22.7: End-of-Appendix Materials
- Page ID
- 57828
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)R Functions
In this appendix, we were introduced to several R functions that will be useful. These are listed here.
Packages
KnoxStats
This is a "book package" with several supporting functions to make learning statistics a bit easier. See here for how to install it.
Statistics
runs.test(r, order=x)
The runs test tests if the residualsr, as ordered byx, are sufficiently distributed around the zero line to suggest that they are independent. This is the version in theKnoxStatspackage.
Exercises
- Prove the Mean Deviance Lemma.
- Prove Lemma 22.2.5.
- Prove Lemma 22.2.6.
- Prove Lemma 22.2.7.
- Prove Lemma 22.2.8.
- Prove Lemma 22.2.9.
- Prove Lemma 22.2.10.
- Prove Lemma 22.2.11. Note that this proof usually requires the Cauchy-Schwarz Inequality.
- Prove that the Cauchy distribution is equivalent to the Student's \(t\) distribution with 1 degree of freedom.
- Prove that the Normal distribution is equivalent to the Student's \(t\) distribution as \(\nu \to \infty\).
- Prove that the interquartile range of the standard Cauchy is 2.
- Prove \(E[S^2] = \sigma^2\) when \(Y \sim N(\mu, \sigma^2)\).
- Prove that the first central moment is always \(0\), as long as \(E[X]\) exists.
- Prove that the third central moment (skew) for the Normal distribution is zero.
- Use moment generating functions to determine when the Binomial distribution has zero skew.
- Calculate the moment generating function for the Poisson distribution. Check that it generates the first two moments. Use it to determine the variance of a Poisson random variable.
- Prove Theorem: MGF of a Normal.
- Use the moment generating function of the Bernoulli distribution to determine the moment generating function of the degenerate distribution.
- Use moment generating functions to calculate the variance of a degenerate distribution.
Theory Readings
- Carl E. Bonferroni (1936). "Teoria statistica delle classi e calcolo delle probabilità." Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze (in Italian). 8: 3–62.
- James V. Bradley (1968). Distribution-Free Statistical Tests, Chapter 12. Prentice-Hall.
- Harald Cramér (1936). "Über eine Eigenschaft der Normalen Verteilungsfunktion." Mathematische Zeitschrift (in German). 41(1): 405–414.
doi:10.1007/BF01180430. - Olive Jean Dunn (1958). "Estimation of the Means for Dependent Variables." Annals of Mathematical Statistics. 29(4): 1095–1111.
doi:10.1214/aoms/1177706374. - Olive Jean Dunn (1961). "Multiple Comparisons Among Means." Journal of the American Statistical Association. 56(293): 52–64.
doi:10.1080/01621459.1961.10482090. - David K. Hildebrand (1986). Statistical Thinking for Behavioral Scientists. Duxbury Press.
- Alexander M. Mood (1940). "The Distribution Theory of Runs." Annals of Mathematical Statistics. 11(4): 367–392.
https://www.jstor.org/stable/pdf/2235718.pdf. - George W. Snedecor. (1934). "Calculation and Interpretation of Analysis of Variance and Covariance." Agronomy Journal 26(3): 255–56.
https://doi.org/10.2134/agronj1934.0...2002600030019x. - Abraham Wald and Jacob Wolfowitz (1940). "On a test whether two samples are from the same population." Annals of Mathematical Statistics. 11(2): 147–162.
doi: 10.1214/aoms/1177731909.