Package 'ArvindRF'

The Arvind Distribution

Description

Density, distribution function, quantile function, random generation, hazard function, and survival function for the Arvind distribution with parameter theta.

Usage

darvind(x, theta, log = FALSE)

parvind(q, theta, lower.tail = TRUE, log.p = FALSE)

qarvind(p, theta, lower.tail = TRUE, log.p = FALSE)

rarvind(n, theta)

harvind(x, theta)

sarvind(x, theta)

Arguments

x, q	Vector of quantiles.
theta	Scale parameter ($\theta > 0$).
log, log.p	Logical or character string indicating boolean evaluation (TRUE or "TRUE" / FALSE or "FALSE"). If TRUE, probabilities/densities are given as log(p).
lower.tail	Logical or character string indicating boolean evaluation. If TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
p	Vector of probabilities.
n	Number of observations. If length(n) > 1, the length is taken to be the number required.

Details

The Arvind distribution is a continuous distribution on $[0, \infty)$ with scale parameter $\theta > 0$. Its probability density function (PDF) is given by:

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

Its cumulative distribution function (CDF) is:

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

Its survival function is:

$S(x; \theta) = \frac{\exp(-\theta x^2)}{1 + \theta x}, \quad x \ge 0$

Its hazard function is:

$h(x; \theta) = \frac{\theta (1 + 2x + 2\theta x^2)}{1 + \theta x}, \quad x \ge 0$

Quantiles are calculated using root-finding via uniroot, and random values are generated via inversion sampling.

Value

darvind gives the density, parvind gives the distribution function, qarvind gives the quantile function, rarvind generates random deviates, harvind gives the hazard function, and sarvind gives the survival function.

Examples

# Evaluate PDF and CDF at x = 1, theta = 0.5
darvind(1, theta = 0.5)
parvind(1, theta = 0.5)

# Quantile function
qarvind(0.95, theta = 0.5)

# Survival and hazard functions
sarvind(1, theta = 0.5)
harvind(1, theta = 0.5)

# Generate random samples
set.seed(123)
r_samples <- rarvind(10, theta = 0.5)
print(r_samples)

---

Goodness-of-Fit Diagnostics and S3 Methods for RF-Arvind Models

Description

Evaluates Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) goodness-of-fit diagnostic tests, information-theoretic criteria (AIC, BIC), and provides summary, print, and plot methods for rf_arvind objects.

Usage

ks_arvind(r, theta)

ad_arvind(r, theta)

## S3 method for class 'rf_arvind'
print(x, ...)

## S3 method for class 'rf_arvind'
summary(object, ...)

## S3 method for class 'summary.rf_arvind'
print(x, ...)

## S3 method for class 'rf_arvind'
plot(x, ...)

Arguments

r	Vector of absolute residuals.
theta	Estimated Arvind distribution parameter $\theta$.
...	Additional arguments passed to default methods.
object, x	Object of class rf_arvind.

Details

Goodness-of-fit of absolute residuals $r_i = |y_i - \hat{y}_i|$ to the Arvind distribution is evaluated via:

• Kolmogorov-Smirnov Test: Compares empirical CDF $F_n(x)$ against Arvind CDF $F(x; \hat{\theta})$.

• Anderson-Darling Test: Evaluates tail-weighted goodness-of-fit using ad.test.

• AIC and BIC: Calculated as $\mathrm{AIC} = 2k - 2\ell(\hat{\theta})$ and $\mathrm{BIC} = k\log n - 2\ell(\hat{\theta})$ with parameter count $k = 1$.

Value

For ks_arvind and ad_arvind, a list containing test statistics and p-values. For summary.rf_arvind, a summary list object printed cleanly. For plot.rf_arvind, produces diagnostic plots.

Examples

set.seed(123)
df <- data.frame(
  y = 5 + 2 * rnorm(50),
  x1 = rnorm(50),
  x2 = runif(50)
)
fit <- rf_arvind(y ~ x1 + x2, data = df, num.trees = 50)
summary(fit)
plot(fit)

---

Bayesian MCMC Estimation and Highest Posterior Density (HPD) Intervals

Description

Estimates the Arvind distribution parameter theta using Markov Chain Monte Carlo (MCMC) sampling, and computes Highest Posterior Density (HPD) intervals at 90

Usage

mcmc_arvind(
  r,
  n_iter = 5000,
  burn_in = 1000,
  thin = 2,
  init_theta = NULL,
  seed = NULL
)

Arguments

r	Vector of absolute residuals.
n_iter	Total number of MCMC iterations (default 5000).
burn_in	Number of initial burn-in iterations to discard (default 1000).
thin	Thinning interval for posterior chain (default 2).
init_theta	Initial value for $\theta$. If NULL, calculated via arvind_mle.
seed	Optional random seed.

