What about discrete distributions?

The K-S test is only appropriate for continuous distirbutions (the hypothesized distribution is continuous).

But what about if our hypothesis if for a discrete distribution, e.g.:

• Discrete Uniform
• Bernoulli
• Poisson

Setting

Population: \(Y \sim\) some discrete population distribution with p.m.f \(p(y) = P(Y = y)\)

Sample: n i.i.d from population, \(Y_1, \ldots, Y_n\)

Parameter: Whole p.m.f

Hypotheses: \(H_0: P(Y = y) = p_0(y)\), versus \(H_A: P(Y = y) \ne p_0(y)\)

Sample estimate of the p.m.f

The discrete sample based estimate of the probability mass function:

\[ \hat{p}(y) = \frac{1}{n} \sum_{i = 1}^{n} \pmb{1} \left\{ Y_i = y \right\} \]

The Chi-square Goodness of Fit test

Chi-square goodness of fit compares the estimated p.m.f to the hypothesized one.

Pearson’s Chi-square statistic: \[ X(p_0) = \sum_y \frac{n\left( \hat{p}(y) - p_0(y) \right)^2}{p_0(y)} \]

Under null hypothesis: \(X(p_0)\) converges in distibrution (as n goes to infinity) to \(\chi^2\) with \(k-1\) degrees of freedom

\(k=\) number of possible values for \(Y\).

An alternative presentation

• \(j = 1, \ldots, n\) indexes possible values/categories for \(Y\)
• \(O_j\) be the observed number of values in category \(j\)
• \(E_j = np_0(j)\) be the expected number of values in category \(j\), based on the hypothesized distribution.

Pearson’s Chi-square statistic: \[ X(p_0) = \sum_{j = 1}^{k} \frac{(O_j - E_j)^2}{E_j} \]

Example: Dice rolling

I rolled a die 60 times and recorded how many times I got each side: 1, 2, 3, 4, 5, 6

Question: Is the die fair? That is, is \(p_0(j) = 1/6\) for \(j = 1, 2, 3, 4, 5, 6\)?

Rejection region

Reject \(H_0\) for \(X(p_0) > \chi^2_{(k-1)} (1-\alpha)\)

Rejection region always to the right, p-values always area to right. Why? We usualy are only interested in evidence of poor fit (not evidence of extra good fit)

Example: Dice rolling cont.

Compare to \(\chi_{(5)}^2\)

qchisq(0.95, df = 5)

## [1] 11.0705

1 - pchisq(chi_sq, df = 5)

## [1] 0.01438768

In R:

rolls

##  [1] 6 5 2 2 6 2 1 1 5 1 2 2 1 1 3 4 5 1 2 4 1 6 6
## [24] 2 4 1 6 6 1 6 1 1 6 1 3 1 3 4 1 1 5 3 4 2 5 1
## [47] 6 3 1 4 3 2 5 1 6 2 1 4 1 2

chisq.test(table(rolls), p = rep(1/6, 6))

## 
##  Chi-squared test for given probabilities
## 
## data:  table(rolls)
## X-squared = 14.2, df = 5, p-value = 0.01439

Estimation of parameters

If the null hypothesis doesn’t completely specify the distribution \(p_0\), but specifes a family of distributions, \(p_0(\theta_1, \theta_2, \ldots, \theta_d)\) where the \(\theta\) are unknown parameters.

You can still use the Chi-square test with some modification

1. Estimate the parameters \(\theta_1, \theta_2, \ldots, \theta_d\)
2. Find \(E_j\) based on estimated parameters and \(p_0\).
3. Compute Pearson’s \(\chi^2\) statistic as usual
4. Compare statistic to a \(\chi^2\) with \(k - d - 1\) degrees of freedom, where \(d\) is the number of parameters that were estimated.

Example: Poisson

I counted the number of passengers in \(n = 40\) vehicles passing through an intersection.

Question: Is the number of passengers per vehicle distributed according to a Poisson distribution?

mean(passengers)

## [1] 1.875

Example: Poisson

p_0 <- dpois(0:6, lambda = mean(passengers))
(E <- p_0 * n)

## [1]  6.1341987 11.5016225 10.7827711  6.7392319
## [5]  3.1590150  1.1846306  0.3701971

# For 7+ category
(E_7 <- (1 - sum(p_0)) * n)

## [1] 0.1283331

Example: Poisson

Test statistic:

\[ \frac{(6 - 6.13)^2}{6.13} + \frac{(11 - 11.5)^2}{11.5} + \ldots + \frac{(0 - 0.13)^2}{0.13} = 2.66 \]

Compare to \(\chi^2_{k - d - 1}\)

1 - pchisq(X, df)

## [1] 0.8504434

Other points

• For binary data, the \(X(p_0)\) statistic is equal to the square of the Z-statistic for testing a hypothesis regarding a binary proportion.

  Therefore, for two sided hypothesis testing \(H_0: p = p_0\) vs \(H_A: p \ne p_0\), the \(\chi^2\) test and the z-test give the exact same result.

• The \(\chi^2\) statistic has an asymptotic \(\chi^2\) distribution.

  Therefore this test is approximate: the test is asymptotically exact.

  The approximation is generally considered appropriate when \(E_j > 5\) for all \(j\).

Next time

Starting two sample inference…