Package {ArvindSt}

Title:	Five Novel Stochastic Regression Models with Arvind-Distributed Errors and Effects
Version:	1.0.0
Description:	Implements the 'Arvind' distribution and five novel stochastic regression models that replace the traditional Gaussian error assumption with 'Arvind'-distributed errors. The 'Arvind' distribution is a flexible single-parameter continuous distribution on the positive real line characterised by a polynomial numerator with Gaussian-type decay. The package provides complete distribution functions (darvind(), parvind(), qarvind(), rarvind()), maximum likelihood estimation via fit_arvind_mle(), and five model-fitting routines: Random Walk on Coefficients via fit_rw1(), Time-Varying Coefficient Linear Model via fit_tvlm(), Simulation-Extrapolation via fit_simex(), Mixed-Effects Regression via fit_mixed(), and Regime-Switching Hidden Markov Model via fit_hmm(). Additionally provides Monte Carlo forecasting with prediction intervals via forecast_arvind(), comprehensive goodness-of-fit diagnostics (21 metrics and 25 plots) via diagnostics_arvind() and plot_arvind(), k-fold and rolling-window cross-validation via cv_arvind(), and unified model comparison via summary_arvind(). For more details see Pandey, Singh, Tyagi, and Tyagi (2024) "Modelling climate, COVID-19, and reliability data: A new continuous lifetime model under different methods of estimation", Statistics and Applications, 22(2), https://ssca.org.in/journal.html.
License:	MIT + file LICENSE
Depends:	R (≥ 4.0.0)
Imports:	stats, graphics, grDevices, utils, ggplot2, forecast, tvReg, lme4, depmixS4, reshape2, rlang
Suggests:	testthat (≥ 3.0.0), knitr, rmarkdown
Config/testthat/edition:	3
Encoding:	UTF-8
Language:	en-US
RoxygenNote:	7.3.3
VignetteBuilder:	knitr
NeedsCompilation:	no
Packaged:	2026-05-05 17:16:50 UTC; 30017827
Author:	Shikhar Tyagi [aut, cre], Arvind Pandey [aut]
Maintainer:	Shikhar Tyagi <shikhar1093tyagi@gmail.com>
Repository:	CRAN
Date/Publication:	2026-05-11 18:20:02 UTC

ArvindSt: Five Novel Stochastic Regression Models with Arvind-Distributed Errors and Effects

Description

Implements the 'Arvind' distribution and five novel stochastic regression models that replace the traditional Gaussian error assumption with 'Arvind'-distributed errors. The 'Arvind' distribution is a flexible single-parameter continuous distribution on the positive real line characterised by a polynomial numerator with Gaussian-type decay. The package provides complete distribution functions (darvind(), parvind(), qarvind(), rarvind()), maximum likelihood estimation via fit_arvind_mle(), and five model-fitting routines: Random Walk on Coefficients via fit_rw1(), Time-Varying Coefficient Linear Model via fit_tvlm(), Simulation-Extrapolation via fit_simex(), Mixed-Effects Regression via fit_mixed(), and Regime-Switching Hidden Markov Model via fit_hmm(). Additionally provides Monte Carlo forecasting with prediction intervals via forecast_arvind(), comprehensive goodness-of-fit diagnostics (21 metrics and 25 plots) via diagnostics_arvind() and plot_arvind(), k-fold and rolling-window cross-validation via cv_arvind(), and unified model comparison via summary_arvind(). For more details see Pandey, Singh, Tyagi, and Tyagi (2024), "Modelling climate, COVID-19, and reliability data: A new continuous lifetime model under different methods of estimation", 'Statistics and Applications', 22(2).

Author(s)

Maintainer: Shikhar Tyagi shikhar1093tyagi@gmail.com (ORCID)

Authors:

• Arvind Pandey arvindmzu@gmail.com

References

Pandey, A., Singh, R.P., Tyagi, S., and Tyagi, A. (2024). Modelling climate, COVID-19, and reliability data: A new continuous lifetime model under different methods of estimation. Statistics and Applications, 22(2).

Mean of the Arvind Distribution

Description

Computes the theoretical mean of the Arvind distribution with parameter theta by numerical integration.

Usage

arvind_mean_fn(theta)

Arguments

Value

A numeric scalar giving the theoretical mean, or NA if integration fails.

Examples

arvind_mean_fn(1)
arvind_mean_fn(2)

Variance of the Arvind Distribution

Description

Computes the theoretical variance of the Arvind distribution with parameter theta by numerical integration.

Usage

arvind_var_fn(theta)

Arguments

Value

A numeric scalar giving the theoretical variance, or NA if integration fails.

Examples

arvind_var_fn(1)
arvind_var_fn(2)

K-Fold and Rolling-Window Cross-Validation

Description

Performs k-fold cross-validation and optionally rolling-window (expanding-window) cross-validation for an ArvindFit model.