Details

MCMC sampling is performed using an adaptive Metropolis-Hastings algorithm on the log-scale of $\theta$ ($\phi = \log\theta$) to ensure positivity and numerical stability. A weakly informative Gamma(0.001, 0.001) prior is placed on $\theta$. The posterior chain is analyzed using HPDinterval to extract HPD intervals at 90

Value

A list containing:

posterior_mean

Posterior mean estimate of $\theta$.

posterior_median

Posterior median estimate of $\theta$.

hpd_90

Highest Posterior Density interval at 90% level.

hpd_95

Highest Posterior Density interval at 95% level.

hpd_99

Highest Posterior Density interval at 99% level.

acceptance_rate

Acceptance rate of the MCMC sampler.

mcmc_chain

mcmc object of posterior samples.

Examples

set.seed(123)
r_sample <- rarvind(50, theta = 0.5)
mcmc_fit <- mcmc_arvind(r_sample, n_iter = 2000, burn_in = 500)
print(mcmc_fit$posterior_mean)
print(mcmc_fit$hpd_95)

---

Maximum Likelihood Estimation and Bootstrap Inference for Arvind Distribution

Description

Functions to compute the Maximum Likelihood Estimate (MLE) / Expectation-Maximization (EM) point estimate for the Arvind distribution parameter theta, and nonparametric bootstrap confidence intervals.

Usage

arvind_mle(r, tol = 1e-09, max_iter = 100)

bootstrap_ci(
  y,
  X,
  n_boot = 100,
  conf_levels = c(0.9, 0.95, 0.99),
  num.trees = 100,
  seed = NULL
)

Arguments

r	Vector of non-negative absolute residuals.
tol	Numerical tolerance for convergence (default 1e-9).
max_iter	Maximum number of iterations (default 100).
y	Response vector for bootstrap estimation.
X	Predictor matrix or data frame.
n_boot	Number of bootstrap samples (default 100).
conf_levels	Vector of confidence levels (default c(0.90, 0.95, 0.99)).
num.trees	Number of trees for ranger in bootstrap fits (default 100).
seed	Optional random seed.

Details

The score equation for $\theta$ given absolute residuals $r_i = |y_i - \hat{y}_i|$ is:

$\frac{n}{\theta} = \sum_{i=1}^n \left[ r_i^2 + \frac{2 r_i}{1 + \theta r_i} - \frac{2 r_i^2}{1 + 2 r_i + 2 \theta r_i^2} \right]$

This equation is solved iteratively using Newton-Raphson / EM optimization. Under the EM/MLE method, confidence intervals for parameters and performance metrics (bias, MSE, risk) are obtained using non-parametric bootstrap resampling at 90

Value

For arvind_mle, a single scalar estimate of $\theta$. For bootstrap_ci, a list containing point estimates, bootstrap samples, and lower/upper bounds for confidence intervals at the specified confidence levels (90

Examples

set.seed(123)
r_sample <- rarvind(50, theta = 0.5)
est_theta <- arvind_mle(r_sample)
print(est_theta)

---

Predict Method for RF-Arvind Models

Description

Obtains point predictions and homoscedastic prediction intervals (at 90

Usage

## S3 method for class 'rf_arvind'
predict(object, newdata = NULL, conf_levels = c(0.9, 0.95, 0.99), ...)

Arguments

object	Object of class rf_arvind.
newdata	Data frame of new predictor observations. If omitted, fitted training values are used.
conf_levels	Vector of confidence levels for prediction intervals (default c(0.90, 0.95, 0.99)).
...	Additional arguments passed to predict.ranger.

Details

Under the Arvind error distribution, the $(1-\alpha)100\%$ prediction interval for a new response $Y^*$ at predictor $\mathbf{x}^*$ is given by:

$\mathrm{PI}_{1-\alpha}(\mathbf{x}^*) = [\hat{f}(\mathbf{x}^*) - Q_{1-\alpha}(\hat{\theta}), \; \hat{f}(\mathbf{x}^*) + Q_{1-\alpha}(\hat{\theta})]$

where $Q_{1-\alpha}(\hat{\theta})$ is the $(1-\alpha)$-th quantile of the Arvind distribution computed via qarvind.

