% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/dmtl.R
\name{ModToppLeone}
\alias{ModToppLeone}
\alias{dmtl}
\alias{pmtl}
\alias{qmtl}
\alias{rmtl}
\alias{smtl}
\alias{hmtl}
\title{The Modified Topp-Leone Distribution}
\usage{
dmtl(x, alpha, log = FALSE)

pmtl(q, alpha, lower.tail = TRUE, log.p = FALSE)

qmtl(p, alpha, lower.tail = TRUE, log.p = FALSE)

rmtl(n, alpha)

smtl(x, alpha, log.p = FALSE)

hmtl(x, alpha, log = FALSE)
}
\arguments{
\item{x, q}{Vector of quantiles.}

\item{alpha}{Shape parameter of the MTL distribution (\code{alpha > 0}).}

\item{log, log.p}{Logical; if \code{TRUE}, probabilities/densities \eqn{p} are given as \eqn{\log(p)}. Default is \code{FALSE}.}

\item{lower.tail}{Logical; if \code{TRUE} (default), probabilities are \eqn{P[X \le x]}, otherwise \eqn{P[X > x]}.}

\item{p}{Vector of probabilities.}

\item{n}{Number of observations. If \code{length(n) > 1}, the length is taken to be the number required.}
}
\value{
\code{dmtl} gives the density, \code{pmtl} gives the distribution function,
\code{qmtl} gives the quantile function, \code{rmtl} generates random deviates,
\code{smtl} gives the survival function, and \code{hmtl} gives the hazard rate function.
}
\description{
Density, distribution function, quantile function, random generation,
survival function, and hazard rate function for the Modified Topp-Leone (MTL)
distribution with shape parameter \code{alpha}.
}
\details{
The probability density function (PDF) of the Modified Topp-Leone distribution is given by:
\deqn{f(x; \alpha) = 2 \alpha (1 + x)^{-(2\alpha + 1)} (2x + x^2)^{\alpha - 1}, \quad x > 0, \; \alpha > 0}
The cumulative distribution function (CDF) is given by:
\deqn{F(x; \alpha) = \left( \frac{2x + x^2}{(1 + x)^2} \right)^\alpha = \left( 1 - \frac{1}{(1 + x)^2} \right)^\alpha, \quad x > 0, \; \alpha > 0}
The quantile function is derived by inverting the CDF:
\deqn{Q(p; \alpha) = (1 - p^{1/\alpha})^{-1/2} - 1, \quad 0 < p < 1}
}
\examples{
# Evaluate PDF and CDF at x = 1 with alpha = 1.5
dmtl(x = 1, alpha = 1.5)
pmtl(q = 1, alpha = 1.5)

# Quantile function and random generation
p_seq <- c(0.25, 0.50, 0.75)
qmtl(p = p_seq, alpha = 1.5)
set.seed(123)
x_sample <- rmtl(n = 10, alpha = 1.5)
x_sample

# Survival and Hazard functions
smtl(x = 1, alpha = 1.5)
hmtl(x = 1, alpha = 1.5)

}
\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
}
