Nothing
Introduction to nisone
In nisone: Inference for Samples of Size at Least One
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
set.seed(1)
library(nisone)
This document introduces the main functions in the {nisone} R package. The most important of which are:
ci1(): Frequentist $n=1$ confidence intervals.wc_width(): The half-width (in units of $|X-A|$) for the $n=1$ confidence intervals.aug_t(): Augmented $t$-interval, the generalization of ci1() for $n\geq 2$.bci1(): Bayesian version of ci1().blevel(): The true confidence level of the credible interval from bci1().bcin(): Bayesian interval for $n\geq 2$ assuming the coefficient of variation is known.bft(): Bayes factor given a $t$-statistic.dn1post(), pn1post(), qn1post(), rn1post(), nun1post(): The marginal posterior distribution of the location parameter using the $n=1$ priors.dinvnorm(), pinvnorm(), qinvnorm(), rinvnorm(): The inverse normal distribution.dginvnorm(), pginvnorm(), qginvnorm(), rginvnorm(): The generalized inverse normal distribution.
These methods are all described in detail in Gerard (2026).
$n=1$ Confidence Intervals
Suppose we have a single observations from a symmetric location-scale family. That is, $\rho()$ is any symmetric density function and the PDF of $X$ is $\frac{1}{\sigma}\rho\left(\frac{X-\mu}{\sigma}\right)$ for some center $\mu$ and some scale $\sigma$. For example, $X \sim N(\mu, \sigma^2)$. Given a prespecified value $A$, valid confidence intervals of the form
$$
X \pm \eta |X - A|
$$
and
$$
\frac{X + A}{2} \pm \eta |X - A|
$$
can be constructed to produce valid $(1 - \alpha)100\%$ confidence intervals. This is done by making $\eta$ large enough to bound the coverage probability below by $(1 - \alpha)$. See Blachman and Machol (1987).
The function ci1() produces these confidence intervals where $\rho()$ is the density of the (i) standard normal, (ii) Cauchy, and (iii) uniform. Surprisingly, constructing intervals using a uniform distribution makes them valid $(1 - \alpha)100\%$ confidence intervals when $X$ has any unimodal density.
Suppose we observe $X = 2$ and we use $A = 1$, then, centered at $X$, the various 95\% intervals are:
ci1(x = 2, A = 1, type = "x", family = "cauchy")
ci1(x = 2, A = 1, type = "x", family = "normal")
ci1(x = 2, A = 1, type = "x", family = "uniform")
Centered at $(X + A) / 2$, they are
ci1(x = 2, A = 1, type = "ave", family = "cauchy")
ci1(x = 2, A = 1, type = "ave", family = "normal")
ci1(x = 2, A = 1, type = "ave", family = "uniform")
If you are interested, you can get the values of $\eta$ for the normal, Cauchy, and uniform via wc_width(). E.g., for a normal distribution 95\% confidence interval of the form $(X + A)/2 \pm \eta|X-A|$, the value of $\eta$ is
wc_width(alpha = 0.05, center = "ave", family = "normal")
$n=1$ Bayesian Intervals
In Gerard (2026), we showed that (for $n=1$) using the following prior produces posterior credible intervals that are asymptotically valid confidence intervals.
$$
\pi(\mu,\nu) = |\mu - A|^{-1}\delta_{\nu^}(\nu), \text{ where}\
\nu = (\mu - A) / \sigma,\
\nu^ = \mathrm{argmax}{\nu>0}\nu\rho\left(\nu\right),
$$
and $\delta{\nu^}(\nu)$ is a pointmass at $\nu^$. This is asymptotic not in the sample size, but in the confidence level. So the credible intervals are approximate confidence intervals for small $\alpha$, which is the typical case.
In the normal case, this corresponds to the prior
$$
\pi(\mu) = |\mu - A|^{-1} \text{ and }
\sigma^2 = (\mu - A)^2 \text{ w.p. 1}.
$$
The resulting posterior distribution is inverse normal (see below). We have implemented this Bayesian approach in bci1(). Though, the asymptotics are so good that you end up getting almost the exact same interval as the frequentist $n=1$ interval,
bci1(x = 1, A = 0)
ci1(x = 1, A = 0)
The true level of the Bayesian credible intervals can be found by blevel(). It's almost exactly the correct level for 95\% confidence intervals. But they can dip a little further below the nominal level for smaller levels:
blevel(level = 0.7, nu = 0.8)
$n \geq 1$ Augmented t-intervals
Suppose we have data $X_1,\ldots,X_n \sim N(\mu, \sigma^2)$. Let $A$ be some prespecified value. Let $\hat{\mu}$ and $\hat{\sigma}^2$ be the
sample mean and sample variance using the augmented data (all the $X$'s and $A$). We consider intervals of the form
$$
\hat{\mu} \pm \eta \hat{\sigma}/\sqrt{n + 1},
$$
where $\eta$ is chosen to control the error probability. We call these "augmented $t$-interval". The function that calculates them is aug_t().
