sumsmnw {lazy.irt}

R Documentation

Distribution of the Weighted Sum of Several Independent Scored Multinomial Distributions

Description

Japanese help file: (sumsmnw_JPH)

Usage

sumsmnw(
  P,
  V = NULL,
  w = rep(1, ncol(P)),
  ncat = NULL,
  compress = 0,
  print = 0,
  plot = 0,
  debug = 0
)

Arguments

P	matrix of probabilities (max # of categories x # of r.v.)
V	matrix of domain values (max # of categories x # of r.v.) or NULL
w	vector of weights (1 x # of r.v.)
ncat	max # of categories (1 x # of r.v) or NULL
compress	= 1 to remove the zero probability categories
print	= 1 to print result
plot	= 1 to plot result
debug	= 1 to print intemediate result

Details

( V[,i], P[,i], w[i] ), i=1,2,...,n is the set of
( domain or category weight ,probability, and weight ) for the i-th r.v.

Non integer V and w will be first converted to integer by linear transformation and converted back at the very end.

ncat[i] = max # of categories for the i-th r.v and
P[(ncat[i]+1):nrow(P),i] == NA

This program calculats the distrobution of
X = \sum_{i=1}^n w[i] X_i
where
X_i is distributed as Scored Multinomial with (V[,i],P[,i]) ,i=1,2,...,n
V[1,i] <= X_i <= V[ncat[i],i] or 0 <= X_i <= ncat[i]
That is, this program calculates the probability that
Pr( X = a ) ,
where
\sum_{i=1}^n V[1,i]*w[i] <= a <= \sum_{i=1}^n V[ncat[i],i]*w[i]

Value

A matrix of (score, prob)

References

Mayekawa, S., & Arai, S. (2008). Distribution of the Sum of Scored Multinomial Random Variables and Its Application to the Item Response Theory. In K. Shigemasu, A. Okada, T.Imaizumi, & T. Hoshino (Eds.) New Trends in Psychometrics. Tokyo: University Academic Press.

Examples

# category x variable matrix of probability:  colSums(P)=c(1,1,1...)
P <- matrix(c(1,2,3,2,   1,2,1,0,  1,2,0,0), 4,3)
P <- t(t(P)/colSums(P))
ncat <- c(4,3,2)
# category x variable matrix of natural category weight
V <- NULL
# variable weight
w <- c(.5,1,1)
res <- sumsmnw( P, V, w, compress=0, print=1, plot=1, ncat=ncat )

# category x variable matrix of category weight
V <- matrix(c(0,1,2,3,   1,2,3,0,  1,1,0,0), 4,3)
# variable weight
w <- c(.5,1,1)
res <- sumsmnw( P, V, w, compress=0, print=1, plot=1, ncat=ncat )

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