I know about PIT, but this works only when you know the distribution, or at least have a strong hint. What I am trying to achieve is to transform a given sample into an equivalent sample with continuous standard uniform distribution.

My approach is described below:

Denote with $S_n = \{ x_1, x_2, .., x_n\}$ my initial sample, with size $n$. Now, I choose arbitrary a value $m$, and estimate $m+1$ quantiles from the sample (for example, if $m=4$, I compute quantiles for $\{0.0, 0.25, 0.5, 0.75, 1.0\}$). I use basically the same procedure described in Wikipedia.

Using quantiles I transform each $x_i$. If $x_i$ happens to be exactly a computed quantile, than I know precisely its equivalent value. If $x_i$ is between two estimated quantiles, I interpolate the equivalent value linearly.

The question is: can I use Kolmogorov-Smirnov one sample goodness-of-fit to find a proper value for $m$ (the number computed quantiles)?

I know that I do a lot of estimations, one over the top of other, so I was wondering if will work something like this. The reason why I would want to save some computation time for a give transformation.

I have done a small simulation. I build a random sample from standard normal with $10^6$ values. I applied the described transformation and make KS test for some values of $m$. The results looks like:

  m         D   p-value
100  0.006090   0.000000000000 ***
200  0.003151   0.000000004733 ***
300  0.001991   0.000720875707 ***
400  0.001484   0.024403417075 *
500  0.001057   0.213437843144

I know about PIT, but this works only when you know the distribution, or at least have a strong hint. What I am trying to achieve is to transform a given sample into an equivalent sample with continuous standard uniform distribution.

I have a sample of size $n$. I choose arbitrary a value $m$, and estimate $m+1$ quantiles (for example, if $m=4$, I compute quantiles for $\{0, .25, .5, .75, 1\}$). The procedure is described in Wikipedia.

Using quantiles I transform each $x_i$. If $x_i$ happens to be exactly a computed quantile, than I know precisely its equivalent value, otherwise I interpolate the equivalent value linearly.

I've done a small simulation. I build a random sample from standard normal with $10^6$ values. I applied the described transformation and make KS test for some values of $m$. The results looks like:

  m         D   p-value
100  0.006090   0.000000000000 ***
200  0.003151   0.000000004733 ***
300  0.001991   0.000720875707 ***
400  0.001484   0.024403417075 *
500  0.001057   0.213437843144

It looks like I can do a PIT on sample with only 500 interpolation points.

The question is: can I use Kolmogorov-Smirnov one sample goodness-of-fit to find a proper value for $m$ (the number computed quantiles)?