We really do not see uniform residuals very often in nature, but it will depends on the peculiar characteristics of your problem. Does it make sense to consider a uniform distribution? Or, at least, does it make sense to consider residuals only in an interval [a, b]? My first guess is that there is still some bias or trend remaining after trend and harmonics removal. Another idea is that somehow your data was sorted before testing. Hope this helps.

Dear Rogerio, i have estimated parameter b using an estimation s^2 (estimation of residuals variance). I used residuals sets of 25 elements. Of course there were outcomes -about 1(more often) or 2(less often) per set, outcomes did not take part in the chi-square test.

Probably You are right and uniform distribution of residuals is because some undeleted component. I wonder if there is some connection between the results of ch-square test and "variance" of existing trend component?

To check the presence of other components you can use the Durbin-Watson test. However, the presence of unaccounted components likely will shift the mean value from zero than the appearance of the residuals uniform distribution.

Why did you choose the chi-square test of the distribution model? For numerical data Kvamer - von Mises and Kolmogorov - Smirnov tests are more appropriate.

Yes, Petr, it is possible to find such a connection. For instance, supose that your data support linear and quadratic trends but you only remove the linear one. Residuals from this fit would favor an uniform distribution, even if you try the tests suggested by Vladimir.

Dear Vladimir and Rogerio, thank You for the advices! As You correctly supposed, i consider time series of a signal and, after deleting the trend component, i mix the residuals and then consider this set without time information. Now i work with fragments with length of 25 observations, so the errors set is also consist of 25 observations.

I attach an example of such interesting data. Before the considering i deleted linear trend from the data. Then i have found main oscillatory trend, here is it:

31.18*sin(2.23*t+0.6)-24.96*sin(2.69*t+0.2), t=0...24

The error after deleting data from this trend is in the last column. This error have the least variance (of course with some precision because of numerirical method of deterination of it). And the trend expains about 3/4 of data variance (in first column)

So You can see that I can not accept the hypothesis of normal error distribution anyway. Actually i do not think that Kolmogorov-Smirnov test should give some critically another result,. Of course firstly i supposed the like-normal distribution of errors, but this result discouraged me.

Dear Peter.

I checked your data by the Kolmogorov-Smirnov test. Calculated value for the normal model is 0.57, and for the uniform - 0.51, when the critical value for 0.05 significance level is 0.895. Thus, both hypotheses are acceptable.

Dear Vladimir, here i have the sample set of 25 observations, it is actually too small to use Kolmogorov-Smirnov test (N.V. Smirnov told about this in his book "Bolshev Smirnov Tables of Mathematical Statistics", for example, Moscow, 1983).

The chi-square statistics gives values of 0.36 for uniform distribution and 3.47 for normal distribution respectively (i used 4 intervals). The critical value for significant level 0.1 is 2.71, for 0.05 is 3.84. So, i'd have to prefer the hypothesis of uniform distribution for errors.

Dear Petr, I agree that the correct using of the Kolmogorov - Smirnov test requires more data. But the same is true for the chi-square test. In this situation, it is advisable to start with a visual comparison of empirical distribution function with the theoretical. If we build such function for your data and compare it with the functions of the normal and the uniform distribution, we can see that in both cases is approximately equivalent systematic deviations.

This confirms the absence of significant difference between two models. Thus, there are three main alternatives:

1) to accept the hypothesis of normal distribution, since there is no sufficient grounds to reject it;

2) to accept the hypothesis of uniform distribution and find plausible explanation for such character of residues;

3) to accept the hypothesis of a uniform distribution due to the presence of unaccounted component.

In the last case, you have to complicate the model, but you have no enough data for it.