# Hyper-Erlang Distribution

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### Description

Density function, distribution function and random generation for the hyper-Erlang distribution, and a function to generate an object of herlang.

### Usage

 1 2 3 4 5 herlang(shape, mixrate = rep(1/length(shape), length(shape)), rate = rep(1, length(shape))) dherlang(x, herlang = herlang(shape = c(1)), log = FALSE) pherlang(q, herlang = herlang(shape = c(1)), lower.tail = TRUE, log.p = FALSE) rherlang(n, herlang = herlang(shape = c(1))) 

### Arguments

 shape an integer vector of shape parameters of Erlang components. mixrate a vector for the initial probabilities of hyper-Erlang distribution. rate a vector of rate parameters of Erlang components. x, q vectors of quantiles. p a vector of probabilities. n number of observations. herlang an object of S4 class of hyper Erlang (herlang). log logical; if TRUE, the log density is returned. lower.tail logical; if TRUE, probabilities are P[X <= x], otherwise, P[X > x]. log.p logical; if TRUE, the log probability is returned.

### Details

The hyper-Erlang distribution with parameters m_i (mixrate), s_i (shape) and r_i (rate): Cumulative probability function;

F(q) = ∑_i \int_0^q m_i \frac{r_i^{s_i} x^{s_i-1} e^{-r_i x}}{(s_i - 1)!} dx

Probability density function;

f(x) = ∑_i m_i \frac{r_i^{s_i} x^{s_i-1} e^{-r_i x}}{(s_i - 1)!}

### Value

herlang gives an object of hyper-Erlang distribution. dherlang gives the density function, pherlang gives the distribution function, and rherlang generates random samples.

### Note

herlang requires shape parameters.

ph, herlang
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ## create a hyper Erlang consisting of two Erlang ## with shape parameters 2 and 3. (param1 <- herlang(c(2,3))) ## create a hyper Erlang consisting of two Erlang ## with shape parameters 2 and 3. (param1 <- herlang(shape=c(2,3))) ## create a hyper Erlang with specific parameters (param2 <- herlang(shape=c(2,3), mixrate=c(0.3,0.7), rate=c(1.0,10.0))) ## convert to a general PH as(param2, "ph") ## p.d.f. for 0, 0.1, ..., 1 (dherlang(x=seq(0, 1, 0.1), herlang=param2)) ## c.d.f. for 0, 0.1, ..., 1 (pherlang(q=seq(0, 1, 0.1), herlang=param2)) ## generate 10 samples (rherlang(n=10, herlang=param2))