sumsmnw {lazy.irt}R Documentation

Distribution of the Weighted Sum of Several Independent Scored Multinomial Distributions

Description

Distribution of the Weighted Sum of Several Independent Scored Multinomial Distributions

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 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 )


[Package lazy.irt version 0.1.3 Index]