Package 'ArvindSt' reference manual

Title:	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), <https://ssca.org.in/journal.html>.
Authors:	Shikhar Tyagi [aut, cre] (ORCID: <https://orcid.org/0000-0003-1606-0844>), Arvind Pandey [aut]
Maintainer:	Shikhar Tyagi <[email protected]>
License:	MIT + file LICENSE
Version:	1.0.0
Built:	2026-05-11 20:38:01 UTC
Source:	https://github.com/cran/ArvindSt

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)
arvind_mean_fn(theta)

Arguments

theta

positive numeric scalar; the distribution parameter.

Value

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

Examples

arvind_mean_fn(1)
arvind_mean_fn(2)

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)
arvind_var_fn(theta)

Arguments

theta

positive numeric scalar; the distribution parameter.

Value

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

Examples

arvind_var_fn(1)
arvind_var_fn(2)

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

Arguments

fit

an object of class "ArvindFit".

k_folds

integer; number of cross-validation folds (default: 5).

rolling

logical; if TRUE (default), also performs rolling-window cross-validation.

n0_frac

numeric; fraction of data used as initial training window for rolling CV (default: 0.5).

seed

integer; random seed for reproducibility (default: 42).

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).

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

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)
darvind(x, theta, log = FALSE)

Arguments

x

numeric vector of quantiles.

theta

positive numeric scalar; the distribution parameter.

log

logical; if TRUE, log-density is returned. Default FALSE.

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)

# 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)
diagnostics_arvind(fit)

Arguments

fit

an object of class "ArvindFit" returned by any of the model-fitting functions.

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)

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)
fit_arvind_mle(e_pos)

Arguments

e_pos

numeric vector of strictly positive observations.

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)

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

Arguments

formula

an object of class formula.

data

a data frame containing the variables in the formula.

nstates

integer; number of hidden states (default: 2).

seed

integer; random seed for reproducibility (default: 42).

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.

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

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

Arguments

formula

an object of class formula specifying the fixed-effects structure.

data

a data frame containing the variables in the formula and the grouping variable.

group_var

character string; the name of the grouping variable in data (default: "Season").

re_formula

optional random-effects formula (e.g., (1 + X1 | group)). If NULL (default), a random intercept model (1 | group_var) is used.

seed

integer; random seed for reproducibility (default: 42).

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.

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

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

Arguments

formula

an object of class formula specifying the model (e.g., Y ~ X1 + X2).

data

a data frame containing the variables in the formula.

theta_innov

positive numeric; the Arvind parameter for state innovations (default: 2.0).

rw_scale

numeric; proportion of OLS coefficients used as innovation scale (default: 0.01).

seed

integer; random seed for reproducibility (default: 42).

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.

Examples

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

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
)
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

formula

an object of class formula.

data

a data frame containing the variables in the formula.

me_vars

character vector of covariate names measured with error. If NULL (default), the first two term labels are used.

me_frac

numeric; fraction of marginal variance used as measurement error variance (default: 0.05).

lambda_grid

numeric vector; SIMEX lambda grid (default: c(0.5, 1, 1.5, 2)).

n_sim

integer; number of SIMEX simulation replicates (default: 100).

theta_me

positive numeric; Arvind parameter for measurement error (default: 2.0).

seed

integer; random seed for reproducibility (default: 123).

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.

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

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

Arguments

formula

an object of class formula.

data

a data frame containing the variables in the formula.

bw

numeric or NULL; the bandwidth for kernel smoothing. If NULL (default), bandwidth is selected automatically via leave-one-out cross-validation.

seed

integer; random seed for reproducibility (default: 42).

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.

Examples

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

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

Arguments

fit

an object of class "ArvindFit".

newdata_sims

optional named list of pre-computed covariate simulation matrices, each of dimension h x nsim.

h

integer; forecast horizon in time steps (default: 120).

nsim

integer; number of Monte Carlo replicates (default: 5000).

covariate_models

optional list of fitted SARIMA models for covariates (auto-fitted if NULL).

seed

integer; random seed for reproducibility (default: 123).

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.

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)


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)

Arvind Distribution Function (CDF)

Description

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

Usage

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

Arguments

q

numeric vector of quantiles.

theta

positive numeric scalar; the distribution parameter.

lower.tail

logical; if TRUE (default), probabilities are $P(X \le q)$ ; otherwise $P(X > q)$ .

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)

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

Arguments

fit

an object of class "ArvindFit".

output_dir

character; directory where plots are saved. Defaults to a temporary directory.

prefix

character or NULL; prefix for plot filenames. If NULL, derived from the model type.

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())


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)
qarvind(p, theta)

Arguments

p

numeric vector of probabilities ( $0 \le p \le 1$ ).

theta

positive numeric scalar; the distribution parameter.

Value

A numeric vector of quantiles.

Examples

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

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)
rarvind(n, theta)

Arguments

n

positive integer; number of random variates to generate.

theta

positive numeric scalar; the distribution parameter.

Value

A numeric vector of length n containing positive random variates.

Examples

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

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)
rarvind_centred(n, theta)

Arguments

n

positive integer; number of random variates to generate.

theta

positive numeric scalar; the distribution parameter.

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

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)
simulate_arvind_data(n = 60, seed = 42)

Arguments

n

integer; number of observations to generate (default: 60).

seed

integer; random seed for reproducibility (default: 42).

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)

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

Arguments

...

one or more objects of class "ArvindFit".

comparison_plots

logical; if TRUE (default), generate comparison plots.

output_dir

character; directory to save comparison plots. Defaults to a temporary directory.

Value

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

Examples

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

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

Package 'ArvindSt'

Help Index

Mean of the Arvind Distribution

Description

Usage

Arguments

Value

Examples

Variance of the Arvind Distribution

Description

Usage

Arguments

Value

Examples

K-Fold and Rolling-Window Cross-Validation

Description

Usage

Arguments

Value

See Also

Examples

Arvind Distribution Density Function

Description

Usage

Arguments

Details

Value

Examples

Goodness-of-Fit Diagnostics for Arvind Models

Description

Usage

Arguments

Details

Value

Examples

Maximum Likelihood Estimation for the Arvind Distribution

Description

Usage

Arguments

Value

Examples

Fit Regime-Switching Regression (HMM)

Description

Usage

Arguments

Value

See Also

Examples

Fit Mixed-Effects Regression with Arvind Errors

Description

Usage

Arguments

Value

See Also

Examples

Fit Random Walk on Coefficients Model (RW1-approx)

Description

Usage

Arguments

Value

See Also

Examples

Fit Simulation-Extrapolation (SIMEX) Model

Description

Usage

Arguments

Value

See Also

Examples

Fit Time-Varying Coefficient Linear Model (tvLM)

Description

Usage

Arguments

Value

See Also

Examples

Monte Carlo Forecasting for Arvind Models

Description

Usage

Arguments