Package {nisone}

Title:	Inference for Samples of Size at Least One
Version:	0.0.2
Description:	Provides different interval estimates of a location parameter when the sample size is one or more. These include classical methods when n=1, new Bayesian analogues of these classical methods, and extensions of these methods to larger sample sizes in the normal case. Other functions calculate Bayes factors based on t-statistics, and implement the (generalized) inverse normal distribution. These methods are described in Gerard (2026) <doi:10.48550/arXiv.2607.25007>.
License:	GPL (≥ 3)
Encoding:	UTF-8
URL:	https://dcgerard.github.io/nisone/
BugReports:	https://github.com/dcgerard/nisone/issues
Suggests:	knitr, rmarkdown, testthat (≥ 3.0.0)
Config/testthat/edition:	3
Depends:	R (≥ 3.5)
Config/roxygen2/version:	8.1.0
VignetteBuilder:	knitr
NeedsCompilation:	no
Packaged:	2026-08-25 18:56:16 UTC; dgerard
Author:	David Gerard [aut, cre]
Maintainer:	David Gerard <gerard.1787@gmail.com>
Repository:	CRAN
Date/Publication:	2026-09-08 12:40:02 UTC

Augmented t-interval

Description

Calculates a t-interval using augmented data c(x,A). The multiplier of this interval bounds the level above level, so these intervals are typically conservative. This method is very fast for n <= 100 and level %in% c(0.8, 0.9, 0.95, 0.99) because I saved those multipliers in an internal dataset. But it can be slow (on the order of a second) for other confidence levels or larger sample sizes.

Usage

aug_t(x, A = 0, level = 0.95, wt = 1)

Arguments

Details

Suppose X_1,\ldots,X_n \sim N(\mu,\sigma^2). Suppose we have prior value A. This provides intervals of the form

\hat{\mu} \pm \eta \hat{\sigma}/\sqrt{n + 1},

where \hat{\mu} and \hat{\sigma} are the sample mean and sample standard deviation of the augmented data X_1,\ldots,X_n,A, and \eta is chosen large enough to maintain the confidence level at all parameter values.

The wt argument allows for more copies of A to be included in the data augmentation. But it doesn't work well with more data augmentation so you should not set it above 1. Though, you can set wt to be between 0 and 1 (to have less data augmentation) and this does seem to work pretty well, but I haven't studied it extensively, so use at your own risk.

Note that you have to choose A before seeing the data. If you choose it based on the data, then you no longer maintain the confidence level.

Value

The augmented t-interval of the specified level.

Author(s)

David Gerard

Examples

set.seed(1)
## A is exactly correct
x <- rnorm(10)
aug_t(x)
stats::t.test(x)$conf.int

## A is very wrong
x <- rnorm(10)
aug_t(x, A = 100)
stats::t.test(x)$conf.int

Bayesian credible interval when n=1

Description

Suppose X \sim N(\mu, \sigma^2). This gives you credible intervals of a specified level when we use prior \sigma = |\mu| with probability 1, and \pi(\mu) = |\mu - A|^{-1}. See the n1post() for access to the full posterior distribution.

Usage

bci1(x, A = 0, level = 0.95)

Arguments

Value

The credible interval of the specified level.

Author(s)

David Gerard

See Also

n1post(): Functions to access the full posterior distribution.

Examples

bci1(x = 1, A = 0)

Bayesian credible interval when n >= 1

Description

Let X_i \sim N(\mu, \sigma^2). Let Y_i = X_i - A, \beta = \mu - A, and \nu = |\mu - A|/\sigma. Then, for a fixed value of \nu, this comes up with a (1-\alpha)100\% posterior credible interval of \mu where the prior is \pi(\beta) = |\beta|^{-1} or \pi(\mu) = |\mu - A|^{-1}. The full posterior distribution is generalized inverse normal (implemented by ginvnorm) with shape parameter n + 1, inverse mean of sum(y) / sum(y^2), and inverse variance 1 / (nu^2 * sum(y^2)).

Usage

bcin(x, A = 0, nu = 1, level = 0.95)

Arguments

Value

The credible interval of the provided level.

Author(s)

David Gerard

See Also

ginvnorm(): The generalized inverse normal distribution.

Examples

set.seed(1)
x <- stats::rnorm(4, mean = 1, sd = 1)
bcin(x = x, A = 3)

Bayes factor from t-statistic

Description

Given a user-provided prior on the standardized effect size, will calculate the Bayes factor.

Usage

bft(t, nu, nd, prior = NULL)

Arguments

Details

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 arbitrary density \pi(\cdot). Given this setting, the Bayes factor is a function of the t-statistic. We calculate it via numeric integration.

