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---
title: "Introduction to AlphaPowerHazard"
author: "Shikhar Tyagi"
date: "2026-07-20"
output: rmarkdown::html_vignette
vignette: >
%\VignetteIndexEntry{Introduction to AlphaPowerHazard}
%\VignetteEngine{knitr::rmarkdown}
%\VignetteEncoding{UTF-8}
---
# Overview
The **AlphaPowerHazard** package implements the flexible **Alpha-Power Hazard** probability distribution and regression models proposed by *Pal et al. (2026)* (*Journal of the Indian Society for Probability and Statistics*, DOI: [10.1007/s41096-026-00297-5](https://doi.org/10.1007/s41096-026-00297-5)).
The baseline hazard rate function is:
$$h(x; \alpha, \beta) = \alpha^x + x^{\beta-1}, \quad \alpha > 1, \beta > 0, x > 0$$
The baseline cumulative hazard function is:
$$H(x; \alpha, \beta) = \frac{\alpha^x - 1}{\log \alpha} + \frac{x^\beta}{\beta}$$
The baseline survival function is:
$$S(x; \alpha, \beta) = \exp\left(-H(x; \alpha, \beta)\right)$$
The probability density function is:
$$f(x; \alpha, \beta) = (\alpha^x + x^{\beta-1}) \exp\left(-\left[\frac{\alpha^x - 1}{\log \alpha} + \frac{x^\beta}{\beta}\right]\right)$$
# Statistical Properties
## Distribution Functions
``` r
x_vals <- c(0.5, 1.0, 1.5, 2.0)
alpha <- 1.5
beta <- 0.8
# PDF, CDF, Survival, Hazard, Cumulative Hazard
dalpha_power(x_vals, alpha, beta)
#> [1] 0.66505616 0.20869926 0.06212618 0.01622428
palpha_power(x_vals, alpha, beta)
#> [1] 0.7197927 0.9165203 0.9774842 0.9948008
salpha_power(x_vals, alpha, beta)
#> [1] 0.280207319 0.083479705 0.022515808 0.005199173
halpha_power(x_vals, alpha, beta)
#> [1] 2.373443 2.500000 2.759225 3.120551
Halpha_power(x_vals, alpha, beta)
#> [1] 1.272226 2.483152 3.793538 5.259256
# Quantiles
qalpha_power(c(0.25, 0.50, 0.75), alpha, beta)
#> [1] 0.0952583 0.2572982 0.5480327
```
## Distributional Moments & Quantile Statistics
``` r
mom <- moments_alpha_power(k = 4, alpha = 1.5, beta = 1.1)
cat("Mean:", mom$mean, "\n")
#> Mean: 0.4804367
cat("Variance:", mom$variance, "\n")
#> Variance: 0.1775296
cat("Skewness:", mom$skewness, "\n")
#> Skewness: 1.44417
cat("Kurtosis:", mom$kurtosis, "\n")
#> Kurtosis: 5.528196
qs <- quantile_stats_alpha_power(alpha = 1.5, beta = 1.1)
cat("Bowley Skewness:", qs$bowley_skewness, "\n")
#> Bowley Skewness: 0.2191885
cat("Moors Kurtosis:", qs$moors_kurtosis, "\n")
#> Moors Kurtosis: 1.243827
```
## Hazard Minimum & Order Statistics
``` r
# Unique minimum location for beta < 1 (bathtub shape)
hrf_m <- hrf_min_alpha_power(alpha = 1.5, beta = 0.8)
cat("HRF Minimum x:", hrf_m$xmin, "h(x):", hrf_m$hmin, "\n")
#> HRF Minimum x: 0.4729624 h(x): 2.372933
# Order statistics
order_stats_alpha_power(c(0.5, 1.0), r = 2, n = 5, alpha = 1.5, beta = 1.1, type = "pdf")
#> [1] 0.537817662 0.008380725
```
# Classical Baseline Estimation Methods
The package implements five classical parameter estimation methods for the baseline distribution:
1. Maximum Likelihood Estimation (MLE)
2. Least Squares Estimation (LSE)
3. Weighted Least Squares Estimation (WLSE)
4. Maximum Product of Spacings Estimation (MPSE)
5. Cramér-von Mises Estimation (CME)
``` r
set.seed(123)
sim_x <- ralpha_power(50, alpha = 1.5, beta = 0.8)
