VGAM (version 0.8-2)

dcnormal1: Univariate Normal Distribution with Double Censoring

Description

Maximum likelihood estimation of the two parameters of a univariate normal distribution when there is double censoring.

Usage

dcnormal1(r1 = 0, r2 = 0, link.sd = "loge",
          earg=list(), isd = NULL, zero = NULL)

Arguments

r1, r2

Integers. Number of smallest and largest values censored, respectively.

link.sd

Parameter link function applied to the standard deviation. See Links for more choices. Being a positive quantity, a log link is the default.

earg

List. Extra argument for the link. See earg in Links for general information.

isd

Numeric. Initial value for the standard deviation. The default value NULL means an initial value is obtained internally from the data.

zero

An integer with value 1 or 2. If so, the mean or standard deviation respectively are modelled as an intercept only. Usually, setting zero=2 will be used, if used at all. The default value NULL means both linear/additive p

Value

• An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Details

This family function uses the Fisher information matrix given in Harter and Moore (1966). The matrix is not diagonal if either r1 or r2 are positive.

By default, the mean is the first linear/additive predictor and the log of the standard deviation is the second linear/additive predictor.

References

Harter, H. L. and Moore, A. H. (1966) Iterative maximum-likelihood estimation of the parameters of normal populations from singly and doubly censored samples. Biometrika, 53, 205--213.

See Also

normal1, tobit.

Examples

# Repeat the simulations described in Harter and Moore (1966)
SIMS = 100   # Number of simulations (change this to 1000)
mu.save = sd.save = rep(NA, len=SIMS)
r1 = 0; r2 = 4; n = 20  
for(sim in 1:SIMS) {
    y = sort(rnorm(n))
    y = y[(1+r1):(n-r2)]  # Delete r1 smallest and r2 largest
    fit = vglm(y ~ 1, dcnormal1(r1=r1, r2=r2))
    mu.save[sim] = predict(fit)[1,1]
    sd.save[sim] = exp(predict(fit)[1,2])   # Assumes a log link and ~ 1
}
# Now look at the results
c(mean(mu.save), mean(sd.save))  # Should be c(0,1)
c(sd(mu.save), sd(sd.save))

# Data from Sarhan and Greenberg (1962); MLEs are mu=9.2606, sd=1.3754
strontium90 = data.frame(y = c(8.2, 8.4, 9.1, 9.8, 9.9))
fit = vglm(y ~ 1, dcnormal1(r1=2, r2=3, isd=6), strontium90, trace=TRUE)
coef(fit, matrix=TRUE)
Coef(fit)