Value

The Bayes factor to a corresponding t-statistic.

Author(s)

David Gerard

References

• Gronau, Q. F., Ly, A., & Wagenmakers, E. J. (2020). Informed Bayesian t-tests. The American Statistician. doi:10.1080/00031305.2018.1562983

Examples

# One sample t, n = 10, t-statistic = 2
bft(t = 2, nu = 10 - 1, nd = 10)

# Two sample t, n1 = 10, n2 = 8, t-statistic = 2
bft(t = 2, nu = 10 + 8 - 2, nd = 1 / (1 / 10 + 1 / 8))

True level of the Bayesian credible interval at a given coefficient of variation

Description

True level of the Bayesian credible interval at a given coefficient of variation

Usage

blevel(level, nu)

Arguments

Value

The true level of the (1-alpha) credible interval.

Author(s)

David Gerard

Examples

blevel(0.95, 0.5)
blevel(0.95, 1)
blevel(0.95, 1.5)

n=1 confidence interval

Description

When you have one observation, X from some symmetric distribution with center \mu, for a pre-specified A, produces confidence intervals for the center form X \pm \eta |X - A| (type = "x") or (X + A)/2 \pm \eta |X - A| (type = "ave"). The average intervals have smaller width on average, and so are the default. We allow X to either come from a normal distribution, a Cauchy, or a uniform. Note that if you use family = "uniform" then these intervals are valid confidence intervals for the median/mode for any symmetric unimodal density, which I think is really amazing.

Usage

ci1(
  x,
  A = 0,
  type = c("ave", "x"),
  family = c("normal", "cauchy", "uniform"),
  level = 0.95
)

Arguments

Value

A matrix of with 4 columns: the data, the center, the lower bound, the upper bound.

Author(s)

David Gerard

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

Examples

ci1(c(1, 2, 10), type = "x")
ci1(c(1, 2, 10), type = "ave")

Confidence density based on n=1 confidence interval

Description

Let X \sim N(\mu, \sigma^2). Consider intervals of the form X \pm \eta |X| or X / 2 \pm \eta |X|. These intervals define a confidence distribution of \mu. This is the density of the confidence distribution of (\mu - X)/|X| or (\mu - X/2)/|X|. Please note that \mu is random here not X. Also note that this confidence distribution does not exist between the 25th and 75th percentiles.

Usage

d_wc(x, center = c("X", "ave"))

Arguments

Value

n=1 confidence density.

Author(s)

David Gerard

See Also

p_wc()

Examples

xseq <- seq(-10, 10, length.out = 500)
dseq <- d_wc(xseq)
plot(xseq, dseq, type = "l")

dseq <- d_wc(xseq, center = "ave")
plot(xseq, dseq, type = "l")

The generalized inverse normal distribution

Description

Density, distribution function, quantile function, and random generation for the generalized inverse normal distribution when parameterized by the shape, mean, and standard deviation of the inverse (reciprocal).

Usage

dginvnorm(x, alpha, mu = 0, tau = 1, log = FALSE)

pginvnorm(q, alpha, mu = 0, tau = 1, lower.tail = TRUE, subdivisions = 500L)

qginvnorm(p, alpha, mu = 0, tau = 1)

rginvnorm(n, alpha, mu = 0, tau = 1)

mginvnorm(alpha, mu = 0, tau = 1)

xginvnorm(alpha, mu = 0, tau = 1)

Arguments

Value

Either a random sample (rginvnorm), the density (dginvnorm), the tail probability (pginvnorm), or the quantile (qginvnorm) of the inverse normal distribution.

Functions

• dginvnorm(): Density function.

• pginvnorm(): Probability function.

• qginvnorm(): Quantile function.

• rginvnorm(): Random generation.

• mginvnorm(): Mean.

• xginvnorm(): Modes.

Author(s)

David Gerard

References

• Robert, C. (1991). Generalized inverse normal distributions. Statistics & Probability Letters, 11(1), 37-41. doi:10.1016/0167-7152(91)90174-P

Examples

set.seed(50)
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)
graphics::hist(
  samp,
  freq = FALSE,
  breaks = 100,
  xlab = "x",
  main = "Generalized Inverse Normal Density")
graphics::lines(x, y, col = "#E69F00")
graphics::abline(v = modes, col = "#56B4E9", lty = 2)

General inverse distribution

Description

Let X come from some location-scale family

f(x) = \frac{1}{\sigma}\rho\left(\frac{X - \mu}{\sigma}\right),

for some standard distribution \rho(). You provide \rho(), \mu and \sigma and these functions will provide the density, distribution, quantile, and random generation for Z = 1/X. Convenience arguments for the normal, Cauchy, and uniform are provided.

