Package {PBGoF}

Skew-Normal Goodness-of-Fit Tests Using Precomputed Quantiles

Description

Computes a skew-normal goodness-of-fit statistic and obtains an approximate p-value from a precomputed 100,000-replicate quantile table.

Usage

PBGoF_ks_test(data, ks_table = NULL)

PBGoF_cvm_test(data, cvm_table = NULL)

Arguments

Details

The fitted absolute CP skewness is rounded to two decimal places and truncated to [0.01, 0.99]. P-values are conservative step-function approximations between 0.01 and 0.99. No interpolation is performed.

If the fitted gamma1 is negative, table lookup uses abs(gamma1). Reflection of a skew-normal variable changes the signs of the DP shape parameter and CP skewness, but the null distributions of the EDF statistics are invariant to this sign change (Mateu-Figueras et al., 2007). The bundled tables therefore require only non-negative skewness values.

For samples larger than 500, all observations remain in the fit and empirical distribution function, while the n = 500 row is used for table lookup. The KS comparison uses \sqrt{n}D_n. For CvM, the stored n = 500 quantiles are divided by \sqrt{500} and compared with W_n^2.

Value

A list containing statistic, actual sample size n, gamma1_hat, gamma1_used, and p.value.

References

Mateu-Figueras, G., Puig, P., and Pewsey, A. (2007). Goodness-of-fit tests for the skew-normal distribution when the parameters are estimated from the data. Communications in Statistics—Theory and Methods, 36(9), 1735–1755. doi:10.1080/03610920601126217

Examples

set.seed(123)
x <- sn::rsn(50, xi = 0, omega = 1, alpha = 3)
PBGoF_ks_test(x)
PBGoF_cvm_test(x)

QQ Plot for a Fitted Skew-Normal or Skew-t Distribution

Description

Draws a QQ plot for a fitted skew-normal or skew-t distribution.

Usage

QQ.plot(data, model = "SN", main = NULL,
  ylab = "Observed quantiles", pch = 16, cex = 0.65, lwd = 2)

Arguments

Details

This function is intended only for checking skew-normal and skew-t models. The reference line uses color "#D55E00". The horizontal-axis label is automatically set to Theoretical SN quantiles or Theoretical ST quantiles.

Value

Invisibly returns a list containing the model, fitted parameters, theoretical quantiles, observed quantiles, and reference-line color.

Examples

set.seed(123)
x <- sn::rsn(200, xi = 0, omega = 1, alpha = 5)
QQ.plot(x, model = "SN")

Graphical Check Using Empirical and Fitted Distribution Functions

Description

Plots the empirical cumulative distribution function together with the CDF of a fitted skew-normal or skew-t distribution.

Usage

cdf.plot.check(data, model = "SN", curve_col = NULL, lwd = 2,
  main = NULL, xlab = "Values", ylab = "Distribution function",
  legend = FALSE, ...)

Arguments

Details

The fitted CDF is overlaid on the ECDF with the standard horizontal reference lines at y = 0 and y = 1.

Value

Invisibly returns a list containing the model, fitted parameters, grid values, fitted CDF values, and ECDF object.

Examples

set.seed(123)
x <- sn::rsn(200, xi = 0, omega = 1, alpha = 5)
cdf.plot.check(x, model = "SN")

Unified parametric-bootstrap goodness-of-fit test

Description

Perform a parametric-bootstrap goodness-of-fit test for either a skew-normal or a skew-t distribution. The fitted model is re-estimated in every bootstrap sample.

Usage

para.bootstrap.test(
  data = NULL,
  model = c("SN", "ST"),
  statistic = c("KS", "CvM"),
  B = 1000L,
  seed = 103L,
  verbose = FALSE
)

Arguments

Details

This function dispatches to the existing model- and statistic-specific implementations, so the estimation and bootstrap calibration are unchanged. The legacy functions sn.para.bootstrap.ks.test(), sn.para.bootstrap.cvm.test(), st.para.bootstrap.ks.test(), and st.para.bootstrap.cvm.test() remain available for compatibility.

Value

A numeric bootstrap p-value with attributes containing the observed statistic and bootstrap diagnostics. For the skew-t model, the fitted parameters are also returned in the fit attribute.

Examples

set.seed(123)
x_sn <- sn::rsn(100, alpha = 3)
para.bootstrap.test(x_sn, model = "SN", statistic = "KS", B = 100)

x_st <- sn::rst(100, alpha = 3, nu = 8)
para.bootstrap.test(x_st, model = "ST", statistic = "CvM", B = 100)

Parametric Bootstrap Goodness-of-Fit Tests for a Skew-Normal Distribution

Description

Tests skew-normal goodness of fit using a parametric bootstrap, re-estimating the model in each bootstrap sample with sn.fit.robust().

