alpha)\%\)-confidence interval about \(\mu\): \[ \left[\bar y - t^\ast_{n-1}(\alpha)\frac{\hat\sigma}{\sqrt{n}}, \, \bar y + t^\ast_{n-1}(\alpha)\frac{\hat\sigma}{\sqrt{n}} \right]. \]

The \(H_1\) hypothesis is accepted at level \(\alpha\) when this interval falls into the interval \([\Delta_1, \Delta_2]\).

This can be written as follows: \[ T_1 := \frac{\bar y - \Delta_1}{\hat\sigma/\sqrt{n}} \geq t^\ast_{n-1}(\alpha) \quad \text{and} \quad T_2 := \frac{\bar y - \Delta_2}{\hat\sigma/\sqrt{n}} \leq - t^\ast_{n-1}(\alpha). \]

Observe that \[ T_1 = \frac{z - \delta_1}{\dfrac{\sqrt{n-1}\hat\sigma/\sigma}{\sqrt{n-1}}} \] where \[ z = \frac{\sqrt{n}}{\sigma}(\bar y - \mu) \sim {\cal N}(0,1) \quad \text{and} \quad \delta_1 = \frac{\mu - \Delta_1}{\sigma/\sqrt{n}}. \]

By reasoning in the same way for \(T_2\), we find that the pair \((T_1, T_2)\) follows the Owen distribution with degrees of freedom \(\nu = n-1\), and non-centrality parameters \(\delta_1\) given above and \(\delta_2 = \tfrac{\mu - \Delta_2}{\sigma/\sqrt{n}}\).

Therefore the power of the test - *i.e.* the probability to accept \(H_1\) - is given by \[
O_4\bigl(n-1, t^\ast_{n-1}(\alpha), -t^\ast_{n-1}(\alpha), \delta_1, \delta_2\bigr),
\] and then can be evaluated thanks to `powen4`

.

The result of the equivalence test is said to be *inconclusive* when only one of the bounds of the confidence interval falls into the interval \([\Delta_1, \Delta_2]\).

The probability to get an inconclusive result can be obtained with the `OwenQ`

package. We show and check that below with the help of simulations.

```
<- -2; Delta2 <- 2
Delta1 <- 1; sigma <- 6; n <- 30L
mu <- 0.05
alpha <- 1e6L
nsims <- inconclusive <- numeric(nsims)
equivalence for (i in 1L:nsims) {
<- rnorm(n, mu, sigma)
y <- t.test(x = y, conf.level = 1-2*alpha)$conf.int
CI <- (CI[1] > Delta1) && (CI[2] < Delta2)
equivalence[i] <- ((CI[1] < Delta1) && (CI[2] > Delta1)) ||
inconclusive[i] 1] < Delta2) && (CI[2] > Delta2))
((CI[ }
```

```
<- n-1
dof <- qt(1-alpha, dof)
q <- sqrt(1/n)*sigma
se <- (mu-Delta1)/se; delta2 <- (mu-Delta2)/se
delta1 # probability to get equivalence
mean(equivalence)
## [1] 0.093075
powen4(dof, q, -q, delta1, delta2)
## [1] 0.09300963
# probability to get inconclusive
mean(inconclusive)
## [1] 0.901323
ptOwen(q, dof, delta2) - ptOwen(-q, dof, delta1) - powen4(dof, q, -q, delta1, delta2)
## [1] 0.90139
```

Now consider two independent samples \(x_i \sim_{\text{iid}} {\cal N}(\mu, \sigma^2)\) for \(i=1, \ldots, n_1\). and \(y_i \sim_{\text{iid}} {\cal N}(\nu, \sigma^2)\) for \(i=1, \ldots, n_2\) and the alternative hypothesis \(H_1\colon\bigl\{|\mu-\nu| \leq \Delta\bigr\}\). The power of this test is evaluated by the function `powerTOST`

below. The parameter `delta0`

corresponds to the difference \(\mu-\nu\).

```
<- function(alpha, delta0, Delta, sigma, n1, n2) {
powerTOST <- sqrt(1/n1 + 1/n2) * sigma
se <- (delta0 + Delta) / se
delta1 <- (delta0 - Delta) / se
delta2 <- n1 + n2 - 2
dof <- qt(1 - alpha, dof)
q powen4(dof, q, -q, delta1, delta2)
}
```

The `OwenQ`

package also provides an implementation of the Owen \(T\)-function, under the name `OwenT`

. This is a port of the function `owens_t`

of `boost`

, the peer-reviewed collection of C++ libraries.

The Owen cumulative functions are based on the two Owen \(Q\)-functions \[ Q_1(\nu, t, \delta, R) = \frac{1}{\Gamma\left(\frac{\nu}{2}\right)2^{\frac12(\nu-2)}} \int_0^R \Phi\left(\frac{tx}{\sqrt{\nu}}-\delta\right) x^{\nu-1} e^{-\frac{x^2}{2}} \mathrm{d}x \] and \[ Q_2(\nu, t, \delta, R) = \frac{1}{\Gamma\left(\frac{\nu}{2}\right)2^{\frac12(\nu-2)}} \int_R^\infty \Phi\left(\frac{tx}{\sqrt{\nu}}-\delta\right) x^{\nu-1} e^{-\frac{x^2}{2}} \mathrm{d}x. \]

They are implemented in the `OwenQ`

package under the respective names `OwenQ1`

and `OwenQ2`

(for integer values of \(\nu\)).

Consider the statistical model given by a sample \(y_i \sim_{\text{iid}} {\cal N}(\mu, \sigma^2)\) for \(i=1, \ldots, n\).

An equal-tailed \((p,\alpha)\)-tolerance interval is an interval containing at least \(100p\%\) of the “center data” with \(100(1-\alpha)\%\) confidence.

The natural choice for such an interval is, as shown by Owen (1965), \[ \bigl[\bar y - k_e \hat\sigma, \bar y + k_e \hat\sigma] \] where \(k_e\) satisfies \[ O_2(n-1, k_e\sqrt{n}, -k_e\sqrt{n}, \delta, -\delta) = 1-\alpha \] where \(\delta = \sqrt{n}z_{(1+p)/2}\).

Therefore, the tolerance factor \(k_e\) can be determined with the help of the `powen2`

function of the `OwenQ`

package. But it is more efficient to use the function `spowen2`

for this purpose; `spowen2(nu, t, delta)`

has the same value as `powen2(nu, t, -t, delta, -delta)`

but it is evaluated more efficiently.

```
<- 0.9; alpha <- 0.05
p <- 100
n <- sqrt(n)*qnorm((1+p)/2)
delta uniroot(function(ke) spowen2(n-1, ke*sqrt(n), delta) - (1-alpha),
lower=qnorm(1-alpha), upper=5, extendInt = "upX", tol=1e-9)$root
## [1] 1.981513
```

The \(k_e\) factors are tabulated in Krishnamoorthy & Mathew’s book.

The `OwenQ`

package provides three internal functions, `iOwenQ1`

, `iOwenQ2`

and `ipowen4`

, which respectively perform the evaluation of \(Q_1\), \(Q_2\) and \(O_4\) by numerical integration using the `RcppNumerical`

package. They can be called with a positive non-integer value of \(\nu\). The evaluation of \(O_4\) with `ipowen4`

is correct only for \(t_1 > t_2\) and \(\delta_1 > \delta_2\).

Owen, D. B. (1965).

*A special case of a bivariate noncentral t-distribution.*Biometrika 52, 437-446.Krishnamoorthy & Mathew (2009).

*Statistical Tolerance Regions*. Wiley.