If $A$ is close to $\mu$, the augmented $t$-intervals tend to be smaller than the Student $t$-intervals on average. For $n=2$, you only need $A$ to be within about 4 standard deviations of $\mu$ to see large improvements.
n <- 2
mu <- 10
sigma <- 4
A <- 5
x <- rnorm(n = n, mean = mu, sd = sigma)
aug_t(x = x, A = A)
t.test(x)$conf.int
An augmented $t$-interval can be viewed as an inverted frequentist test that uses a Bayes factor as a frequentest test statistic. Gronau and Wagenmakers (2020) studied the connection between Bayes factors and t-statistics. Let $\delta$ be the standardized effect size, let $\sigma^2$ be the variance (assumed equal in two-sample case). In the one-sample case, $\delta = \frac{\mu - \mu_0}{\sigma}$, where $\mu_0$ is the null value. In the two-sample case, $\delta = \frac{\mu_1 - \mu_2}{\sigma}$, where $\mu_1$ and $\mu_2$ are the means of the two-samples. We place the prior $1/\sigma^2$ under the null and $\pi(\delta)/\sigma^2$ under the alternative, for some prior density $\pi(\cdot)$. Given this setting, the Bayes factor is a function of the $t$-statistic.
$$
\mathrm{BF} = \frac{\int_{\delta}T_{\nu}(t|\sqrt{n_\delta}\delta)\pi(\delta)\mathrm{d}\delta}{T_{\nu}(t)},
$$
where $T_{\nu}()$ is the central $t$-density with $\nu$ degrees of freedom, and $T_{\nu}(|a)$ is the non-central $t$-density with $\nu$ degrees of freedom and non-centrality parameter $a$. In the one-sample case, $n_{\delta} = n$ and $\nu = n - 1$, and in the two-sample case $n_{\delta} = \left(\frac{1}{n_1} + \frac{1}{n_2}\right)^{-1}$ and $\nu = n_1 + n_2 - 2$. The bft() function will calculate this Bayes factor given a $t$-statistic and any prior distribution (absolutely continuous with respect to Lebesgue measure) you provide over $\delta$ (with Cauchy as a default).
tstat <- mean(x) / (sd(x) / length(x))
bft(t = tstat, nu = length(x) - 1, nd = length(x))
To get the Bayes factor using augmented data, you assume that $\mu$ has a normal prior with mean $A$ and variance $\sigma^2$, then you just include $A$ in calculating the augmented $t$-statistic
x_aug <- c(x, A)
tstat <- mean(x_aug) / (sd(x_aug) / length(x_aug))
bft(t = tstat, nu = length(x_aug) - 1, nd = length(x_aug))
Generalized Inverse Normal Distribution
Robert (1991) considered the generalized inverse normal distribution of the form
$$
f(x|\alpha,\mu,\tau) = K(\alpha,\mu,\tau)|x|^{-\alpha}\exp\left{-\frac{1}{2\tau^2}\left(\frac{1}{x}-\mu\right)^2\right},
$$
where $K(\alpha,\mu,\tau)$ is the proportionality constant. When $\alpha = 2$, this corresponds to the inverse normal distribution (different from the inverse Gaussian distribution), where $\frac{1}{X} \sim N(\mu, \tau^2)$. The generalized inverse normal distribution shows up in Bayesian analysis when you parameterize the normal distribution in terms of its mean $\mu$ and its coefficient of variation $\nu = \mu / \tau$.
The nisone() R package contains density, distribution, quantile, and random generation functions for the (generalized) inverse normal distribution. You can use this to get the full posterior distribution for $n=1$ intervals when $\rho()$ is normal. In the normal case, the marginal posterior of $\mu - A$ is inverse normal with inverse mean $\frac{1}{X - A}$ and inverse variance $\frac{1}{(X - A)^2}$. E.g., if we observed $X = 2$ and use $A = 0.75$, the full posterior density is
museq <- seq(-2, 5, length.out = 500)
X <- 2
A <- 0.75
density <- dinvnorm(x = museq - A, imean = 1 / (X - A), isd = 1 / abs(X - A))
graphics::plot(
museq,
density,
type = "l",
xlab = expression(mu),
ylab = "Posterior Density"
)
You can get 95\% Credible intervals via
qinvnorm(p = c(0.025, 0.975), imean = 1 / (X - A), isd = 1 / abs(X - A)) + A
Which agree with bci1()
bci1(x = X, A = A)
Interfacing with the posterior is provided by the n1post functions so that you don't need to manually calculate the posterior above.