Usage

cv_arvind(fit, k_folds = 5, rolling = TRUE, n0_frac = 0.5, seed = 42)

Arguments

Value

A list with components:

cv_rmse

numeric vector of length k_folds; per-fold RMSE.

cv_mae

numeric vector of length k_folds; per-fold MAE.

mean_cv_rmse

numeric; average k-fold RMSE.

mean_cv_mae

numeric; average k-fold MAE.

roll_rmse

numeric; rolling-window RMSE (or NA).

See Also

diagnostics_arvind(), forecast_arvind()

Examples

dat <- simulate_arvind_data(n = 50, seed = 1)
m1 <- fit_rw1(Y ~ X1 + X2 + X3, dat, seed = 42)
cv <- cv_arvind(m1, k_folds = 3, rolling = FALSE, seed = 42)
cv$mean_cv_rmse

Arvind Distribution Density Function

Description

Computes the probability density function (PDF) of the Arvind distribution.

Usage

darvind(x, theta, log = FALSE)

Arguments

Details

The Arvind distribution with parameter \theta > 0 has PDF

f(x; \theta) = \frac{\theta(1 + 2x + 2\theta x^2)}{(1 + \theta x)^2} \exp(-\theta x^2), \quad x > 0.

Value

A numeric vector of density values (or log-density values when log = TRUE).

Examples

# Evaluate the PDF at several points
darvind(c(0.5, 1, 2), theta = 1)

# Log-density
darvind(1, theta = 2, log = TRUE)

# Returns 0 for x <= 0
darvind(-1, theta = 1)

Goodness-of-Fit Diagnostics for Arvind Models

Description

Computes 21 goodness-of-fit metrics for any fitted ArvindFit object, including MSE, RMSE, MAE, MAPE, R-squared, AIC, BIC, Kolmogorov-Smirnov test, Anderson-Darling statistic, and more.

Usage

diagnostics_arvind(fit)

Arguments

Details

The following metrics are computed:

Model

character; the model type.

MSE

Mean Squared Error.

RMSE

Root Mean Squared Error.

MAE

Mean Absolute Error.

MAPE

Mean Absolute Percentage Error.

R2

R-squared.

AdjR2

Adjusted R-squared.

AIC

Akaike Information Criterion.

AICc

Corrected AIC.

BIC

Bayesian Information Criterion.

LogLik

Log-likelihood at the MLE.

Bias

Mean residual.

MASE

Mean Absolute Scaled Error.

DW

Durbin-Watson statistic.

LjungBox_stat

Ljung-Box test statistic.

LjungBox_p

Ljung-Box p-value.

Theta

Estimated Arvind parameter.

KS_stat

Kolmogorov-Smirnov test statistic.

KS_pvalue

Kolmogorov-Smirnov p-value.

AD_stat

Anderson-Darling test statistic.

CvM_stat

Cramer-von Mises test statistic.

Value

A data frame with one row and 21 columns of diagnostics metrics. See Details for the full list.

Examples

dat <- simulate_arvind_data(n = 50, seed = 1)
m1 <- fit_rw1(Y ~ X1 + X2 + X3, dat, seed = 42)
diagnostics_arvind(m1)

Maximum Likelihood Estimation for the Arvind Distribution

Description

Fits the Arvind distribution to a vector of positive observations by maximum likelihood. Optimisation is performed on the log-scale via the Brent method.

Usage

fit_arvind_mle(e_pos)

Arguments

Value

A list with components:

theta

numeric; the MLE of theta.

negloglik

numeric; the minimised negative log-likelihood.

Examples

set.seed(42)
x <- rarvind(200, theta = 2)
fit_arvind_mle(x)

Fit Regime-Switching Regression (HMM)

Description

Fits a hidden Markov model with state-dependent coefficients and Arvind-distributed errors. The EM algorithm with forward-backward recursions is used for parameter estimation, and the Viterbi algorithm decodes the most likely state sequence.

Usage

fit_hmm(formula, data, nstates = 2, seed = 42)

Arguments

Value

An object of class "ArvindFit", a list containing the same standard fields as fit_rw1(), plus:

hmm_fit

the fitted depmixS4 object.

nstates

integer; number of hidden states.

states

integer vector; Viterbi-decoded state sequence.

trans_probs

matrix; estimated transition probability matrix.

state_betas

list of numeric vectors; state-specific coefficients.

state_sigmas

numeric vector; state-specific standard deviations.

See Also

diagnostics_arvind(), forecast_arvind(), cv_arvind()

Examples

dat <- simulate_arvind_data(n = 50, seed = 1)
m5 <- fit_hmm(Y ~ X1 + X2 + X3, dat, nstates = 2, seed = 42)
m5$states
m5$states

Fit Mixed-Effects Regression with Arvind Errors

Description

Fits a mixed-effects regression model with Arvind-distributed random effects and observation-level errors. Estimation uses a two-stage approach: REML initialisation via lme4, followed by Arvind MLE on the residuals.

