Goodness-of-Fit Tests

Let Y₁, Y ₂, . . ., Y _n be a set of independent and identically distributed random variables. Assume that the probability distribution of the Y _i's has the density function f _o (y). We can divide the set of all possible values of Y_i,
i ÃŽ {1, 2, ..., n}, into m non-overlapping intervals D₁, D₂, ...., D_m. Define the probability values p₁, p₂ , ..., p_m as;

p₁ = P(Y_i ÃŽ D₁)
p₂ = P(Y_i ÃŽ D₂)

:
:

p_m = P(Y_i ÃŽ D_m)

Since the union of the mutually exclusive intervals D₁, D₂, ...., D_m is the set of all possible values for the Y_i's, (p₁ + p₂ + .... + p_m) = 1. Define the set of discrete random variables X₁, X₂, ...., X_m, where

X₁= number of Y_i's whose value ÃŽ D₁
X₂= number of Y_i's whose value ÃŽ D₂

:
:

X_m= number of Y_i's whose value ÃŽ D_m

and (X₁+ X₂+ .... + X_m) = n. Then the set of discrete random variables X₁, X₂, ...., X_mwill have a multinomial probability distribution with parameters n and the set of probabilities {p₁, p₂, ..., p_m}. If the intervals D₁, D₂, ...., D_m are chosen such that np_i³ 5 for i = 1, 2, ..., m, then;

For the goodness-of-fit sample test, we formulate the null and alternative hypothesis as

Ho : f_Y(y) = f_o(y)
H1 : f_Y(y) ¹ f_o(y)

At the a level of significance, Ho will be rejected in favor of H₁ if

However, it is possible that in a goodness-of-fit test, one or more of the parameters of f_o(y) are unknown. Then the probability values p₁, p₂, ..., p_m will have to be estimated by assuming that H_o is true and calculating their estimated values from the sample data. That is, another set of probability values p'₁, p'₂, ..., p'_m will need to be computed so that the values (np'₁, np'₂, ..., np'_m) are the estimated expected values of the multinomial random variable (X₁, X₂, ...., X_m). In this case, the random variable C will still have a chi-square distribution, but its degrees of freedom will be reduced. In particular, if the density function f_o(y) has r unknown parameters,

For this goodness-of-fit test, we formulate the null and alternative hypothesis as

H_o: f_Y(y) = f_o(y)
H₁: f_Y(y) ¹ f_o(y)

At the a level of significance, H_o will be rejected in favor of H₁ if