Author: Shikhar Tyagi, Abhishek Tyagi, Arvind Pandey, Bhupendra Singh, Vrijesh Tripathi
Date: 2026-08-05

# Modified Topp-Leone Distribution: Properties, Estimation, and Applications

#### Shikhar Tyagi, Abhishek Tyagi, Arvind Pandey, Bhupendra Singh, Vrijesh Tripathi

#### 2026-08-05

## Introduction

The `ModToppLeone` package provides comprehensive tools for working with the Modified Topp-Leone (MTL) distribution, introduced by Singh, Tyagi, Singh, and Tyagi (2025). The MTL distribution is a flexible single-parameter lifetime model obtained via the transformation \(Y = X / (1 - X)\) where \(X\) follows the classical Topp-Leone distribution.

The probability density function (PDF) and cumulative distribution function (CDF) of the MTL distribution with shape parameter \(\alpha > 0\) are given by:

\[f(y; \alpha) = 2 \alpha (1 + y)^{-(2\alpha + 1)} (2y + y^2)^{\alpha - 1}, \quad y > 0\]

\[F(y; \alpha) = \left( 1 - \frac{1}{(1 + y)^2} \right)^\alpha, \quad y > 0\]

## Core Distribution Functions

The package provides standard distribution functions: `dmtl`, `pmtl`, `qmtl`, `rmtl`, `smtl`, and `hmtl`.

    # Density and CDF
    dmtl(x = 1.0, alpha = 1.5)
    #> [1] 0.3247595
    pmtl(q = 1.0, alpha = 1.5)
    #> [1] 0.6495191

    # Quantile function and Random Generation
    qmtl(p = c(0.25, 0.50, 0.75), alpha = 1.5)
    #> [1] 0.2876192 0.6439022 1.3937547

    set.seed(123)
    sample_data <- rmtl(n = 10, alpha = 1.5)
    sample_data
    #>  [1] 0.33118428 1.61135481 0.49233090 2.54454282 3.99419343 0.07060969
    #>  [7] 0.69846089 2.69933597 0.74728669 0.56742655

    # Survival and Hazard Rate Functions
    smtl(x = 1.0, alpha = 1.5)
    #> [1] 0.3504809
    hmtl(x = 1.0, alpha = 1.5)
    #> [1] 0.9266111

## Statistical Properties

The package includes helper functions to derive theoretical statistical properties:

    # Mode and Mean
    mode_mtl(alpha = 2.5)
    #> [1] 0.553774
    mean_mtl(alpha = 1.5)
    #> [1] 1.356194

    # Quantiles summary (Median, Skewness, Kurtosis)
    quantiles_mtl(alpha = 1.5)
    #>        Q1    Median        Q3      Mode  Skewness  Kurtosis
    #> 0.2876192 0.6439022 1.3937547 0.4678898 0.3558059 1.6156166

    # Mean Deviations about mean and median
    meandev_mtl(alpha = 1.5)
    #>   MD_mean MD_median
    #> 1.2537654 0.3417418

    # Stress-Strength Reliability P(Y2 < Y1)
    ssr_mtl(alpha1 = 2, alpha2 = 3)
    #> [1] 0.4

## Classical Estimation Methods

The parameter \(\alpha\) can be estimated using five classical point estimation procedures: Maximum Likelihood (MLE), Ordinary Least Squares (OLS), Weighted Least Squares (WLS), Cramér-von Mises (CVM), and Maximum Product of Spacings (MPS).

    set.seed(42)
    sim_data <- rmtl(n = 50, alpha = 2.0)

