#StackBounty: #distributions #p-value #goodness-of-fit #kolmogorov-smirnov Goodness-of-fit test on arbitrary parametric distributions w…

Bounty: 100

There have been many questions regarding this topic already addressed on CV. However, I was still unsure if this question was addressed directly.

1. Is it possible, for any arbitrary parametric distribution, to properly calculate the p-value for a Kolmogorov-Smirnov test where the parameters of the null distribution are estimated from the data?
2. Or does the choice of parametric distribution determine if this can be achieved?
3. What about the Anderson-Darling, Cramer von-Mises tests?
4. What is the general procedure for estimating the correct p-values?

My general understanding of the procedure would be the following. Assume we have data \$X\$ and a parametric distribution \$F(x;theta)\$. Then I would:

• Estimate parameters \$hattheta_{0}\$ for \$F(x;theta)\$.
• Calculate Kolmogorv-Smirnov, Anderson-Darling, Cramer von-Mises test statistics: KS\$_{0}\$, AD\$_{0}\$ and CVM\$_{0}\$.
• For \$i=1,2,ldots,n\$
1. Simulate data \$y\$ from \$F(;hattheta_{0})\$
2. Estimate \$hattheta_{i}\$ for \$F(y;theta_{i})\$
3. Calculate KS\$_{i}\$, AD\$_{i}\$ and CVM\$_{i}\$ statistics for \$F(y;hattheta_{i})\$
• Calculate \$p\$-values as the proportion of these statistics that are more extreme than KS\$_{0}\$, AD\$_{0}\$ and CVM\$_{0}\$, respectively.

Is this correct?

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