exactMeDb: Exact moment estimates for the db distribution.
In dbd: Discretised Beta Distribution

Description Usage Arguments Details Value Author(s) See Also Examples

Attempts to calculate “exact” moment estimates of the parameters of a db distribution. This is done by minimising the sum of squared differences between the sample mean and variance (xbar and s2) and the theoretical mean and variance. Calls upon optim() with the "BFGS" method.

1	exactMeDb(x, ntop, zeta=FALSE, par0 = NULL, maxit = 1000)

`x`	A random sample from the db distribution whose parameters are being estimated. Missing values are allowed.
`ntop`	The `ntop` parameter of the db distribution whose parameters are being estimated. I.e. it is the maximum possible value of the distribution, whose values are integers between 1 and `ntop`, or between 0 and `ntop` if `zeta` (see below) is `TRUE`.
`zeta`	See `ddb()`.
`par0`	Optional starting values for the iterative estimation procedure. A vector with entries `alpha` and `beta`. Ideally this vector should be named; if not it is assumed that the entries are in the order `alpha`, `beta`. If not supplied starting values are calculated using the undocumented function `meDb()`.
`maxit`	Integer scalar. The maximum number of iterations to be undertaken by `optim()`. What happens if this number is exceeded depends on the value of `options()[["maxitErrorOrWarning"]]`. This may be `"error"` (in which case an error is thrown if `maxit` is exceeded) or `"warning"` (in which case a warning is issued). The values is set equal to `"error"` at startup. It may be switched, from on possibility to the other, by means of the function `set.eow()`.

This function is really an “intellectual curiosity”. The results produced may be compared with those produced via maximum likelihood (using mleDb()) which in theory should be “better”. Since numerical optimisation has to be applied to calculate the “exact” moment estimates, there is no real saving in terms of computation cost.

An object of class "exactMeDb". Such an object consists of a named vector with entries "alpha" and "beta", which are the “exact” moment estimates of the corresponding parameters. It has a number of attributes:

"ntop" The value of the ntop argument.
"zeta" The value of the zeta argument.
"minSqDiff" The (minimised) value of the sum of the squared differences between the sample mean and variance (xbar and s2) and the theoretical mean and variance. Ideally this minimised value should be zero.
ndata The number of non-missing values in the data set for which the likelihood was maximised, i.e. sum(!is.na(x)).

Rolf Turner r.turner@auckland.ac.nz

ddb meDb() mleDb() expValDb() varDb() optim()

set.seed(42)
x <- rdb(500,3,5,2)
eMom <- exactMeDb(x,ntop=2,zeta=FALSE)
eMle <- mleDb(x,ntop=2)

# Get much better results using true parameter values
# as starting values; pity we can't do this in real life!
eMom <- exactMeDb(x,ntop=2,zeta=FALSE,par0=c(alpha=3,beta=5))
eMle <- mleDb(x,2,par0=c(alpha=3,beta=5))

# Larger ntop value
x <- rdb(500,3,5,20)
eMom <- exactMeDb(x,ntop=20,zeta=FALSE)
eMle <- mleDb(x,ntop=20)

# Binomial, n = 10, p = 0.3.
set.seed(42)
x    <- rbinom(1000,10,0.3)
eMom <- exactMeDb(x,ntop=10,zeta=TRUE)
eMle <- mleDb(x,ntop=10,zeta=TRUE)
p1   <- dbinom(0:10,10,0.3)
p2   <- dbinom(0:10,10,mean(x)/10)
p3   <- table(factor(x,levels=0:10))/1000
p4   <- ddb(0:10,alpha=eMom["alpha"],beta=eMom["beta"],ntop=10,zeta=TRUE)
plot(eMle,obsd=x,legPos=NULL,ylim=c(0,max(p1,p2,p3,p4)))
lines(0.2+(0:10),p1,col="orange",type="h",ylim=c(0,max(p1,p2)))
lines(0.3+(0:10),p2,col="green",type="h")
legend("topright",lty=1,col=c("red","blue","orange","green","black"),
       legend=c("dbMle","observed","true binomial","fitted binomial","dbMom"),bty="n")