Eleisha's Segment 24: Goodness of Fit

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To Calculate

1. Let X be an R.V. that is a linear combination (with known, fixed coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_k } ) of twenty Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(0,1) } deviates. That is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = \sum_{k=1}^{20} \alpha_k T_k where T_k \sim N(0,1) } . How can you most simply form a t-value-squared (that is, something distributed as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Chisquare}(1) } from X? For some particular choice of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_k's } (random is ok), generate a sample of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} 's, plot their histogram, and show that it agrees with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Chisquare}(1)} .

2. From some matrix of known coefficients \alpha_{ik} with k=1,\ldots,20 and i = 1,\ldots,100, generate 100 R.V.s X_i = \sum_{k=1}^{20} \alpha_{ik} T_k where T_k \sim N(0,1). In other words, you are expanding 20 i.i.d. T_k's into 100 R.V.'s. Form a sum of 100 t-values-squareds obtained from these variables and demonstrate numerically by repeated sampling that it is distributed as \text{Chisquare}(\nu)? What is the value of \nu? Use enough samples so that you could distinguish between \nu and \nu-1.

3. Reproduce the table of critical \Delta\chi^2 values shown in slide 7. Hint: Go back to segment 21 and listen to the exposition of slide 7. (My solution is 3 lines in Mathematica.)


To Think About 1. Design a numerical experiment to exemplify the assertions on slide 8, namely that \chi^2_{min} varies by \pm\sqrt{2\nu} from data set to data set, but varies only by \pm O(1) as the fitted parameters \mathbf b vary within their statistical uncertainty?

2. Suppose you want to estimate the central value \mu of a sample of N values drawn from \text{Cauchy}(\mu,\sigma). If your estimate is the mean of your sample, does the "universal rule of thumb" (slide 2) hold? That is, does the accuracy get better as N^{-1/2}? Why or why not? What if you use the median of your sample as the estimate? Verify your answers by numerical experiments.