dn1post(x = 10, obs = X, A = A, nu = 1, fam = "normal")
dinvnorm(x = 10 - A, imean = 1 / (X - A), isd = 1 / abs(X - A))
qn1post(p = c(0.025, 0.975), A = A, obs = X, nu = 1, fam = "normal")
qinvnorm(p = c(0.025, 0.975), imean = 1 / (X - A), isd = 1 / abs(X - A)) + A
More generally, we have functions for the generalized inverse normal.
#| fig-width: 5
#| fig.height: 3
set.seed(50)
qginvnorm(p = 0.025, alpha = 4, mu = 0.5, tau = 1)
pginvnorm(q = -1.345, alpha = 4, mu = 0.5, tau = 1)
samp <- rginvnorm(n = 1000, alpha = 4, mu = 0.5, tau = 1)
x <- seq(min(samp), max(samp), length.out = 500)
y <- dginvnorm(x = x, alpha = 4, mu = 0.5, tau = 1)
modes <- xginvnorm(alpha = 4, mu = 0.5, tau = 1)
hist(
samp,
freq = FALSE,
breaks = 100,
xlab = "x",
main = "Generalized Inverse Normal Density")
lines(x, y, col = "#E69F00")
abline(v = modes, col = "#56B4E9", lty = 2)
References
- Blachman, N., & Machol, R. (1987). Confidence intervals based on one or more observations. IEEE Transactions on Information Theory, 33(3), 373-382. doi:10.1109/TIT.1987.1057306
- Gerard, D. (2026). Constructing and extending n = 1 Bayesian confidence intervals for location parameters in location-scale families. arXiv Preprint. doi:10.48550/arXiv.2607.25007
- Gronau, Q. F., Ly, A., & Wagenmakers, E. J. (2020). Informed Bayesian t-Tests. The American Statistician, 74(2), 137–143. doi:10.1080/00031305.2018.1562983
- Robert, C. (1991). Generalized inverse normal distributions. Statistics & Probability Letters, 11(1), 37-41. doi:10.1016/0167-7152(91)90174-P
Try the nisone package in your browser
Any scripts or data that you put into this service are public.
nisone documentation built on Sept. 8, 2026, 5:08 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
set.seed(1) library(nisone)
This document introduces the main functions in the {nisone} R package. The most important of which are:
ci1(): Frequentist $n=1$ confidence intervals.wc_width(): The half-width (in units of $|X-A|$) for the $n=1$ confidence intervals.aug_t(): Augmented $t$-interval, the generalization ofci1()for $n\geq 2$.bci1(): Bayesian version ofci1().blevel(): The true confidence level of the credible interval frombci1().bcin(): Bayesian interval for $n\geq 2$ assuming the coefficient of variation is known.bft(): Bayes factor given a $t$-statistic.dn1post(),pn1post(),qn1post(),rn1post(),nun1post(): The marginal posterior distribution of the location parameter using the $n=1$ priors.dinvnorm(),pinvnorm(),qinvnorm(),rinvnorm(): The inverse normal distribution.dginvnorm(),pginvnorm(),qginvnorm(),rginvnorm(): The generalized inverse normal distribution.
These methods are all described in detail in Gerard (2026).
$n=1$ Confidence Intervals
Suppose we have a single observations from a symmetric location-scale family. That is, $\rho()$ is any symmetric density function and the PDF of $X$ is $\frac{1}{\sigma}\rho\left(\frac{X-\mu}{\sigma}\right)$ for some center $\mu$ and some scale $\sigma$. For example, $X \sim N(\mu, \sigma^2)$. Given a prespecified value $A$, valid confidence intervals of the form $$ X \pm \eta |X - A| $$ and $$ \frac{X + A}{2} \pm \eta |X - A| $$ can be constructed to produce valid $(1 - \alpha)100\%$ confidence intervals. This is done by making $\eta$ large enough to bound the coverage probability below by $(1 - \alpha)$. See Blachman and Machol (1987).
The function ci1() produces these confidence intervals where $\rho()$ is the density of the (i) standard normal, (ii) Cauchy, and (iii) uniform. Surprisingly, constructing intervals using a uniform distribution makes them valid $(1 - \alpha)100\%$ confidence intervals when $X$ has any unimodal density.