Usage

fit_mixed(formula, data, group_var = "Season", re_formula = NULL, seed = 42)

Arguments

Value

An object of class "ArvindFit", a list containing the same standard fields as fit_rw1(), plus:

lme_model

the fitted lme4::lmer object.

theta_re

numeric; Arvind parameter estimated from random effects.

group_var

character; the grouping variable name.

See Also

diagnostics_arvind(), forecast_arvind(), cv_arvind()

Examples

dat <- simulate_arvind_data(n = 50, seed = 1)
m4 <- fit_mixed(Y ~ X1 + X2 + X3, dat, group_var = "Group", seed = 42)
m4$theta
m4$theta

Fit Random Walk on Coefficients Model (RW1-approx)

Description

Fits a stochastic regression model with time-varying coefficients evolving as a random walk with Arvind-distributed innovations. The observation errors also follow the Arvind distribution.

Usage

fit_rw1(formula, data, theta_innov = 2, rw_scale = 0.01, seed = 42)

Arguments

Value

An object of class "ArvindFit", a list containing:

model_type

character; "RW1-approx".

fitted

numeric vector; fitted values.

residuals

numeric vector; raw residuals.

theta

numeric; estimated Arvind parameter for residuals.

sigma

numeric; residual scale.

shift

numeric; shift applied to residuals.

e_pos

numeric vector; positive standardised residuals.

negloglik

numeric; negative log-likelihood.

beta_t

matrix; time-varying coefficient paths.

beta_final

numeric vector; final coefficient values.

sigma_rw

numeric vector; random walk innovation scales.

theta_innov

numeric; Arvind parameter used for innovations.

n

integer; number of observations.

p

integer; number of parameters.

X

matrix; design matrix.

Y

numeric vector; response variable.

formula

the model formula.

data

the input data frame.

See Also

diagnostics_arvind(), forecast_arvind(), cv_arvind()

Examples

dat <- simulate_arvind_data(n = 50, seed = 1)
m1 <- fit_rw1(Y ~ X1 + X2 + X3, dat, seed = 42)
m1$theta

Fit Simulation-Extrapolation (SIMEX) Model

Description

Fits a regression model correcting for measurement error attenuation using the SIMEX algorithm with Arvind-distributed measurement noise and residuals.

Usage

fit_simex(
  formula,
  data,
  me_vars = NULL,
  me_frac = 0.05,
  lambda_grid = c(0.5, 1, 1.5, 2),
  n_sim = 100,
  theta_me = 2,
  seed = 123
)

Arguments

Value

An object of class "ArvindFit", a list containing the same standard fields as fit_rw1(), plus:

beta

numeric vector; SIMEX-corrected coefficient estimates.

simex_coefs

matrix; coefficient estimates at each lambda level.

lambda_grid

numeric vector; the SIMEX lambda grid used.

me_vars

character vector; covariate names with measurement error.

sigma2_me

named numeric vector; measurement error variances.

See Also

diagnostics_arvind(), forecast_arvind(), cv_arvind()

Examples

dat <- simulate_arvind_data(n = 50, seed = 1)
m3 <- fit_simex(Y ~ X1 + X2 + X3, dat,
                me_vars = c("X1", "X2"),
                n_sim = 20, seed = 123)
m3$beta
m3$beta

Fit Time-Varying Coefficient Linear Model (tvLM)

Description

Fits a time-varying coefficient linear model using kernel-weighted least squares (via the tvReg package) with Arvind-distributed residuals.

Usage

fit_tvlm(formula, data, bw = NULL, seed = 42)

Arguments

Value

An object of class "ArvindFit", a list containing the same standard fields as fit_rw1(), plus:

tv_coefs

matrix; time-varying coefficient estimates.

tv_fit

the fitted tvReg::tvLM object.

See Also

diagnostics_arvind(), forecast_arvind(), cv_arvind()

Examples

dat <- simulate_arvind_data(n = 50, seed = 1)
m2 <- fit_tvlm(Y ~ X1 + X2 + X3, dat, bw = 0.5, seed = 42)
m2$theta

Monte Carlo Forecasting for Arvind Models

Description

Generates Monte Carlo forecasts with 80 percent and 95 percent prediction intervals for any fitted ArvindFit model. Covariates are forecast using SARIMA models (via the forecast package) if not supplied.

Usage

forecast_arvind(
  fit,
  newdata_sims = NULL,
  h = 120,
  nsim = 5000,
  covariate_models = NULL,
  seed = 123
)

Arguments

Value

A list with components:

sims

matrix (h x nsim); full simulation matrix.

mean

numeric vector length h; mean forecast.

median

numeric vector length h; median forecast.

lo80

numeric vector; lower 80 percent prediction interval.

hi80

numeric vector; upper 80 percent prediction interval.

lo95

numeric vector; lower 95 percent prediction interval.

hi95

numeric vector; upper 95 percent prediction interval.

