Fitting a theoretical distribution

Fitting a Theoretical Distribution Essentials

• Fitting a theoretical distribution is a procedure used to describe a data set by finding the probability distribution that best fits the observed data.
• The goodness of fit can be evaluated using a Chi-Squared Test.
• This method can be used to validate statistical models, by comparing observed values with the values expected under the theoretical distribution.

Steps in Fitting a Theoretical Distribution

• Collect data: The first step is to collect a good range of data points. The more the data, the greater the precision of your distribution fitting.
• Select a Distribution: Choose the theoretical distribution that has the best theoretical justification for fitting your data.
• Estimate Parameters: Calculate or estimate the parameters of your chosen distribution. These could be the mean, standard deviation, etc., depending on the distribution.
• Calculate Expected Frequencies: For classes of outcomes, estimate the expected frequencies using the theoretical probability distribution and the parameters you calculated.
• Compare Observed to Expected Frequencies: Apply the Chi-Squared Test to compare observed frequencies to the expected frequencies under the theoretical distribution.

The Chi-Squared Test

• The Chi-Squared test statistic involves observed and expected frequencies. It is calculated as the sum of the squares of the difference between observed and expected frequencies, divided by the expected frequency, for each outcome category.
• The calculated Chi-square statistic is then compared to the Chi-square distribution with (k-p-1) degrees of freedom, where k represents the number of outcome categories, and p indicates the number of parameters estimated from the data.
• If the calculated test statistic is greater than the critical value, then the null hypothesis that the data follows the theoretical distribution is rejected.

Interpreting Results

• Always remember to interpret your results in the context of the problem at hand.
• A low Chi-squared value implies that the observed data fits the expected distribution well, while a high value indicates a poor fit.
• A good model fit alone does not confirm the correctness of the underlying theoretical distribution - it only validates that the model is a plausible explanation for the observed data. This is an example of the broader principle that correlation does not imply causation.
• When rejected, the test suggests that changes could be made to the model or that other models might better explain the data. Always consider the assumptions behind the chosen theoretical model, as these can have a significant impact on its appropriateness to describe the data.
• Consider potential sources of biases and errors in your data collection and analysis process. For instance, sampling error or data entry error may give you a poor fit, even though the theoretical distribution is correct.