Testing Continuous Uniform Models Using Chi-Square Goodness-of-Fit

Apply the chi-square goodness-of-fit test to assess whether a continuous sample is consistent with a fully specified Uniform(a,b) distribution by binning the data into equal-width subintervals, computing expected counts, and comparing the test statistic to a critical value with k-1 degrees of freedom.

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Discretizing a Continuous Uniform Distribution

The chi-square goodness-of-fit test compares observed counts to expected counts in $k$ categories. To apply it to a continuous distribution, we first need to discretize the data by partitioning the sample space into bins.

To test whether a sample $X_1, X_2, \ldots, X_n$ comes from $\text{Uniform}(a,b),$ partition the interval $[a,b]$ into $k$ equal-width subintervals

\Big[a,\; a+\tfrac{b-a}{k}\Big),\; \Big[a+\tfrac{b-a}{k},\; a+\tfrac{2(b-a)}{k}\Big),\; \ldots,\; \Big[a+\tfrac{(k-1)(b-a)}{k},\; b\Big]

and let $O_i$ denote the number of observations falling into the $i$ -th bin.

Under $H_0: X \sim \text{Uniform}(a,b),$ every equal-width subinterval has the same probability:

P(X \in \text{bin}_i) = \frac{1}{k}.

Therefore the expected count in each bin is

E_i = n \cdot \frac{1}{k} = \frac{n}{k}.

For instance, with $n = 60$ observations partitioned into $k = 4$ equal-width bins on $[0,8],$ the expected count in each bin is $E_i = 60/4 = 15.$