See Also

fit_rw1(), diagnostics_arvind(), cv_arvind()

Examples

dat <- simulate_arvind_data(n = 50, seed = 1)
m1 <- fit_rw1(Y ~ X1 + X2 + X3, dat, seed = 42)
fc <- forecast_arvind(m1, h = 12, nsim = 100, seed = 42)
head(fc$mean)

Transform Residuals for Arvind Fitting

Description

Transforms raw residuals to positive values suitable for fitting the Arvind distribution by shifting and standardising.

Usage

make_arvind_resid(resid_raw, Y_ref)

Arguments

Value

A list with components:

shift

numeric; the shift applied.

sigma

numeric; the standard deviation used for standardisation.

e_pos

numeric vector; positive standardised residuals.

theta

numeric; MLE of the Arvind parameter.

negloglik

numeric; negative log-likelihood at the MLE.

Arvind Distribution Function (CDF)

Description

Computes the cumulative distribution function (CDF) of the Arvind distribution.

Usage

parvind(q, theta, lower.tail = TRUE)

Arguments

Details

The CDF is given by

F(x; \theta) = 1 - \frac{1}{1 + \theta x} \exp(-\theta x^2), \quad x > 0.

Value

A numeric vector of probabilities.

Examples

parvind(1, theta = 1)
parvind(c(0.5, 1, 2), theta = 2)
parvind(1, theta = 1, lower.tail = FALSE)

Diagnostic Plots for Arvind Models

Description

Generates up to 25 diagnostic plots for a fitted ArvindFit object, including observed vs fitted, residual histogram with Arvind density overlay, Q-Q plot, ACF, ECDF comparison, and more.

Usage

plot_arvind(fit, output_dir = tempdir(), prefix = NULL)

Arguments

Value

The fit object is returned invisibly.

Examples

dat <- simulate_arvind_data(n = 50, seed = 1)
m1 <- fit_rw1(Y ~ X1 + X2 + X3, dat, seed = 42)
plot_arvind(m1, output_dir = tempdir())

Arvind Distribution Quantile Function

Description

Computes quantiles of the Arvind distribution by numerical inversion of the CDF using uniroot.

Usage

qarvind(p, theta)

Arguments

Value

A numeric vector of quantiles.

Examples

qarvind(0.5, theta = 1)
qarvind(c(0.25, 0.5, 0.75), theta = 2)

Random Generation from the Arvind Distribution

Description

Generates random variates from the Arvind distribution using a rejection sampling algorithm with a half-normal proposal distribution.

Usage

rarvind(n, theta)

Arguments

Value

A numeric vector of length n containing positive random variates.

Examples

set.seed(42)
x <- rarvind(100, theta = 1)
summary(x)

Centred Random Generation from the Arvind Distribution

Description

Generates centred Arvind variates with approximately zero mean, suitable for use as error terms and innovation terms in stochastic regression models.

Usage

rarvind_centred(n, theta)

Arguments

Details

The centred variate is computed as \tilde{\varepsilon} = \varepsilon - \mu_A(\theta), where \varepsilon \sim \mathrm{Arvind}(\theta) and \mu_A(\theta) is the mean of the Arvind distribution.

Value

A numeric vector of length n with approximately zero mean.

Examples

set.seed(42)
eps <- rarvind_centred(1000, theta = 2)
mean(eps)  # approximately 0

Generate Simulated Data for Examples

Description

Creates a small simulated dataset that mimics the structure needed for demonstrating the ArvindSt model-fitting functions. Useful for examples and testing.

Usage

simulate_arvind_data(n = 60, seed = 42)

Arguments

Value

A data frame with columns:

Y

numeric; simulated response variable.

X1

numeric; first covariate.

X2

numeric; second covariate.

X3

numeric; third covariate.

Group

factor; grouping variable with 4 levels.

Examples

dat <- simulate_arvind_data(n = 50, seed = 1)
head(dat)

Summary and Comparison of Multiple Arvind Models

Description

Accepts multiple ArvindFit objects, computes diagnostics for each, produces a unified comparison table, and prints the best model by RMSE, R-squared, and AIC.

Usage

summary_arvind(..., comparison_plots = TRUE, output_dir = tempdir())

Arguments

Value

A data frame of diagnostic metrics (one row per model) is returned invisibly.

See Also

diagnostics_arvind(), plot_arvind()

Examples

dat <- simulate_arvind_data(n = 50, seed = 1)
m1 <- fit_rw1(Y ~ X1 + X2 + X3, dat, seed = 42)
summary_arvind(m1)