pdfcau

The parameters from <code><a href="/link/parcau?package=lmomco&version=1.4.3" rd-options="" data-mini-rdoc="lmomco::parcau">parcau</a></code> or <code><a href="/link/vec2par?package=lmomco&version=1.4.3" rd-options="" data-mini-rdoc="lmomco::vec2par">vec2par</a></code>.

para

This function computes the probability density
of the Cauchy distribution given parameters ($\xi$ and $\alpha$) of the
distribution provided by <code><a href="/link/parcau?package=lmomco&version=1.4.3" rd-options="" data-mini-rdoc="lmomco::parcau">parcau</a></code> or <code><a href="/link/vec2par?package=lmomco&version=1.4.3" rd-options="" data-mini-rdoc="lmomco::vec2par">vec2par</a></code>. The probability density
function of the distribution is$$f(x) = \left(\pi \alpha [1 + \left({\frac{x-\xi}{\alpha}}\right)^2] \right)^{-1} \mbox{,}$$where $f(x)$ is the probability density for quantile $x$,
$\xi$ is a location parameter and $\alpha$ is a scale parameter.

distribution

The package implements the statistical theory of L-moments
        in R including L-moment estimation, probability-weighted moment
        estimation, parameter estimation for numerous familiar and
        not-so-familiar distributions, and L-moment estimation for the
        same distributions from the parameters. L-moments are derived
        from the expectations of order statistics and are linear with
        respect to the probability-weighted moments; choice of either
        can be made by mathematical convenience. L-moments are directly
        analogous to the well-known product moments; however, L-moments
        have many advantages including unbiasedness, robustness, and
        consistency with respect to the product moments. The method of
        L-moments can out perform the method of maximum likelihood. The
        lmomco package historically is oriented around canonical
        FORTRAN algorithms of J.R.M. Hosking, and the nomenclature for
        many of the functions parallels that of the Hosking library,
        which later became available in the lmom package. However, vast
        arrays of various extensions and curiosities are added by the
        author to aid and expand the breadth of L-moment application.
        Such extensions include venerable statistics as Sen weighted
        mean, Gini mean difference, plotting positions, and conditional
        probability adjustment. The plotting of L-moment ratio diagrams
        is directly supported in this package. Computations of
        L-moments for right-tail and left-tail censoring by known or
        unknown censoring threshold and also by indicator variable also
        are available. E.A.H. Elamir and A.H. Seheult have developed
        the trimmed L-moments, which are implemented in this package,
        and numerical integration of quantile functions is used to
        dynamically compute trajectories of select TL-moment ratios for
        the construction of TL-moment ratio diagrams. Robert Serfling
        and Peng Xiao have extended L-moments into multivariate space;
        the so-called sample L-comoments are implemented here and might
        have considerable application in copula theory because they
        measure asymmetric correlation and higher co-moments. The
        supported distributions with moment type shown as "L"
        (L-moments) or "TL" (trimmed L-moments) and additional support
        for right-tail censoring ([RC]) include: Cauchy (TL),
        Exponential (L), Gamma (L), Generalized Extreme Value (L),
        Generalized Lambda (L & TL), Generalized Logistic (L),
        Generalized Normal (L), Generalized Pareto (L[RC] & TL), Gumbel
        (L), Kappa (L), Kumaraswamy (L), Normal (L), 3-parameter
        log-Normal (L), Pearson Type III (L), Rayleigh (L), Reverse
        Gumbel (L[RC]), Rice/Rician (L), Truncated Exponential (L),
        Wakeby (L), and Weibull (L).

pdfcau: Probability Density Function of the Cauchy Distribution

Description

Usage

Arguments

Value

References

See Also

Examples