Usage

didist(
  x,
  center = 0,
  scale = 1,
  fam = c("normal", "cauchy", "uniform"),
  ddist = NULL,
  log = FALSE,
  ...
)

pidist(
  q,
  center = 0,
  scale = 1,
  fam = c("normal", "cauchy", "uniform"),
  pdist = NULL,
  lower.tail = TRUE,
  log.p = FALSE,
  ...
)

qidist(
  p,
  center = 0,
  scale = 1,
  fam = c("normal", "cauchy", "uniform"),
  qdist = NULL,
  pdist = NULL,
  ...
)

ridist(
  n,
  center = 0,
  scale = 1,
  fam = c("normal", "cauchy", "uniform"),
  rdist = NULL,
  ...
)

Arguments

Value

Either a random sample (ridist), the density (didist), the tail probability (pidist), or the quantile (qidist) of the inverse distribution.

Functions

• didist(): Density function.

• pidist(): Distribution function.

• qidist(): Quantile function.

• ridist(): Random generation.

Author(s)

David Gerard

Examples

didist(1, fam = "normal")
didist(1, center = 3, scale = 2, ddist = dt, df = 1)

pidist(1, fam = "normal")
pidist(1, center = 3, scale = 2, pdist = pt, df = 1)

qidist(0.1, fam = "normal")
qidist(0.1, center = 3, scale = 2, qdist = qt, pdist = pt, df = 1)

The inverse normal distribution

Description

Density, distribution function, quantile function, and random generation for the inverse normal distribution when parameterized by the mean and standard deviation of the inverse (reciprocal).

Usage

dinvnorm(x, imean = 0, isd = 1, log = FALSE)

pinvnorm(q, imean = 0, isd = 1, lower.tail = TRUE, log.p = FALSE)

qinvnorm(p, imean = 0, isd = 1)

rinvnorm(n, imean = 0, isd = 1)

Arguments

Value

Either a random sample (rinvnorm), the density (dinvnorm), the tail probability (pinvnorm), or the quantile (qinvnorm) of the inverse normal distribution.

Functions

• dinvnorm(): Density function.

• pinvnorm(): Probability function.

• qinvnorm(): Quantile function.

• rinvnorm(): Random generation.

Author(s)

David Gerard

References

• Robert, C. (1991). Generalized inverse normal distributions. Statistics & Probability Letters, 11(1), 37-41. doi:10.1016/0167-7152(91)90174-P

Examples

x <- seq(-4, 4, length.out = 300)
y <- dinvnorm(x = x, imean = 0.1, isd = 1)
graphics::plot(x, y, type = "l")

p <- pinvnorm(q = x, imean = 0.1, isd = 1)
graphics::plot(x, p, type = "l", ylim = c(0, 1))
graphics::abline(h = c(0, 1), lty = 2)

qinvnorm(p = c(0.025, 0.5, 0.975), imean = 0.1, isd = 1)

s <- rinvnorm(n = 10000, imean = 0.1, isd = 1)
stats::quantile(s, c(0.025, 0.5, 0.975))

Augmented mean

Description

If wt is an integer, this is mean(c(x, rep(A, wt))). But we generalize it to non-integer wt as well.

Usage

mean_aug(x, A, wt = 1)

Arguments

Value

The augmented mean

Author(s)

David Gerard

Examples

mean_aug(c(1, 2, 3), 4, wt = 10)

Posterior Distribution for n=1 Data

Description

Density, probability, quantile, and random generation from the posterior distribution of the location parameter when one has n = 1 observation and uses a particular prior that results in valid confidence intervals (asymptotically in the confidence level).

Usage

nun1post(fam = c("normal", "cauchy"), ddist = NULL, ...)

dn1post(
  x,
  A,
  obs,
  nu = NULL,
  fam = c("normal", "cauchy"),
  ddist = NULL,
  log = FALSE,
  ...
)

pn1post(
  q,
  A,
  obs,
  nu = NULL,
  fam = c("normal", "cauchy"),
  pdist = NULL,
  lower.tail = TRUE,
  log.p = FALSE,
  ...
)

qn1post(
  p,
  A,
  obs,
  nu = NULL,
  fam = c("normal", "cauchy"),
  qdist = NULL,
  pdist = NULL,
  ...
)

rn1post(n, A, obs, nu = NULL, fam = c("normal", "cauchy"), rdist = NULL, ...)

