Binomial-extensions {countreg}R Documentation

Extension of the Binomial Distribution

Description

Score function, hessian, mean, and variance for the binomial distribution with parameters prob and size.

Usage

sbinom(x, prob, size, parameter = "prob", drop = TRUE)
hbinom(x, prob, size, parameter = "prob", drop = TRUE)
mean_binom(prob, size, drop = TRUE)
var_binom(prob, size, drop = TRUE)

Arguments

x

vector of quantiles.

prob

probability of success on each trial.

size

number of trials (zero or more).

parameter

character. Derivatives are computed wrt this paramter. Note: Only "prob" is implemented.

drop

logical. Should the result be a matrix (drop = FALSE) or should the dimension be dropped (drop = TRUE, the default)?

Details

The binomial distribution with size = n and prob = p has density

p(x) = choose(n, x) p^x (1-p)^(n-x)

for x = 0, …, n.

The score function is

s(p) = x/p - (n-x)/(1-p)

The hessian is

h(p) = - x/p^2 - (n-x)/(1-p)^2

Value

sbinom gives the score function, i.e., the 1st derivative of the log-density wrt prob and hbinom gives the hessian, i.e., the 2nd derivative of the log-density wrt prob. mean and var give the mean and variance, respectively.

See Also

Binomial encompassing dbinom, pbinom, qbinom and rbinom.

Examples

## Simulate some data
set.seed(123)
y <- rbinom(50, size = 1, prob = 0.3)

## Plot log-likelihood function
par(mfrow = c(1,3))
ll <- function(x) {sum(dbinom(y, size = 1, prob = x, log = TRUE))}
curve(sapply(x, ll), xlab = expression(pi), ylab = "", main = "Log-likelihood")
abline(v = 0.3, lty = 3)

## Plot score function
curve(sapply(x, function(x) sum(sbinom(y, size = 1, x))),
      xlab = expression(pi), ylab = "", main = "Score")
abline(h = 0, lty = 3)
abline(v = 0.3, lty = 3)

## Plot hessian
curve(sapply(x, function(x) sum(hbinom(y, size = 1, x))),
      xlab = expression(pi), ylab = "", main = "Hessian")
abline(v = 0.3, lty = 3)

[Package countreg version 0.2-1 Index]