Segment 24. Goodness of Fit

Contents

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Links to the slides: PDF file or PowerPoint file

Problems

To Calculate

1. Let $X$ be an R.V. that is a linear combination (with known, fixed coefficients $\alpha_k$) of twenty $N(0,1)$ deviates. That is, $X = \sum_{k=1}^{20} \alpha_k T_k$ where $T_k \sim N(0,1)$. How can you most simply form a t-value-squared (that is, something distributed as $\text{Chisquare}(1)$ from $X$? For some particular choice of $\alpha_k$'s (random is ok), generate a sample of $x$'s, plot their histogram, and show that it agrees with $\text{Chisquare}(1)$.

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.)

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.