Statistical Inference: A Short Course

13.2 The Multinomial Chi-Square Statistic

We found in Chapter 5 that the essential characteristics of a binomial random experiment are

a. We have a discrete random variable X.

b. We have a simple alternative experiment.

c. The n trials are identical and independent.

d. p, the probability of a success, is constant from trial to trial.

With a simple alternative experiment, there are two possible outcomes: X = 0 or X = 1. Hence the elements of the population belong to one of two classes—success or failure.

Let us consider the more general case where the elements of the population are classified as belonging to one of k possible classes (k ≥ 2). That is, we now perform what is called a multinomial random experiment. For this type of experiment, each of the n trials results in a k-fold alternative, that is, each trial results in any one of k mutually exclusive and collectively exhaustive outcomes with respective probabilities , where and the p_i are constant from trial to trial. If the discrete random variable , depicts the number of times the ith outcome type E_i occurs in the n independent ...