The Tobit Distribution

Density, distribution function, quantile function and random generation for the Tobit model.

dtobit(x, mean = 0, sd = 1, Lower = 0, Upper = Inf, log = FALSE)
ptobit(q, mean = 0, sd = 1, Lower = 0, Upper = Inf,
       lower.tail = TRUE, log.p = FALSE)
qtobit(p, mean = 0, sd = 1, Lower = 0, Upper = Inf,
       lower.tail = TRUE, log.p = FALSE)
rtobit(n, mean = 0, sd = 1, Lower = 0, Upper = Inf)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1 then the length is taken to be the number required.

Lower, Upper

vector of lower and upper thresholds.

mean, sd, lower.tail, log, log.p

see rnorm.

Value

dtobit gives the density, ptobit gives the distribution function, qtobit gives the quantile function, and rtobit generates random deviates.

Author

T. W. Yee

Details

See tobit, the VGAM family function for estimating the parameters, for details. Note that the density at Lower and Upper is the the area to the left and right of those points. Thus there are two spikes (but less in value); see the example below. Consequently, dtobit(Lower) + dtobit(Upper) + the area in between equals unity.

See also

tobit, rnorm.

Examples

mu <- 0.5; x <- seq(-2, 4, by = 0.01)
Lower <- -1; Upper <- 2.0

integrate(dtobit, lower = Lower, upper = Upper,
          mean = mu, Lower = Lower, Upper = Upper)$value +
dtobit(Lower, mean = mu, Lower = Lower, Upper = Upper) +
dtobit(Upper, mean = mu, Lower = Lower, Upper = Upper)  # Adds to 1
#> [1] 1

if (FALSE) { # \dontrun{
plot(x, ptobit(x, m = mu, Lower = Lower, Upper = Upper),
     type = "l", ylim = 0:1, las = 1, col = "orange",
     ylab = paste("ptobit(m = ", mu, ", sd = 1, Lower =", Lower,
                  ", Upper =", Upper, ")"),
     main = "Orange is the CDF; blue is density",
     sub = "Purple lines are the 10,20,...,90 percentiles")
abline(h = 0)
lines(x, dtobit(x, m = mu, L = Lower, U = Upper), col = "blue")

probs <- seq(0.1, 0.9, by = 0.1)
Q <- qtobit(probs, m = mu, Lower = Lower, Upper = Upper)
lines(Q, ptobit(Q, m = mu, Lower = Lower, Upper = Upper),
      col = "purple", lty = "dashed", type = "h")
lines(Q, dtobit(Q, m = mu, Lower = Lower, Upper = Upper),
      col = "darkgreen", lty = "dashed", type = "h")
abline(h = probs, col = "purple", lty = "dashed")
max(abs(ptobit(Q, mu, L = Lower, U = Upper) - probs))  # Should be 0

epts <- c(Lower, Upper)  # Endpoints have a spike (not quite, actually)
lines(epts, dtobit(epts, m = mu, Lower = Lower, Upper = Upper),
      col = "blue", lwd = 3, type = "h")
} # }