This paper talks about a statistic that is asympototically chi-square but is robust to non-constant variance as well. Sometimes, this isn't the case, but there are robust alternatives to the standard chi-square. Usually in regression, we assume constant variance. Chi-Square Test with Heterogeneous variance Journal of the American Statistical Association, 64(325), 387-389. On the Kolmogorov-Smirnov test for the exponential distribution with mean unknown. The paper below gives a quick review on this. On the Kolmogorov Smirnoff test, for example, in the empirical-to-theoretical distribution comparison, if you have to estimate parameters to conduct the test, then the p-values are technically not valid. Estimating Parameters in Kolmogorov Smirnov Test Some of the papers below talk about validity and goodness of fit under different assumption violations.
When it comes to goodness of fit tests, there a few different criteria to keep in mind, and assumptions on the estimated distributions tend to matter - the degree to which they matter (i.e., robustness to assumption violations varies). I'm doing this since, if I had a choice between multiple goodness of fit tests, I would do a process of elimination by saying which tests are less "valid" given my data and potential sources of noise. I'll try to complement that by providing a couple papers that talk specifically about properties of tests when assumptions are violated. Whuber gave a comment on what the typical strategy that someone might follow.
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Great question! (Almost a professional statistician here!) Let's see if I can try to answer your question. Is there a go-to R function/package that conveniently performs a range of goodness-of-fit tests all at once? How do we choose which goodness-of-fit to use to compare the fit of different distributions? In straightforward cases, one distribution may have the best fit according to every test across the board, but in more marginal cases, what if one distribution is best according to one test, and another is best according to another test? (Is there a reference, like a book chapter, that explores this question?) Information Criteria: AIC/AIC $_\text$/CAIC/BIC/HQIC.Chi-squared / G-test (designed for discrete data, but may be applied to continuous data if it is divided into arbitrary bins).I have looked around and found numerous goodness-of-fit tests, including However, imagine you are a professional statistician (which I am not) who wants to assess the fit rigorously. Stat_function(fun=dgamma,args=fitdistr(DT4,"gamma")$estimate, color=pal, size=.5) +Īnnotate("text", label="Log-normal", x=250, y=0.025, color=pal) +Īnnotate("text", label="Exponential", x=250, y=0.023, color=pal) +Īnnotate("text", label="Weibull", x=250, y=0.021, color=pal) +Īnnotate("text", label="Gamma", x=250, y=0.019, color=pal)Īll four distributions have a faily good fit, with log-normal perhaps having the best fit, just from a visual assessment.
Stat_function(fun=dweibull,args=fitdistr(DT4,"weibull") $estimate, color=pal, size=.5) + Stat_function(fun=dexp,args=fitdistr(DT4,"exponential")$estimate, color=pal, size=.5) + Stat_function(fun=dlnorm,args=fitdistr(DT4,"log-normal") $estimate, color=pal, size=.5) + ggplot(DT4, aes(TTE)) + geom_histogram(aes(y=.density.), bins=90) +
I make a guess that the data may be fitted to one of the four distributions:Īt first, I assess the distribution visually using geom_histotgram (from package ggplot2) and plot the estimated distributions using stat_function, where the distribution parameters are estimated using fitdistr (from package MASS). I have a one-sample dataset, where the data are strictly positive, and have a right-skewed distribution.
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I notice there are a lot of questions on the topic (most are along the lines of: how to do a Chi-squared goodness-of-fit test), but mine is perhaps more general.
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