Methods for Individual Distribution Identification

Maximum likelihood estimates of the parameters in the distribution are calculated by maximizing the likelihood function with respect to the parameters. For a given data set, the likelihood function of a distribution estimates the probability of generating the data under that distribution.

The Newton-Raphson algorithm is used to calculate maximum likelihood estimates of the parameters which define the distribution. The Newton-Raphson algorithm is a recursive method for calculating the maximum of a function. ¹ The percentiles are then calculated from the distribution.

Minitab uses Anderson-Darling statistics to perform the goodness-of-fit test.

Let Z = F(X), where F(X) is the cumulative distribution function. Suppose that a sample X₁, .., X_n gives values Z_(i) = F(X_i), i=1,.., n. Rearrange Z_(i) in ascending order, Z₍₁₎ < Z₍₂₎ <...<Z_(n). Then the Anderson-Darling statistic (A²) is calculated as follows:

• A² = –n - (1/n) Σ_i[(2i – 1) log Z_(i) + (2n + 1 – 2i) log (1 – Z_(i))]

The modified Anderson-Darling goodness-of-fit test statistic is calculated for each distribution. The p-values are based on tables 4.8−4.22 in D'Agostino and Stephens² If no exact p-value is found in the table, Minitab calculates the p-value based on interpolation using the range of the p-value.

The likelihood-ratio test compares the fit of a larger distribution family with a subset of the same family and determines whether there is a significant improvement in fit with the larger distribution. For instance, for a 2-parameter exponential distribution, the likelihood-ratio test compares the fit of 2-parameter exponential distribution family with the fit of 1-parameter exponential distribution family (a subset with the second parameter being 0). If a 2-parameter exponential distribution significantly improves the fit, then the p-value for likelihood-ratio test statistic is very small.

The likelihood-ratio test statistic is calculated as follows.

Let A be the maximum likelihood estimate (MLE) of the parameter vector for the larger distribution family (for example, the 3-parameter distribution family), and L(A) be the log likelihood. Let B be the MLE of the parameter vector for the corresponding smaller distribution family (for example, the corresponding 2-parameter distribution family), and L(B) be the log likelihood.

Likelihood-ratio test statistic = 2 * L(A) 2 * L(B).

Under the null hypothesis, the smaller distribution family fits the data well. The likelihood-ratio test statistic is chi-square distributed with df = dimension of vector (A) – dimension of vector (B).