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]