Suppose we observe $X = 2$ and we use $A = 1$, then, centered at $X$, the various 95\% intervals are:
ci1(x = 2, A = 1, type = "x", family = "cauchy") ci1(x = 2, A = 1, type = "x", family = "normal") ci1(x = 2, A = 1, type = "x", family = "uniform")
Centered at $(X + A) / 2$, they are
ci1(x = 2, A = 1, type = "ave", family = "cauchy") ci1(x = 2, A = 1, type = "ave", family = "normal") ci1(x = 2, A = 1, type = "ave", family = "uniform")
If you are interested, you can get the values of $\eta$ for the normal, Cauchy, and uniform via wc_width(). E.g., for a normal distribution 95\% confidence interval of the form $(X + A)/2 \pm \eta|X-A|$, the value of $\eta$ is
wc_width(alpha = 0.05, center = "ave", family = "normal")
$n=1$ Bayesian Intervals
In Gerard (2026), we showed that (for $n=1$) using the following prior produces posterior credible intervals that are asymptotically valid confidence intervals. $$ \pi(\mu,\nu) = |\mu - A|^{-1}\delta_{\nu^}(\nu), \text{ where}\ \nu = (\mu - A) / \sigma,\ \nu^ = \mathrm{argmax}{\nu>0}\nu\rho\left(\nu\right), $$ and $\delta{\nu^}(\nu)$ is a pointmass at $\nu^$. This is asymptotic not in the sample size, but in the confidence level. So the credible intervals are approximate confidence intervals for small $\alpha$, which is the typical case.
In the normal case, this corresponds to the prior
$$
\pi(\mu) = |\mu - A|^{-1} \text{ and }
\sigma^2 = (\mu - A)^2 \text{ w.p. 1}.
$$
The resulting posterior distribution is inverse normal (see below). We have implemented this Bayesian approach in bci1(). Though, the asymptotics are so good that you end up getting almost the exact same interval as the frequentist $n=1$ interval,
bci1(x = 1, A = 0) ci1(x = 1, A = 0)
The true level of the Bayesian credible intervals can be found by blevel(). It's almost exactly the correct level for 95\% confidence intervals. But they can dip a little further below the nominal level for smaller levels:
blevel(level = 0.7, nu = 0.8)
$n \geq 1$ Augmented t-intervals
Suppose we have data $X_1,\ldots,X_n \sim N(\mu, \sigma^2)$. Let $A$ be some prespecified value. Let $\hat{\mu}$ and $\hat{\sigma}^2$ be the
sample mean and sample variance using the augmented data (all the $X$'s and $A$). We consider intervals of the form
$$
\hat{\mu} \pm \eta \hat{\sigma}/\sqrt{n + 1},
$$
where $\eta$ is chosen to control the error probability. We call these "augmented $t$-interval". The function that calculates them is aug_t().
If $A$ is close to $\mu$, the augmented $t$-intervals tend to be smaller than the Student $t$-intervals on average. For $n=2$, you only need $A$ to be within about 4 standard deviations of $\mu$ to see large improvements.
n <- 2 mu <- 10 sigma <- 4 A <- 5 x <- rnorm(n = n, mean = mu, sd = sigma) aug_t(x = x, A = A) t.test(x)$conf.int
An augmented $t$-interval can be viewed as an inverted frequentist test that uses a Bayes factor as a frequentest test statistic. Gronau and Wagenmakers (2020) studied the connection between Bayes factors and t-statistics. Let $\delta$ be the standardized effect size, let $\sigma^2$ be the variance (assumed equal in two-sample case). In the one-sample case, $\delta = \frac{\mu - \mu_0}{\sigma}$, where $\mu_0$ is the null value. In the two-sample case, $\delta = \frac{\mu_1 - \mu_2}{\sigma}$, where $\mu_1$ and $\mu_2$ are the means of the two-samples. We place the prior $1/\sigma^2$ under the null and $\pi(\delta)/\sigma^2$ under the alternative, for some prior density $\pi(\cdot)$. Given this setting, the Bayes factor is a function of the $t$-statistic.
$$
\mathrm{BF} = \frac{\int_{\delta}T_{\nu}(t|\sqrt{n_\delta}\delta)\pi(\delta)\mathrm{d}\delta}{T_{\nu}(t)},
$$
where $T_{\nu}()$ is the central $t$-density with $\nu$ degrees of freedom, and $T_{\nu}(|a)$ is the non-central $t$-density with $\nu$ degrees of freedom and non-centrality parameter $a$. In the one-sample case, $n_{\delta} = n$ and $\nu = n - 1$, and in the two-sample case $n_{\delta} = \left(\frac{1}{n_1} + \frac{1}{n_2}\right)^{-1}$ and $\nu = n_1 + n_2 - 2$. The bft() function will calculate this Bayes factor given a $t$-statistic and any prior distribution (absolutely continuous with respect to Lebesgue measure) you provide over $\delta$ (with Cauchy as a default).
