Bhattacharjee distribution
Density, distribution function, and random generation for the Bhattacharjee distribution.
dbhatt(x, mu = 0, sigma = 1, a = sigma, log = FALSE) pbhatt(q, mu = 0, sigma = 1, a = sigma, lower.tail = TRUE, log.p = FALSE) rbhatt(n, mu = 0, sigma = 1, a = sigma)
x, q |
vector of quantiles. |
mu, sigma, a |
location, scale and shape parameters. Scale and shape must be positive. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x]. |
n |
number of observations. If |
If Z ~ Normal(0, 1) and U ~ Uniform(0, 1), then Z+U follows Bhattacharjee distribution.
Probability density function
f(z) = 1/(2*a) * (Φ((x-μ+a)/σ) - Φ((x-μ+a)/σ))
Cumulative distribution function
F(z) = σ/(2*a) * ((x-μ)*Φ((x-μ+a)/σ) - (x-μ)*Φ((x-μ-a)/σ) + φ((x-μ+a)/σ) - φ((x-μ-a)/σ))
Bhattacharjee, G.P., Pandit, S.N.N., and Mohan, R. (1963). Dimensional chains involving rectangular and normal error-distributions. Technometrics, 5, 404-406.
x <- rbhatt(1e5, 5, 3, 5) hist(x, 100, freq = FALSE) curve(dbhatt(x, 5, 3, 5), -20, 20, col = "red", add = TRUE) hist(pbhatt(x, 5, 3, 5)) plot(ecdf(x)) curve(pbhatt(x, 5, 3, 5), -20, 20, col = "red", lwd = 2, add = TRUE)
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