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Maximum Likelihood Estimators

Maximum Likelihood Estimators
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2014-09-29 08:08
FunctionCalculusProbability & StatisticsOptimizationGraphing
This example demonstrates the maximum likelihood estimator and log-likelihood function for various statistical models, covering core concepts like maximum likelihood estimation and probability density functions. It is constructed using GeoGebra commands like Point, Segment, Textfield, Uniform, Ray, and involves probability density functions such as doubleExpPDF, uniformPDF, and normPDF. Suitable for high school students and statistics learners.

GeoGebra Commands & Parameters

  1. Point(xAxis) → x10
  2. Point(xAxis) → x9
  3. Point(xAxis) → x8
  4. Point(xAxis) → x7
  5. Point(xAxis) → x6
  6. Point(xAxis) → x5
  7. Point(xAxis) → x4
  8. Point(xAxis) → x3
  9. Point(xAxis) → x2
  10. Point(xAxis) → x1
  11. Point(xAxis) → thetaPoint
  12. Segment(x1, (x(x1), f(x(x1)))) → linex1
  13. Textfield(thetaDummy) → thetaInput
  14. Segment(x2, (x(x2), f(x(x2)))) → linex2
  15. Segment(x3, (x(x3), f(x(x3)))) → linex3
  16. Segment(x4, (x(x4), f(x(x4)))) → linex4
  17. Segment(x5, (x(x5), f(x(x5)))) → line5
  18. Segment(x6, (x(x6), f(x(x6)))) → line6
  19. Segment(x7, (x(x7), f(x(x7)))) → line7
  20. Segment(x8, (x(x8), f(x(x8)))) → line8
  21. Segment(x9, (x(x9), f(x(x9)))) → line9
  22. Segment(x10, (x(x10), f(x(x10)))) → line10
  23. Segment(thetaPoint, (x(thetaPoint), f(x(thetaPoint)))) → linetheta
  24. Uniform(0, thetaDummy, x) → funif
  25. Ray(thetahatPoint, (x(thetahatPoint), -1)) → thetahatray1
  26. Point({theta, L(theta)}) → Ltheta
  27. Ray(thetaPoint, (x(thetaPoint), -1)) → thetaray1
  28. Point({theta, LogL(theta)}) → LogLtheta
  29. Ray(LogLtheta, (x(thetaPoint), 1)) → thetaraylog
  30. Point({x(thetahatPoint), LogL(x(thetahatPoint))}) → LogLthetahat
  31. Point({x(thetahatPoint), L(x(thetahatPoint))}) → Lthetahat
  32. Ray(Ltheta, (x(thetaPoint), 1)) → thetaray2
  33. Ray(LogLthetahat, (x(thetahatPoint), yscale)) → thetahatraylog
  34. Ray((x(thetahatPoint), L(x(thetahatPoint))), (x(thetahatPoint), yscale)) → thetahatray2

AI Drawing Prompt Reference

Draw a graph of maximum likelihood estimation for statistical models, including 10 points on the x-axis with corresponding lines representing probability density functions for double exponential, uniform, and normal distributions. Include a text input for parameter θ and its estimate θ̂. Show the likelihood and log-likelihood functions for θ̂, including the likelihood line for θ̂, the log-likelihood line for θ̂, the intersection point of the likelihood and log-likelihood functions for θ̂, and the slope lines for the likelihood and log-likelihood functions of θ̂.
The prompt above was compiled from the original drawing commands and is for reference only.