GoF ICF Laplace & Cauchy

We now extend the goodness-of-fit test based on the characteristic function to Laplace and Cauchy distributions.

Laplace Distribution

We now show how to test whether the data set X = {x1, …, xn} follows a Laplace distribution with pdf

Laplace distribution pdf

Now we use the following estimates (these are the MLE estimates when n is odd)

Laplace parameters

The test statistic and critical values for this test are

GoF test statistic Laplace

Critical values Laplace distribution

Example

Example 1: Decide whether the data in B3:B12 of Figure 1 fits a Laplace distribution.

This example comes from the referenced textbook. Figure 1 describes the analysis. Note that cell C3 contains the formula =(B3-C15)/C16 and cell D3 contains the formula =IF(D$1<$A3,(ABS($C3-D$2)+1)*EXP(-ABS($C3-D$2)),””). The other formulas in C3:M12 are filled in as explained for Example 1 of Goodness-of-Fit Test based on the Characteristic Function.

Laplace distribution fitting example

Figure 1 – GoF for Laplace distribution

We see from Figure 1 that I = .263573 > .256842 = I.10, and so we have a significant result at the .10 level. But also I = .263573 < .308083 = I.05, which is not significant at the .05 level. In fact, the data comes from a Cauchy distribution.

Note that we obtain the same result using the formula =ICF_GOF(B3:B12,”laplace”,TRUE). See Goodness-of-Fit Test based on the Characteristic Function for a description of the ICF_GOF function and its arguments.

Cauchy Distribution

We now show how to test whether the data set X = {x1, …, xn} follows a Laplace distribution with pdf

Cauchy distribution pdf

We can estimate the parameters by

Cauchy distribution parameters

This is the approach used by CAUCHY_FITX (see Fitting a Cauchy Distribution) to estimate the parameters, except that there the first n+1 in the formula used to estimate μ is replaced by n.

Here it is important that values of X must be used in (ascending or descending) order.

The test statistic is given by

Test statistic Cauchy distribution

For even values of n starting with n = 16 for alpha .05, and n = 14 for alpha = .10, the critical values are:

Cauchy distribution critical values

Example

Example 2: Decide whether the data in B3:B18 of Figure 8 fits a Cauchy distribution.

Figure 2 describes the analysis. Note that cell C3 contains the formula =(B3-C20)/C21 and cell D3 contains the formula =IF(D$1<$A3,8/(($C3-D$2)^2+4),””). The other formulas in C3:S12 are filled in as explained for Example 1 of Goodness-of-Fit Test based on the Characteristic Function.

Cauchy distribution GoF example

Figure 2 – GoF for a Cauchy distribution

We see from Figure 2 that I = .339629 < .770906 = I.10, and so we can’t reject the null hypothesis that the data comes from a Cauchy distribution.

This is not surprising since the data was constructed by inserting the formula =T3_INV(RAND(),1,2,1) in cell B3, highlighting B3:B18, and pressing Ctrl-D.

Note that we obtain the same result using the formula =ICF_GOF(B3:B18,”cauchy”,TRUE,-3). See Goodness-of-Fit Test based on the Characteristic Function for a description of the ICF_GOF function and its arguments.

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

Reference

Epps, T. W. (2014) Probability and statistical theory for applied researchers
https://books.google.co.uk/books?id=NCs8DQAAQBAJ&pg=PR4&lpg=PR4&dq=Epps,+T.+W.+Probability+and+statistical+theory+for+applied+researchers&source=bl&ots=GxU40vCNHu&sig=ACfU3U0vgZZndBfjMMmqYPQuCXAJf2jrow&hl=en&sa=X&ved=2ahUKEwiw5_3R-4KCAxVCgFwKHa2fA384FBDoAXoECAQQAw#v=onepage&q=Epps%2C%20T.%20W.%20Probability%20and%20statistical%20theory%20for%20applied%20researchers&f=false

 

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