Probability density, cumulative distribution function, quantile function, and random generation for the Pakes distribution.
dpakes(x, zeta)
ppakes(q, zeta)
qpakes(p, zeta)
rpakes(n, zeta)A numeric vector.
Numeric vector of quantiles.
Numeric vector of probabilities
Number of observations.
Mean of distribution. A single, non-negative, numeric value.
Adrian Baddeley.
These functions concern the probability distribution of the random variable $$ X = \sum_{n=1}^\infty \prod_{j=1}^n U_j^{1/\zeta} $$ where \(U_1, U_2, \ldots\) are independent random variables uniformly distributed on \([0,1]\) and \(\zeta\) is a parameter.
This distribution arises in many contexts. For example, for a homogeneous Poisson point process in two-dimensional space with intensity \(\lambda\), the standard Gaussian kernel estimator of intensity with bandwidth \(\sigma\), evaluated at any fixed location \(u\), has the same distribution as \((\lambda/\zeta) X\) where \(\zeta = 2 \pi \lambda\sigma^2\).
Following the usual convention,
dpakes computes the probability density,
ppakes the cumulative distribution function,
and qpakes the quantile function,
and rpakes generates random variates with this distribution.
The computation is based on a recursive integral equation for the cumulative distribution function, due to Professor Tony Pakes, presented in Baddeley, Moller and Pakes (2008). The solution uses the fact that the random variable satisfies the distributional equivalence $$ X \equiv U^{1/\zeta} (1 + X) $$ where \(U\) is uniformly distributed on \([0,1]\) and independent of \(X\).
Baddeley, A., Moller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes, Annals of the Institute of Statistical Mathematics 60, 627--649.
curve(dpakes(x, 1.5), to=4)
rpakes(3, 1.5)
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