To Simulate A Multinomial Distribution
Various methods may be used to simulate a multinomial distribution. A very simple one is to use a random number generator to generate numbers between 0 and 1. First, we divide the interval from 0 to 1 in k subintervals equal in size to the probabilities of the k categories. Then, we generate a random number for each of n trials and use a logical test to classify the virtual measure or observation in one of the categories.
Example
If we have :
| Categories | 1 | 2 | 3 | 4 | 5 | 6 |
| Probabilities | 0.15 | 0.20 | 0.30 | 0.16 | 0.12 | 0.07 |
| Superior limits of subintervals | 0.15 | 0.35 | 0.65 | 0.81 | 0.93 | 1.00 |
Then, with a software like Excel, we may use the following recipe:
| Cells : | Ai | Bi | Ci | ... | Gi |
| Formulae : | Alea | =If($Ai<0.15;1;0) | =If(And($Ai>=0.15;$Ai<0.35);1;0) | ... | =If($Ai>=0.93;1;0) |
After that, we will use functions such as SumIf to accumulate the observed results by category and to calculate the estimated covariance matrix for each simulated sample.
Another way with Excel, is to use the discrete random number generator. In that case, the categories must be label or relabel with numeric values.
In the two cases, the result is a multinomial distribution with k categories without any correlation. This is equivalent, with a continuous random distribution, to simulate k independent standardized normal distributions, or a multinormal distribution N(0,I) having k components identically distributed and statistically independent.
Read more about this topic: Multinomial Distribution
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