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I have 116 Bessel-corrected sample variances (average of squared distances from sample mean), each from a sample of three measurements. All measurements were done using the same method.

I had predicted that the empirical cumulative distribution would match that of a scaled chi-squared curve with 2 degrees of freedom, so I tried plotting them on the same graph.

For the chi-squared distribution's scaling, I scaled according to the degree of freedom and my prediction for the population variance (0.01192), in the form (for y-values) =CHISQDIST(2*E2/0.01192,2,1) , where E2 corresponding to an x-value point. For the PDF later, the form was =CHISQDIST(2*F2/0.01192,2,0) * (2/0.01192) .

For the sample variances curve, I sorted the sample variances by size and for the y-values used 1/116, 2/116, ..., 116/116.

My intention was that both curves would represent distributions with the mean at the population variance (of the original, normal distribution).

In Image 1 below, the yellow line represents my uncorrected sample variances CDF, the blue line represents my 2-degrees-of-freedom scaled chi-squared distribution CDF, and the red line represents a 1-degree-of-freedom scaled chi-squared distribution CDF after I began trying to rule out possibilities.

To my eyes, the yellow line seems to have a mostly-consistent shape of its own, distinct from both the blue and red lines, but I have no idea how to identify what CDF that is. My knowledge of CDFs is shallow and so I could also be imagining something to hold import that holds none.

In Image 2 below, remembering the PDF to be the slope of the CDF, I have approximated the PDF of the samples' distribution by dy/dx using adjacent points (so for point 2, using $\frac{y_3 - y_1}{x_3 - x_1}). The y-axis is logarithmic to better include all points shown. I again am concerned by that the shape does not seem to match the 2-degrees-of-freedom curve and not knowing why.

[I have somewhat rewritten/redone the above question before reading welcome offered information in more detail!]

Image 1, CDFs Image 2, PDFs

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  • $\begingroup$ One method of graphical comparison is illustrated in my answer at stats.stackexchange.com/a/17148/919. It's difficult to learn much from the empirical cumulative distributions you use: consider probability plots and density plots (such as KDEs or histograms). $\endgroup$
    – whuber ♦
    Nov 30, 2019 at 16:13
  • $\begingroup$ one remark for synthetic data (blue and green curves) plot only line and don't plot points markers so you will have cleaner plots (usually setting point shape as null or single quotes '') $\endgroup$
    – quester
    Nov 30, 2019 at 17:19
  • $\begingroup$ Thank you both for your comments! @whuber , may I ask for your input on why the MLE and MMSE sample variance corrections are different? ( stats.stackexchange.com/questions/436370/… ) $\endgroup$
    – MCC
    Dec 4, 2019 at 16:55
  • $\begingroup$ The comment by BruceET to your question is the most revealing: the MMSE of the variance depends only on the fourth moment of the underlying distribution while the MLE depends (strongly) on the underlying distribution. $\endgroup$
    – whuber ♦
    Dec 4, 2019 at 17:11
  • $\begingroup$ Thank you for your continued replies. I think I have passed the point at which (due to my foundation being too shallow) I should declare victory and retreat; nevertheless, my questions spring forth endlessly. @whuber would you then say that the MMSE estimate has the lowest estimated variance relative to the true parameter (variance + bias^2), but that the MLE estimate is more reliable/accurate from that it takes into account more information about the distribution? If I were to point to a distribution of possible MMSE results and a distribution of possible MLE results, is there [continued] $\endgroup$
    – MCC
    Dec 5, 2019 at 16:03

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