Usage

sn.para.bootstrap.ks.test(data = NULL, B = 1000L, seed = 103L,
  verbose = FALSE)

sn.para.bootstrap.cvm.test(data = NULL, B = 1000L, seed = 103L,
  verbose = FALSE)

Arguments

Details

The KS function uses \sqrt{n}D; the CvM function uses the Cramér–von Mises statistic. Failed fits are excluded and the p-value uses the finite-simulation correction (1 + \sum I(T_b >= T_0))/(B_{valid}+1). The names ending in .1 are compatibility aliases.

Value

A single numeric bootstrap p-value with attributes statistic, B, valid, and failed.

Examples

set.seed(123)
x <- sn::rsn(50, xi = 0, omega = 1, alpha = 3)
sn.para.bootstrap.ks.test(x, B = 99)
sn.para.bootstrap.cvm.test(x, B = 99)

Numerically Robust Fitting of the Skew-Normal Distribution

Description

Fits a skew-normal distribution by MLE, then falls back to default penalized MLE and matching-prior penalized MLE when necessary.

Usage

sn.fit.robust(data = NULL, para_form = c("DP", "CP"))

Arguments

Details

Robustness here concerns numerical fitting failures, not resistance to outliers or model contamination.

Value

A named numeric vector containing parameter estimates and standard errors. If all fitting procedures fail, all entries are NA.

Examples

set.seed(123)
x <- sn::rsn(100, xi = 0, omega = 1, alpha = 5)
sn.fit.robust(x, "DP")
sn.fit.robust(x, "CP")

Graphical Check of a Fitted Skew-Normal Distribution

Description

Draws a density-scale histogram of the observed data and overlays a skew-normal density fitted by sn.fit.robust().

Usage

sn.plot.check(
  data,
  breaks = "FD",
  col = "grey83",
  border = "grey83",
  curve_col = "#6CC6C6",
  lwd = 2,
  main = NULL,
  xlab = "Values",
  add_rug = TRUE,
  ...
)

Arguments

Details

This visual model check complements, but does not replace, the formal goodness-of-fit tests provided by PBGoF.

Value

Invisibly returns a list containing the fitted DP parameters, histogram object, and coordinates of the fitted density curve.

Examples

set.seed(123)
x <- sn::rsn(100, xi = 0, omega = 1, alpha = 4)
sn.plot.check(x)

Parametric Bootstrap Goodness-of-Fit Tests for a Skew-t Distribution

Description

Tests a fitted univariate skew-t distribution using KS or Cramer–von Mises statistics. All four skew-t parameters are re-estimated in each bootstrap sample with st.fit.robust().

Usage

st.para.bootstrap.ks.test(data = NULL, B = 1000L, seed = 103L, verbose = FALSE)

st.para.bootstrap.cvm.test(data = NULL, B = 1000L, seed = 103L, verbose = FALSE)

Arguments

Details

The KS statistic is sqrt(n) D. The CvM statistic is 1/(12n)+\sum_i[U_i-(2i-1)/(2n)]^2. Failed fits are excluded, and the finite-simulation correction is used.

Value

A numeric bootstrap p-value with attributes statistic, B, valid, failed, and fit.

Examples

set.seed(123)
x <- sn::rst(100, alpha=3, nu=8)
st.para.bootstrap.ks.test(x, B=100)
st.para.bootstrap.cvm.test(x, B=100)

Numerically Robust Fitting of a Univariate Skew-t Distribution

Description

Fits a univariate skew-t distribution using ordinary maximum likelihood, followed by Q-penalized maximum likelihood with alternative optimizers when needed.

Usage

st.fit.robust(data = NULL, para_form = c("DP", "CP"))

Arguments

Details

DP contains xi, omega, alpha, and nu. CP contains the mean, standard deviation, skewness, and kurtosis when these moments exist. Robust refers to numerical fallback, not resistance to outliers.

Value

A named numeric vector containing four estimates and their standard errors.

Examples

set.seed(123)
x <- sn::rst(100, alpha=3, nu=8)
st.fit.robust(x, "DP")

Graphical Check of a Fitted Skew-t Distribution

Description

Draws a density-scale histogram and overlays the skew-t density fitted by st.fit.robust().

Usage

st.plot.check(data, breaks = "FD", col = "grey83", border = "grey83",
  curve_col = "#6CC6C6", lwd = 2, main = NULL, xlab = "Values",
  add_rug = TRUE, ...)

Arguments

Value

Invisibly returns fitted DP parameters, histogram information, and fitted curve coordinates.

Examples

set.seed(123)
x <- sn::rst(100, alpha=3, nu=8)
st.plot.check(x)