It is very common to see articles and theses where statistical tests were used, although the sample is not probabilistic. Is there any theoretical reference to support this practice?
I think no, the theory provides a possibility to inference but when hypothesis are not complimented the inference only could be an approximation to a perfect inference.
If the data is supposed to be drawn from a specific distribution with known parameters, then it can be tested whether or not it is a random sample for the specific distribution by using chi-square tests.
I think there is no way to check whether or not the data is real.
Statistical tests are different from probability. Probability is the study of chance, while statistics is concerned with how we handle data using different analysis techniques and collection methods. I think you are asking about the distribution of the data, whether it is normally distributed or not.You can use non parametric statistical analysis if the data is non-normal in its distribution
You may be able to model using Ordinary Least Square and not Maximum Likelihood Estimates and the must the relationship that exist amount the error terms (coefficient of determination) to evaluate the model - then by jack-knifing the data you can find the probability distribution of the parameter estimates which in turn would allow a test of significance for the parameter estimates.
Thank you all for your answers.
Andrew, what I want to know is why so many people use inferencial statistic (like tests) when they have a non-probabilistic sample, drawn from a population whith unknown parameters? What is the purpose?
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