Equation 19 of the paper

eq_19(x, G, p_n, N = 50L, K = 50L, alpha, theta, parallel = FALSE, ...)

Arguments

x	Support of the probability density function.
G	Baseline cumulative distribution function.
p_n	Zero-truncated power series distribution.
N	Maximum number of sums of the external sum of Eq. 19 of the paper.
K	Maximum number of sums of the interior sum of Eq. 19 of the paper.
alpha	Parameter added by the proposed family.
theta	Parameter added by the proposed family.
parallel	If `parallel = TRUE`, the code will be executed in parallel, taking advantage of the predecessor's multicolors, if any. If `parallel = FALSE`, the code will be executed serially.
...	Baseline distribution parameters.

Examples

# Implementing the Weibull distribution function (baseline). This function is
# used in the eq_19 function. Another baseline G can be passed and tested in
# eq_19.
cdf_w <- function(x, beta, lambda) {
   1 - exp(-(lambda * x)^beta)
}

pdf_aprox <- function(x, alpha, theta, lambda, beta) {
  eq_19(
    x = x,
    G = cdf_w,
    p_n = pf_ztp,
    N = 200L,
    K = 100L,
    alpha = alpha,
    theta = theta,
    beta = beta,
    lambda = lambda
  )
}

# True parameters
x = 1.28
alpha = 1.2
theta = 1.5
beta = 1.33
lambda = 2

# Checking theorical density and approximate density for a finite sum of Eq 19 of the paper
pdf_aprox(
   x = x,
   alpha = alpha,
   theta = theta,
   lambda = lambda,
   beta = beta
)
#> [1] 0.2395632
pdf_theorical(
   x = x,
   alpha = alpha,
   theta = theta,
   beta = beta,
   lambda = lambda
)
#> [1] 0.2395632