Author: Shikhar Tyagi, Arvind Pandey
Date: 2026-05-05

# Introduction to ArvindSt

#### Shikhar Tyagi, Arvind Pandey

#### 2026-05-05

## Overview

The ArvindSt package provides a unified implementation of 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 \((0, \infty)\) with PDF:

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

## Distribution Functions

    library(ArvindSt)

    # PDF at several points
    darvind(c(0.5, 1, 2), theta = 1)
    #> [1] 0.86533420 0.45984930 0.02645592

    # CDF
    parvind(1, theta = 2)
    #> [1] 0.9548882

    # Quantiles
    qarvind(c(0.25, 0.5, 0.75), theta = 1)
    #> [1] 0.2515668 0.5223750 0.8715184

    # Random generation
    set.seed(42)
    x <- rarvind(1000, theta = 2)
    summary(x)
    #>      Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
    #> 0.0001702 0.1495542 0.3086652 0.3835834 0.5492802 1.8545692

## Visualising the Arvind Distribution

    x_seq <- seq(0.01, 4, length.out = 300)
    thetas <- c(0.5, 1, 2, 5)

    plot(NULL, xlim = c(0, 4), ylim = c(0, 1.5),
         xlab = "x", ylab = "f(x)", main = "Arvind PDF Family")
    cols <- c("red", "blue", "darkgreen", "purple")
    for (i in seq_along(thetas)) {
      lines(x_seq, darvind(x_seq, thetas[i]), col = cols[i], lwd = 2)
    }
    legend("topright", paste("theta =", thetas),
           col = cols, lwd = 2, cex = 0.8)

## MLE Estimation

    set.seed(42)
    x <- rarvind(500, theta = 2)
    fit <- fit_arvind_mle(x)
    cat("Estimated theta:", fit$theta, "\n")
    #> Estimated theta: 2.236184
    cat("True theta: 2\n")
    #> True theta: 2

## Model Fitting Example

    # Generate simulated data
    dat <- simulate_arvind_data(n = 60, seed = 1)

    # Fit RW1 model
    m1 <- fit_rw1(Y ~ X1 + X2 + X3, dat, seed = 42)
    cat("Model:", m1$model_type, "\n")
    #> Model: RW1-approx
    cat("Theta:", m1$theta, "\n")
    #> Theta: 0.1631118
    cat("R-squared:", 1 - sum(m1$residuals^2) / sum((m1$Y - mean(m1$Y))^2), "\n")
    #> R-squared: 0.935366

## Diagnostics

    d1 <- diagnostics_arvind(m1)
    d1[, c("Model", "RMSE", "R2", "AIC", "KS_pvalue")]
    #>        Model     RMSE       R2      AIC KS_pvalue
    #> 1 RW1-approx 4.954889 0.935366 179.2226  0.614732

## Cross-Validation

    cv1 <- cv_arvind(m1, k_folds = 3, rolling = FALSE, seed = 42)
    #> Warning: 'newdata' had 20 rows but variables found have 40 rows
    #> Warning: 'newdata' had 20 rows but variables found have 40 rows
    #> Warning: 'newdata' had 20 rows but variables found have 40 rows
    cat("Mean CV RMSE:", cv1$mean_cv_rmse, "\n")
    #> Mean CV RMSE: 27.39963

## Summary

The ArvindSt package provides:

- Distribution functions: `darvind()`, `parvind()`, `qarvind()`, `rarvind()`
- Simulated data generation: `simulate_arvind_data()`
- Five model-fitting functions: `fit_rw1()`, `fit_tvlm()`, `fit_simex()`, `fit_mixed()`, `fit_hmm()`
- Diagnostics: `diagnostics_arvind()`, `plot_arvind()`
- Forecasting: `forecast_arvind()`
- Cross-validation: `cv_arvind()`
- Model comparison: `summary_arvind()`