By David Ruppert (auth.)
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1 Normal probability plots Many statistical models assume that a random sample comes from a normal distribution. Normal probability plots are used to check this assumption. If the assumption is true, then the qth sample quantile will be approximately equal to μ + σ Φ−1 (q), which is the population quantile. Therefore, except for sampling variation, a plot of the sample quantiles versus Φ−1 will be linear. The normal probability plot is a plot of X(i) versus Φ−1 {i/(n + 1)} (these are the i/(n + 1) sample and population quantiles, respectively).
Because there are only two components, the conditional variance is discrete, in fact, with only two possible values, and the example was easy to analyze. The marginal distributions of the GARCH processes studied in Chapter 12 are also normal mixtures, but with infinitely many components and a continuous distribution of the conditional variance. Although GARCH processes are more complex than the simple mixture model in this section, the same theme applies — a nonconstant conditional variance of a mixture distribution induces heavy-tailed marginal distributions even though the conditional distributions are normal distributions, which have relatively light tails.
5 − x)quantile is below the median. 11. 4. The parameter μ is a scale parameter and σ is a shape parameter. The lognormal distribution does not have a location parameter since its location is fixed to start at 0. MATLAB functions The MATLAB functions logncdf, lognpdf, and logninv in the Statistics Toolbox compute the lognormal PDF, CDF, and inverse CDF, respectively. The arguments are the argument of the PDF, CDF, and inverse CDF, μ, and σ. 05 μ=1, σ=1 0 0 2 4 6 8 10 12 14 16 18 x Fig. 4. Examples of lognormal probability densities.