Value

A data frame containing:

fit	Point predictions $\hat{y}^*$.

Columns for lower and upper prediction interval bounds for each requested level (e.g. PI_90_Lower, PI_90_Upper, PI_95_Lower, PI_95_Upper, PI_99_Lower, PI_99_Upper).

Examples

set.seed(123)
df <- data.frame(
  y = 5 + 2 * rnorm(50),
  x1 = rnorm(50),
  x2 = runif(50)
)
fit <- rf_arvind(y ~ x1 + x2, data = df, num.trees = 50)
preds <- predict(fit, newdata = df[1:5, ])
print(preds)

---

Random Forest Regression with Arvind Distribution Error Model

Description

Fits a Random Forest regression model combined with an Arvind distribution error model. Supports parameter estimation via EM algorithm (with bootstrap confidence intervals at 90

Usage

rf_arvind(
  formula,
  data,
  method = c("em", "mcmc"),
  num.trees = 500,
  n_boot = 100,
  n_iter = 5000,
  burn_in = 1000,
  conf_levels = c(0.9, 0.95, 0.99),
  na.action = stats::na.omit,
  seed = NULL,
  ...
)

Arguments

formula	Object of class formula specifying model formula.
data	Data frame containing the variables in the model.
method	Estimation method: "em" (default) or "mcmc".
num.trees	Number of trees in the Random Forest ensemble (default 500).
n_boot	Number of bootstrap iterations for EM confidence intervals (default 100).
n_iter	Number of MCMC iterations for Bayesian estimation (default 5000).
burn_in	Number of burn-in MCMC iterations (default 1000).
conf_levels	Confidence levels for intervals (default c(0.90, 0.95, 0.99)).
na.action	Function specifying action for missing values (default na.omit).
seed	Optional integer seed for reproducibility.
...	Additional arguments passed to ranger.

Details

The RF-Arvind regression model assumes:

$y_i = f(\mathbf{x}_i) + \varepsilon_i, \quad i = 1, \ldots, n$

where $f(\mathbf{x})$ is estimated by a Random Forest predictor, and absolute errors $|\varepsilon_i|$ follow the Arvind distribution with scale parameter $\theta > 0$.

In Stage 1, Random Forest regression is fitted using ranger under the mean squared error criterion to predict point estimates $\hat{y}_i = \hat{f}(\mathbf{x}_i)$. In Stage 2, absolute residuals $r_i = |y_i - \hat{y}_i|$ are modeled using the Arvind error distribution.

Parameter estimation can be performed via:

• EM Algorithm (method = "em"): Computes maximum likelihood point estimate $\hat{\theta}$ and uses non-parametric bootstrap resampling to evaluate confidence intervals at 90

• Bayesian MCMC (method = "mcmc"): Draws MCMC posterior samples for $\theta$ via adaptive Metropolis-Hastings and computes Highest Posterior Density (HPD) intervals at 90

Missing values are processed according to the specified na.action. Logical parameters accept both standard R boolean flags (TRUE/FALSE) and character representations ("TRUE"/"FALSE").

Value

An S3 object of class "rf_arvind", containing:

rf_model

The fitted ranger Random Forest object.

theta

Estimated Arvind scale parameter $\theta$.

method

Estimation method used ("em" or "mcmc").

fitted_values

Fitted point predictions $\hat{y}$.

residuals

Raw residuals $y - \hat{y}$.

abs_residuals

Absolute residuals $|y - \hat{y}|$.

bias

Estimated model bias.

mse

Mean squared error.

rmse

Root mean squared error.

mae

Mean absolute error.

risk

Model risk value (empirical squared loss).

intervals

Bootstrap confidence intervals (for EM) or HPD intervals (for MCMC) at 90%, 95%, and 99% levels.

bootstrap_results

Detailed bootstrap outputs if method = "em".

mcmc_results

Detailed MCMC sampler outputs if method = "mcmc".

formula

Model formula.

data

Processed training data frame.

response_name

Name of response variable.

predictor_names

Vector of predictor variable names.

Examples

# Fit RF-Arvind model on synthetic regression data
set.seed(123)
df <- data.frame(
  y = 5 + 2 * rnorm(50),
  x1 = rnorm(50),
  x2 = runif(50)
)
fit_em <- rf_arvind(y ~ x1 + x2, data = df, method = "em", num.trees = 50, n_boot = 30, seed = 123)
print(fit_em)

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