fit_mle <- fit_alpha_power_base(sim_x, method = "mle")
fit_lse <- fit_alpha_power_base(sim_x, method = "lse")
fit_wlse <- fit_alpha_power_base(sim_x, method = "wlse")
fit_mpse <- fit_alpha_power_base(sim_x, method = "mpse")
fit_cme <- fit_alpha_power_base(sim_x, method = "cme")
fit_mle
#> Alpha-Power Hazard Base Parameter Estimation (MLE)
#> ====================================================
#> Estimate StdErr
#> alpha 1.38760 0.58822
#> beta 0.86409 0.11333
#>
#> Objective Value: -6.59988
#> Convergence Code: 0
```
# Hazard Regression Models (M1-M4)
The package supports four hazard regression models (M1, M2, M3, M4) across uncensored data, right censoring, left censoring, interval censoring, and progressive censoring schemes:
- **Model M1**: $h(y|Z) = (\alpha^y + y^{\beta-1}) \exp(Z^\top \theta)$
- **Model M2**: $h(y|Z) = \alpha^y + y^{\exp(Z^\top \theta)-1}$
- **Model M3**: $h(y|Z) = (1 + \exp(Z^\top \theta))^y + y^{\beta-1}$
- **Model M4**: $h(y|Z) = (1 + \exp(Z^\top \theta))^y + y^{\exp(Z^\top \theta)-1}$
``` r
set.seed(42)
n <- 60
df <- data.frame(
time = ralpha_power(n, alpha = 1.5, beta = 0.8),
status = sample(c(1, 1, 1, 0), n, replace = TRUE),
age = rnorm(n),
bmi = rnorm(n)
)
fit_m1 <- alpha_power_hazard(survival::Surv(time, status) ~ age + bmi, data = df, model = "M1")
summary(fit_m1)
#> Call:
#> alpha_power_hazard(formula = survival::Surv(time, status) ~ age +
#> bmi, data = df, model = "M1")
#>
#> =========================================================
#> Alpha-Power Hazard Regression Model (Model M1)
#> =========================================================
#> Sample size : 60
#> No. of covariates : 2
#> No. of baseline pars : 2
#>
#> ----- Parameter Estimates --------------------------------
#> Estimate StdErr t_stat p_value CI_lower CI_upper Signif
#> alpha 1.2723 0.6197 2.053 0.044742 1.00010 2.48684 *
#> beta 1.2455 0.3979 3.130 0.002773 0.46567 2.02530 **
#> age -0.4556 0.2271 -2.006 0.049669 -0.90079 -0.01051 *
#> bmi 0.3870 0.1593 2.429 0.018398 0.07466 0.69925 *
#>
#> ----- Model Fit ------------------------------------------
#> Log-Likelihood : -20.9800
#> AIC : 49.9600
#> BIC : 58.3374
#>
#> ----- Overall F-test (covariates) ------------------------
#> F-stat = 5.1235, p-value = 0.009049
#>
#> ----- Multicollinearity Diagnostics ----------------------
#> VIF Tolerance
#> age 1.002 0.998
#> bmi 1.002 0.998
```
# Model Diagnostics & Predictions
``` r
res <- residuals_alpha_power(fit_m1)
head(res$cox_snell)
#> [1] 1.2701583 1.1401953 0.1364794 0.4670338 0.9322080 0.2069983
head(res$martingale)
#> [1] -0.27015830 -0.14019529 0.86352056 0.53296623 0.06779196 0.79300174
# Predictions
pred_s <- predict_alpha_power(fit_m1, type = "survival")
head(pred_s$survival[, 1:4])
#> [,1] [,2] [,3] [,4]
#> [1,] 0.28078717 0.23375690 0.8936540 0.4223338
#> [2,] 0.36920939 0.31975657 0.9155756 0.5085588
#> [3,] 0.21400373 0.17131072 0.8724243 0.3512434
#> [4,] 0.50247593 0.45496584 0.9408973 0.6268589
#> [5,] 0.06955134 0.04733927 0.7898032 0.1638179
#> [6,] 0.40806221 0.35854573 0.9237210 0.5442893
```
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
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