Confidence that data belongs to distribution

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SUMMARY

This discussion focuses on determining the probability that a set of non-uniformly spaced data points originates from a specific piecewise probability density function (pdf) f(x;γ), where γ is known. The user seeks to employ statistical methods, specifically maximum likelihood estimation (MLE), to estimate the optimal shift value (x_shift) that maximizes the likelihood of the data fitting the distribution. The referenced pdf provides a foundational basis for the analysis, emphasizing the need for precise statistical techniques to validate the data's alignment with the distribution.

PREREQUISITES
  • Understanding of piecewise probability density functions (pdf)
  • Knowledge of maximum likelihood estimation (MLE)
  • Familiarity with statistical hypothesis testing
  • Basic proficiency in statistical software or programming languages (e.g., R, Python)
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  • Research "maximum likelihood estimation for piecewise distributions"
  • Explore "statistical hypothesis testing methods for distribution fitting"
  • Learn "how to implement MLE in Python using SciPy"
  • Investigate "numerical optimization techniques for parameter estimation"
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Statisticians, data analysts, and researchers working with non-standard distributions who need to validate data against theoretical models and optimize parameter estimates for better data fitting.

coolnessitself
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Hi all,
This seems like a simple question, but I'm just not too knowledgeable about statistical methods.

I have a piecewise pdf [tex]f(x;\gamma)[/tex] (not any regular distribution) where [tex]\gamma[/tex] is known, and a set of non-uniformly spaced data points I obtained that somewhat resemble [tex]f(x;\gamma)[/tex]. How do I go about finding some numeric value that shows the probability that my data comes from this distribution?
Also, once I'm able to do this, is there a methodology that allows me to find the best way the data fits, i.e. find [tex]x_\mathrm{shift}[/tex] such that if all data points are shifted right by [tex]x\rightarrow x+x_\mathrm{shift}[/tex], the data has the highest probability of coming from [tex]f(x;\gamma)[/tex]?

(If it makes a difference, the pdf is http://nvl.nist.gov/pub/nistpubs/jres/106/2/j62mil.pdf" )
 
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Would that be "maximum likelihood estimation", where the parameter to be estimated is xshift?
 

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