Quantile-Quantile Plots

• If X is a random variable with cdf F, then the q-quantile of X is the value $\gamma$ such that $F(\gamma) = P(X \le \gamma) = q$ for $0 < q < 1$.
• When $F$ has an inverse, we write $\gamma = F^{-1} (q)$

• Now we discuss how to use Q-Q plots to draw distributions.
  • Let $\{x_i, i = 1,2,...n \}$ be a sample data from X.
  • Re-order the $x_i$s as $y_i$s such that $y_1 \le y_2 \le ... \le y_n$.

  • The q-q plot is based on the fact that $y_i$ is an estimate of the (j-1/2)/n quantile of X, that is,
    
    $$y_i \approx F^{-1} ( \frac{j - \frac{1}{2}} {n} )$$
    
    
    where F is the distribution function under the consideration that fits the samples.

  • Plot $y_i$ against $F^{-1}(\frac{j - \frac{1}{2}} {n} )$. If the distribution fits, the line would form a straight line. Use the measurement of deviation to decide whether or not the set of samples fit the distribution function.

• Example 10.4 on page 365.