dHoyt(x, qpar, omega)
pHoyt(q, qpar, omega, lower.tail = TRUE)
qHoyt(p, qpar, omega, lower.tail = TRUE, loUp = NULL)
rHoyt(n, qpar, omega, method = c('eigen', 'chol', 'cdf'), loUp = NULL)
length(n) > 1
, the length is taken to be the number required.TRUE
(default), probabilities are $P[X \le x]$ otherwise, $P[X > x]$.dHoyt
gives the density, pHoyt
gives the cumulative distribution function, qHoyt
gives the quantile function, rHoyt
generates random deviates.The length of the result is determined by n
for rHoyt
, and is the maximum of the lengths of the numerical parameters for the other functions.In dHoyt
, pHoyt
and qHoyt
, the numerical parameters are recycled to the length of the result. Only the first element of the logical parameters is used. In rHoyt
, only the first element of qpar
and omega
is used.
qpar
and omega
may be determined with getHoytParam
.dHoyt
involves the modified Bessel function of the first kind and order 0 (Chew & Boye, 1962; Hoyt, 1947). pHoyt
is implemented as the symmetric difference between two Marcum Q-functions (Paris, 2009). The Marcum Q-function is the cdf of a non-central chi^2 variable (Nuttall, 1975).
qHoyt
is implemented through numerical root finding of pHoyt
. If no search interval for uniroot
is provided, the quantiles of an approximating central chi^2 distribution are used to determine the search intervals.
rHoyt
with method='eigen'
or with method='chol'
simulates 2D normal deviates based on the covariance matrix corresponding to parameters qpar
and omega
, and then determines the radius. rHoyt
with method='cdf'
is much slower as it performs numerical root finding of pHoyt
given simulated quantiles from a uniform random variable in (0,1). If no search interval for uniroot
is provided, the quantiles of an approximating central chi^2 distribution are used to determine the search intervals.
See Rice
for the distribution of radial error around an offset center for uncorrelated bivariate normal variables with equal variances. See Rayleigh
for the distribution of radial error around the true center of uncorrelated bivariate normal variables with equal variances. See mvnEll
for the distribution of radial error around an offset center for correlated normal variables with unequal variances.
Hoyt, R. S. (1947). Probability functions for the modulus and angle of the normal complex variate. Bell System Technical Journal, 26(2), 318-359.
Nuttall, AH. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96
Paris, JF. 2009. Nakagami-q (Hoyt) distribution function with applications. Electronics Letters, 45(4). 210-211. Erratum: doi:10.1049/el.2009.0828
http://reference.wolfram.com/language/ref/HoytDistribution.html
getHoytParam
,
Rayleigh
,
Rice
,
mvnEll
,
Bessel
,
Chisquare
,
uniroot
dHoyt(1, qpar=c(0.1, 0.5, 0.9), omega=10)
pHoyt(c(0.1, 0.5, 0.9), qpar=0.5, omega=10)
qHoyt(0.5, qpar=0.5, omega=c(5, 10, 15))
rHoyt(5, qpar=0.5, omega=10)
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