tstat <- mean(x) / (sd(x) / length(x)) bft(t = tstat, nu = length(x) - 1, nd = length(x))
To get the Bayes factor using augmented data, you assume that $\mu$ has a normal prior with mean $A$ and variance $\sigma^2$, then you just include $A$ in calculating the augmented $t$-statistic
x_aug <- c(x, A) tstat <- mean(x_aug) / (sd(x_aug) / length(x_aug)) bft(t = tstat, nu = length(x_aug) - 1, nd = length(x_aug))
Generalized Inverse Normal Distribution
Robert (1991) considered the generalized inverse normal distribution of the form $$ f(x|\alpha,\mu,\tau) = K(\alpha,\mu,\tau)|x|^{-\alpha}\exp\left{-\frac{1}{2\tau^2}\left(\frac{1}{x}-\mu\right)^2\right}, $$ where $K(\alpha,\mu,\tau)$ is the proportionality constant. When $\alpha = 2$, this corresponds to the inverse normal distribution (different from the inverse Gaussian distribution), where $\frac{1}{X} \sim N(\mu, \tau^2)$. The generalized inverse normal distribution shows up in Bayesian analysis when you parameterize the normal distribution in terms of its mean $\mu$ and its coefficient of variation $\nu = \mu / \tau$.
The nisone() R package contains density, distribution, quantile, and random generation functions for the (generalized) inverse normal distribution. You can use this to get the full posterior distribution for $n=1$ intervals when $\rho()$ is normal. In the normal case, the marginal posterior of $\mu - A$ is inverse normal with inverse mean $\frac{1}{X - A}$ and inverse variance $\frac{1}{(X - A)^2}$. E.g., if we observed $X = 2$ and use $A = 0.75$, the full posterior density is
museq <- seq(-2, 5, length.out = 500) X <- 2 A <- 0.75 density <- dinvnorm(x = museq - A, imean = 1 / (X - A), isd = 1 / abs(X - A)) graphics::plot( museq, density, type = "l", xlab = expression(mu), ylab = "Posterior Density" )
You can get 95\% Credible intervals via
qinvnorm(p = c(0.025, 0.975), imean = 1 / (X - A), isd = 1 / abs(X - A)) + A
Which agree with bci1()
bci1(x = X, A = A)
Interfacing with the posterior is provided by the n1post functions so that you don't need to manually calculate the posterior above.
dn1post(x = 10, obs = X, A = A, nu = 1, fam = "normal") dinvnorm(x = 10 - A, imean = 1 / (X - A), isd = 1 / abs(X - A)) qn1post(p = c(0.025, 0.975), A = A, obs = X, nu = 1, fam = "normal") qinvnorm(p = c(0.025, 0.975), imean = 1 / (X - A), isd = 1 / abs(X - A)) + A
More generally, we have functions for the generalized inverse normal.
#| fig-width: 5 #| fig.height: 3 set.seed(50) qginvnorm(p = 0.025, alpha = 4, mu = 0.5, tau = 1) pginvnorm(q = -1.345, alpha = 4, mu = 0.5, tau = 1) samp <- rginvnorm(n = 1000, alpha = 4, mu = 0.5, tau = 1) x <- seq(min(samp), max(samp), length.out = 500) y <- dginvnorm(x = x, alpha = 4, mu = 0.5, tau = 1) modes <- xginvnorm(alpha = 4, mu = 0.5, tau = 1) hist( samp, freq = FALSE, breaks = 100, xlab = "x", main = "Generalized Inverse Normal Density") lines(x, y, col = "#E69F00") abline(v = modes, col = "#56B4E9", lty = 2)
References
- Blachman, N., & Machol, R. (1987). Confidence intervals based on one or more observations. IEEE Transactions on Information Theory, 33(3), 373-382. doi:10.1109/TIT.1987.1057306
- Gerard, D. (2026). Constructing and extending n = 1 Bayesian confidence intervals for location parameters in location-scale families. arXiv Preprint. doi:10.48550/arXiv.2607.25007
- Gronau, Q. F., Ly, A., & Wagenmakers, E. J. (2020). Informed Bayesian t-Tests. The American Statistician, 74(2), 137–143. doi:10.1080/00031305.2018.1562983
- Robert, C. (1991). Generalized inverse normal distributions. Statistics & Probability Letters, 11(1), 37-41. doi:10.1016/0167-7152(91)90174-P
Try the nisone package in your browser
Any scripts or data that you put into this service are public.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.