Testing if Data is close to the Exponential Distribution

I want to see if the data that I have has an exponential distribution. I would like an empirical way to see how far from the exponential distribution the data's true distribution is.

Here is my data

X = [1.5236, 1.0138, 1.4330, 0.1246, 0.1251, 1.4544, .3800 ,0.9418,1.3444,1.5806,0.1252, 0.1262,1.4613,0.9354,0.1244,1.2701,0.1251,0.9222,1.4693,1.5419,1.3928,0.1259,0.9297,0.1248,1.4239,0.1248,0.9662,0.9982,0.1245,0.1230]

I tried using the allfitdist() function to test what it would look like compared to all distributions but it did not work well

[D, PD] = allfitdist(X, 'PDF')

I also tried Lilliefors test

[h,p] = lillietest(X,'Distribution','exponential','MCTol',1e-4)

But I know this can only tell me if it has reason to say it is not exponentially distributed. It cannot confirm the data has exponential distribution.

How can I take this data and see if it is exponentially distributed? Also, how can I find how far the true distribution is from the exponential distribution?

Answer 1

[Torsten]

How can I take this data and see if it is exponentially distributed?

There is no way to do so.

Also, how can I find how far the true distribution is from the exponential distribution?

With "true distribution" you mean the empirical distribution ?

The result below doesn't look very convincing, does it ?

X = [1.5236, 1.0138, 1.4330, 0.1246, 0.1251, 1.4544, .3800 ,0.9418,1.3444,1.5806,0.1252, 0.1262,1.4613,0.9354,0.1244,1.2701,0.1251,0.9222,1.4693,1.5419,1.3928,0.1259,0.9297,0.1248,1.4239,0.1248,0.9662,0.9982,0.1245,0.1230];
pd = fitdist(X.','Exponential')

pd = 
  ExponentialDistribution

  Exponential distribution
    mu = 0.811873   [0.584799, 1.20332]

plot(pd)

[Marissa Whitby]

Thank you! I agree. I did mean the empiracle distribution. Is there a way to quantify how bad of a match the exponential distritbution would be here?

[Torsten]

Usually the span of the 95% interval for the estimation of "mu" is taken as a measure on how good the exponential distribution approximates your data - and this span is quite large in your case.

[Marissa Whitby]

Thank you!