dtmixbeta function

Exponentially Tilted Mixture Beta Distribution

Density, distribution function, quantile function and pseudorandom number generation for the exponentially tilted mixture of beta distributions, with shapes $(i+1, m-i+1)$, $i = 0, \ldots, m$, given mixture proportions $p=(p_0,\ldots,p_m)$ and support interval.

dtmixbeta(x, p, alpha, interval = c(0, 1), regr, ...)

ptmixbeta(x, p, alpha, interval = c(0, 1), regr, ...)

qtmixbeta(u, p, alpha, interval = c(0, 1), regr, ...)

rtmixbeta(n, p, alpha, interval = c(0, 1), regr, ...)

Arguments

• x: a vector of quantiles

• p: a vector of m+1 components of p must be nonnegative and sum to one for mixture beta distribution. See 'Details'.

• alpha: regression coefficients

• interval: support/truncation interval [a, b].

• regr: regressor vector function $r(x)=(1,r_1(x),...,r_d(x))$

  which returns n x (d+1) matrix, n=length(x)

• ...: additional arguments to be passed to regr

• u: a vector of probabilities

• n: sample size

Returns

A vector of $f_m(x; p)$ or $F_m(x; p)$ values at $x$. dmixbeta returns the density, pmixbeta returns the cumulative distribution function, qmixbeta returns the quantile function, and rmixbeta generates pseudo random numbers.

Details

The density of the mixture exponentially tilted beta distribution on an interval $[a, b]$ can be written c("$f_m(x; p)=(b-a)^{-1}\\exp(\\alpha'r(x))\n$", "$\\sum_{i=0}^m p_i\\beta_{mi}[(x-a)/(b-a)]/(b-a)$"), where $p = (p_0, \ldots, p_m)$, $p_i\ge 0$, $\sum_{i=0}^m p_i=1$ and $\beta_{mi}(u) = (m+1){m\choose i}u^i(1-x)^{m-i}$, $i = 0, 1, \ldots, m$, is the beta density with shapes $(i+1, m-i+1)$. The cumulative distribution function is $F_m(x; p) = \sum_{i=0}^m p_i B_{mi}[(x-a)/(b-a);alpha]$, where $B_{mi}(u ;alpha)$, $i = 0, 1, \ldots, m$, is the exponentially tilted beta cumulative distribution function with shapes $(i+1, m-i+1)$.

Examples

# classical Bernstein polynomial approximation
a<--4; b<-4; m<-200
x<-seq(a,b,len=512)
u<-(0:m)/m
p<-dnorm(a+(b-a)*u)
plot(x, dnorm(x), type="l")
lines(x, (b-a)*dmixbeta(x, p, c(a, b))/(m+1), lty=2, col=2)
legend(a, dnorm(0), lty=1:2, col=1:2, c(expression(f(x)==phi(x)),
               expression(B^{f}*(x))))

References

Guan, Z., Application of Bernstein Polynomial Model to Density and ROC Estimation in a Semiparametric Density Ratio Model

See Also

mable

Author(s)

Zhong Guan zguan@iu.edu