    # Unified estimation wrapper
    fit_results <- fit_mtl(x = sim_data, method = "all")
    fit_results
    #>   Method Estimate        SE    LogLik      AIC      BIC     CAIC   KS_stat
    #> 1    MLE 2.400166 0.3394347 -94.32128 190.6426 192.5546 190.7259 0.1468569
    #> 2    OLS 3.067035        NA -95.95476 193.9095 195.8215 193.9929 0.1166564
    #> 3    WLS 3.032103        NA -95.79980 193.5996 195.5116 193.6829 0.1181673
    #> 4    CVM 3.078623        NA -96.00760 194.0152 195.9272 194.0985 0.1161557
    #> 5    MPS 2.270047        NA -94.39753 190.7951 192.7071 190.8784 0.1535976
    #>     p.value
    #> 1 0.2093677
    #> 2 0.4692642
    #> 3 0.4530415
    #> 4 0.4747030
    #> 5 0.1700895

## Bayesian Estimation

Bayesian estimation is supported under both informative (Gamma) and non-informative priors with symmetric (SELF) and asymmetric (ELF, PLF, GELF) loss functions, alongside Chen-Shao Highest Posterior Density (HPD) intervals.

    # Non-informative prior Bayesian estimation
    bayes_res <- bayes_mtl(x = sim_data, prior = "noninformative", loss = "all")
    bayes_res$estimates
    #>     SELF      ELF      PLF     GELF
    #> 2.400848 2.351487 2.425335 2.351487
    bayes_res$hpd
    #>    Lower    Upper
    #> 1.742630 3.077446

## Censoring Schemes

The package supports sample generation and parameter estimation under various censoring schemes, including Random Right Censoring, Type-I, Type-II, and Progressive Type-II Censoring.

    # Progressive Type-II Censoring example
    R_scheme <- c(2, 0, 1, 0, 2)
    prog_sample <- rcensor_mtl(n = 10, alpha = 2.0, scheme = "progressive2", m = 5, R = R_scheme)

    mle_censor_mtl(x = prog_sample$x, scheme = "progressive2", R = R_scheme)
    #> $method
    #> [1] "MLE under progressive2 censoring"
    #>
    #> $estimate
    #> [1] 3.172105
    #>
    #> $se
    #> [1] 1.0705
    #>
    #> $conf.level
    #> [1] 0.95
    #>
    #> $ci
    #>    Lower    Upper
    #> 1.073963 5.270248
    #>
    #> $loglik
    #> [1] -6.318677
    #>
    #> attr(,"class")
    #> [1] "mtl_censor_fit"

## Real Datasets

The package includes three benchmark real datasets analyzed in the research paper:

1. `dataset_air`: Air conditioning failure times of Boeing 720 jet airplanes.
2. `dataset_covid_india`: Daily new COVID-19 cases in India.
3. `dataset_covid_france`: Daily new COVID-19 cases in France.

    data(dataset_air)
    mle_mtl(dataset_air)
    #> Warning in ks.test.default(x, "pmtl", alpha = alpha_hat): ties should not be
    #> present for the one-sample Kolmogorov-Smirnov test
    #> $method
    #> [1] "Maximum Likelihood Estimation (MLE)"
    #>
    #> $estimate
    #> [1] 0.878027
    #>
    #> $se
    #> [1] 0.1603051
    #>
    #> $conf.level
    #> [1] 0.95
    #>
    #> $ci
    #>     Lower     Upper
    #> 0.5638349 1.1922192
    #>
    #> $loglik
    #> [1] -11.11799
    #>
    #> $AIC
    #> [1] 24.23599
    #>
    #> $BIC
    #> [1] 25.63718
    #>
    #> $CAIC
    #> [1] 24.37884
    #>
    #> $KS_stat
    #> [1] 0.1488445
    #>
    #> $p.value
    #> [1] 0.5195341
    #>
    #> attr(,"class")
    #> [1] "mtl_fit"

## References

- Singh, B., Tyagi, S., Singh, R. P., and Tyagi, A. (2025). Modified Topp-Leone Distribution: Properties, Classical and Bayesian Estimation with Application to COVID-19 and Reliability Data. Thailand Statistician, 23(1), 72-96.