Arguments

Details

Let X have PDF \frac{1}{\sigma}\rho((x - \mu)/\sigma) for a known and symmetric \rho(\cdot). Let \nu = |\mu - A| / \sigma for a pre-specified A. We place prior \pi(\mu) = |\mu - A|^{-1} and \nu = \text{argmax}_{a > 0}a\rho(a) with probability 1.

These functions will calculate the posterior distribution and allow you to interact with it through the distribution, density, quantile, and random generation functions.

Value

density, distribution, quantile, or random values.

Functions

• nun1post(): Obtains prior value of \nu.

• dn1post(): Density function.

• pn1post(): Distribution function.

• qn1post(): Quantile function.

• rn1post(): Random generation.

Author(s)

David Gerard

Examples

set.seed(1)
# Observe x = 2, assume t with 2 df, prior value is A = 1
nun1post(ddist = dt, df = 2) ## nu should be 1
x <- seq(-10, 10, length.out = 500)
y <- dn1post(x = x, A = 1, obs = 2, nu = 1, ddist = dt, df = 2)
z <- rn1post(n = 10000, A = 1, obs = 2, nu = 1, rdist = rt, df = 2)
z <- z[z >= -10 & z <= 10]
graphics::hist(z, freq = FALSE, breaks = 200, border = "grey", col = "grey")
graphics::lines(x, y, type = "l", col = "#E69F00")
graphics::abline(v = c(1, 2), col = c("#56B4E9", "#009E73"), lty = c(2, 3))
pn1post(q = 2, A = 1, obs = 2, nu = 1, pdist = pt, df = 2)
qn1post(p = 0.7113, A = 1, obs = 2, nu = 1, qdist = qt, pdist = pt, df = 2)

Confidence distribution CDF based on n=1 confidence interval

Description

Let X \sim N(\mu, \sigma^2). Consider intervals of the form X \pm \eta |X| or X / 2 \pm \eta |X|. These intervals define a confidence distribution of \mu. This is the CDF of the confidence distribution of (\mu - X)/|X| or (\mu - X/2)/|X|. Please note that \mu is random here not X. Also note that this confidence distribution does not exist between the 25th and 75th percentiles.

Usage

p_wc(q, center = c("X", "ave"))

Arguments

Value

n=1 confidence distribution CDF.

Author(s)

David Gerard

See Also

d_wc()

Examples

qseq <- seq(-10, 10, length.out = 100)
pseq <- p_wc(qseq)
plot(qseq, pseq, type = "l", xlab = "q", ylab = "CDF")

Augmented variance

Description

If wt is an integer, this is var(c(x, rep(A, wt))). But we generalize it to non-integer wt as well.

Usage

var_aug(x, A, wt = 1)

Arguments

Value

The augmented variance

Author(s)

David Gerard

Examples

var_aug(c(1, 2, 3), 4, wt = 10)

Worst case exclusion probability of a n=1 confidence interval

Description

This assumes the random variable X follows a normal distribution.

Usage

wc_alpha(width, center = c("X", "ave"))

Arguments

Value

The worst case probability (a priori) that a confidence interval of half-width width will not capture the mean.

Author(s)

David Gerard

Examples

wc_alpha(width = 2, center = "X")
wc_alpha(width = 2, center = "ave")

Worst-case coefficient of variation

Description

This assumes the random variable X is normal. Intervals are of the form center +/- width * |X|.

Usage

wc_cov(width, center = c("X", "ave"))

Arguments

Details

We use A = 0 for calculations, but the worst case COV doesn't change if A is non-zero. So you can think of intervals of the form center +/- width * |X - A| where center is either X or (X + A)/2.

Value

Worst case coefficient of variation \nu = |X - A|/\sigma.

Author(s)

David Gerard

Examples

wc_cov(width = 2, center = "X")
wc_cov(width = 2, center = "ave")

Given confidence level, provide half-width of CI

Description

(1-\alpha)100\% confidence intervals are of the form X \pm \eta|X-A| or (X + A)/2 \pm \eta|X-A|. You give \alpha and this function will provide \eta if X follows a Cauchy, normal, or uniform distribution.

Usage

wc_width(
  alpha,
  center = c("X", "ave", "approx"),
  family = c("normal", "cauchy", "uniform")
)

Arguments

Value

Returns half-width of worst-case confidence interval in units of |X - A|.

Author(s)

David Gerard

Examples

wc_width(alpha = 0.2, center = "X")
wc_width(alpha = 0.2, center = "ave")
wc_width(alpha = 0.2, center = "approx")

wc_width(alpha = 0.05, center = "X")
wc_width(alpha = 0.05, center = "ave")
wc_width(alpha = 0.05, center = "approx")