% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/properties.R
\name{properties_mtl}
\alias{properties_mtl}
\alias{mode_mtl}
\alias{mean_mtl}
\alias{moments_mtl}
\alias{quantiles_mtl}
\alias{meandev_mtl}
\alias{inequality_mtl}
\alias{mrl_mtl}
\alias{mpl_mtl}
\alias{pwm_mtl}
\alias{entropy_mtl}
\alias{ssr_mtl}
\alias{order_stats_mtl}
\title{Statistical Properties of the Modified Topp-Leone Distribution}
\usage{
mode_mtl(alpha)

mean_mtl(alpha, n_terms = 100)

moments_mtl(r, alpha, n_terms = 100)

quantiles_mtl(alpha)

meandev_mtl(alpha)

inequality_mtl(y, alpha)

mrl_mtl(y, alpha)

mpl_mtl(y, alpha)

pwm_mtl(r, s, alpha)

entropy_mtl(alpha, type = c("shannon", "renyi", "omega"), gamma = 0.5)

ssr_mtl(alpha1, alpha2)

order_stats_mtl(y, alpha, n, u, type = c("pdf", "cdf"))
}
\arguments{
\item{alpha}{Shape parameter of the MTL distribution (\code{alpha > 0}).}

\item{n_terms}{Number of terms to use in series expansions (default: 100).}

\item{r}{Order of the raw moment or probability-weighted moment.}

\item{y}{Evaluation point for residual life, past life, inequality curves, or order statistics.}

\item{s}{Parameter for probability-weighted moment.}

\item{type}{Type of entropy (\code{"renyi"}, \code{"shannon"}, or \code{"omega"}) or order statistics output (\code{"pdf"} or \code{"cdf"}).}

\item{gamma}{Order parameter for Rényi or \eqn{\omega}-entropy (\code{gamma > 0, gamma != 1}).}

\item{alpha1, alpha2}{Shape parameters for two independent MTL variables in stress-strength reliability.}

\item{n}{Sample size for order statistics.}

\item{u}{Index of the order statistic (\code{1 <= u <= n}).}
}
\value{
Numeric scalar or vector corresponding to the calculated statistical property:
\itemize{
  \item \code{mode_mtl}: Mode of the MTL distribution.
  \item \code{mean_mtl}: Mean of the MTL distribution.
  \item \code{moments_mtl}: $r$-th raw moment (finite for $r = 1$).
  \item \code{quantiles_mtl}: Named vector of quantiles, median, skewness, and kurtosis.
  \item \code{meandev_mtl}: Mean deviation about mean and median.
  \item \code{inequality_mtl}: Lorenz, Bonferroni, and Zenga inequality measures.
  \item \code{mrl_mtl}: Mean residual life function.
  \item \code{mpl_mtl}: Mean past life (expected inactivity time) function.
  \item \code{pwm_mtl}: Probability-weighted moment \eqn{\mu_{r,s}}.
  \item \code{entropy_mtl}: Specified entropy measure.
  \item \code{ssr_mtl}: Stress-strength reliability \eqn{R = P[Y_2 < Y_1]}.
  \item \code{order_stats_mtl}: Density or CDF of the $u$-th order statistic.
}
}
\description{
Functions to calculate various statistical properties of the Modified Topp-Leone (MTL)
distribution including mode, mean, quantiles, mean deviations, inequality measures,
mean residual life, mean past life, probability-weighted moments, entropies,
stress-strength reliability, and order statistics.
}
\examples{
# Mode and Mean
mode_mtl(alpha = 2.5)
mean_mtl(alpha = 1.5)

# Quantiles summary
quantiles_mtl(alpha = 1.5)

# Mean deviations
meandev_mtl(alpha = 1.5)

# Inequality measures at y = 1
inequality_mtl(y = 1, alpha = 1.5)

# Mean residual life and past life
mrl_mtl(y = 1, alpha = 1.5)
mpl_mtl(y = 1, alpha = 1.5)

# Entropies
entropy_mtl(alpha = 1.5, type = "shannon")
entropy_mtl(alpha = 1.5, type = "renyi", gamma = 0.5)

# Stress-Strength Reliability
ssr_mtl(alpha1 = 2, alpha2 = 3)

# Order statistics density for median in n = 5 sample
order_stats_mtl(y = 1, alpha = 1.5, n = 5, u = 3, type = "pdf")

}
\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. \emph{Thailand Statistician}, 23(1), 72-96.
}
\author{
Shikhar Tyagi, Abhishek Tyagi, Arvind Pandey, Bhupendra Singh, Vrijesh Tripathi
}
