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Economic Benefits
www.statslab.cam.ac.uk/~frank/PAPERS/deloitte.html15 May 2023: Measuring the Economic Benefits of Mathematical Sciences Research in the UK. ... Measuring the Economic Benefits of Mathematical Sciences Research in the UK,. -
23 1/I/10H Statistics Use the generalized likelihood-ratio test to ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2004.pdf28 Apr 2023: You should explain carefully the assumptions underlyingthe test. 1/II/21H Statistics. State and prove the Rao–Blackwell Theorem. ... Find the best test of size α, 0 < α < 1, of the hypothesisH0 : θ = θ0 against H1 : θ = θ1, where θ1 > θ0. -
Home Articles Slides Images CV Jason P. Miller University ...
www.statslab.cam.ac.uk/~jpm205/cv.html24 Feb 2023: Stanford University (September 2006 - June 2011) PhD in Mathematics. University of Michigan, Ann Arbor (2002-2006) Bachelors in Mathematics, Computer Science, and Economics (highest distinction, summa cum laude, GPA -
Talks
www.statslab.cam.ac.uk/~frank/TALKS/18 Sep 2023: MITACS meeting on Network and Internet Economics, Vancouver, May 2011. EURANDOM, Netherlands, April 2011. ... The Human Dimension of Risk, Cambridge, December 2010. Economic perspectives on congestion (abstracts). -
47 A1/12 B1/15 Principles of Statistics (i) Explain in ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2002.pdf28 Apr 2023: Show that we may construct a different, randomised, size α test with the samepower function for θ > θ0. ... f(x; θ) = h(x)g(θ) exp{θt(x)} , x R. Explain in detail how you would test. -
PRINCIPLES OF STATISTICS – EXAMPLES 2/4 Part II, Michaelmas ...
www.statslab.cam.ac.uk/~lab85/resources/PoSM23b.pdf31 Oct 2023: From this limiting result: (i) derive a test for the hypothesis H0 : θ = θ0 vs. ... For all these models, derive explicit expressions for the likelihood ratio test statistic ofa simple hypothesis test of H0 : θ = θ0, θ0 Θ, vs. -
26 1/I/3H Statistics State the factorization criterion for sufficient …
www.statslab.cam.ac.uk/~lab85/resources/Stats2002.pdf28 Apr 2023: What is the result of applying this test at the 0.01 level? [ ... tostrength tests, the number out of the six blocks failing the test being recorded in thefollowing table:. -
STATISTICS Part IBExample Sheet 2 (of 3) Lent 2022 ...
www.statslab.cam.ac.uk/~lab85/resources/StatsL23b.pdf27 Apr 2023: By first deriving the MLEs under each hypothesis, show that the likelihood ratio andPearson’s chi-squared tests are identical to those for the independence test. ... Is the test uniformly most powerfulfor testing H0 : θ = θ0 against H1 : θ > θ0? -
59 1/II/27I Principles of Statistics An angler starts fishing ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2008.pdf28 Apr 2023: What arethe size and power of the test in terms of α and β? ... Is φkadmissible as a test of H0 against Hθ for every θ (0, 1)? -
44 Paper 1, Section I 7H StatisticsSuppose that X1, ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2019.pdf27 Apr 2023: test to test the hypothesis H1 : σ2 > σ̃2 against the hypothesis H0 : σ. ... 2 = σ̃2. Does the F-test depend on whether µ, µ̃ are known? -
53 1/II/27I Principles of Statistics State Wilks’ Theorem on ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2005.pdf28 Apr 2023: Explain why for large n thelikelihood-ratio test and the t-test are nearly the same. ... Howcould you use this to make a test of H0 against H1 : µ Θ1? -
42 Paper 1, Section I 7H StatisticsConsider an estimator ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2014.pdf28 Apr 2023: You do not need to derive the test.] What conclusion shouldbe reported to the doctor? ... What is the size of this test? Write down an expression forits power. -
PRINCIPLES OF STATISTICS – EXAMPLES 2/4 Part II, Michaelmas ...
www.statslab.cam.ac.uk/~lab85/resources/PoSM22b.pdf27 Apr 2023: From this limiting result: (i) derive a test for the hypothesis H0 : θ = θ0 vs. ... For all these models, derive explicit expressions for the likelihood ratio test statistic ofa simple hypothesis test of H0 : θ = θ0, θ0 Θ, vs. -
42 Paper 1, Section I 7H StatisticsX1, X2, . ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2018.pdf27 Apr 2023: 8H StatisticsDefine a simple hypothesis. Define the terms size and power for a test of one simple. ... Construct a best test of size 0.05 for the null hypothesis. -
STATISTICS Part IBExample Sheet 3 (of 3) Lent 2022 ...
www.statslab.cam.ac.uk/~lab85/resources/StatsL23c.pdf27 Apr 2023: Find thelikelihood ratio test of size α. (b) The scientist repeats the test in part (a) for each characteristic j = 1,. ... ratio test of equality of means rejects H0 if. 1I1. Ii=1 ni(Y i Y )2. -
80 Paper 3, Section II27I Principles of Statistics What ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2009.pdf28 Apr 2023: For a problem with random variable X and real parameter θ, define monotonelikelihood ratio (MLR) and monotone test. ... Let φ be a monotonetest, with power function γ() , and let φ′ be any other test, with power function γ′(). -
53 1/II/27J Principles of Statistics (a) What is a ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2006.pdf28 Apr 2023: Itis required to test H0 : θ = 12 against the alternative H1 : θ (0, 1). ... Show that theconstruction of a likelihood-ratio test leads us to the statistic. -
40 Paper 1, Section I 7H Statistics(a) State and ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2017.pdf28 Apr 2023: Find a most powerful test of size α of H0 : θ = 0 versus H1 : θ = 1. ... Find a uniformly most powerful test of size α of H0 : θ = 0 versus H1 : θ > 0. -
41 A1/12 B1/15 Principles of Statistics (i) What are ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2001.pdf28 Apr 2023: Part II. 43. A3/12 B3/15 Principles of Statistics. (i) Explain what is meant by a uniformly most powerful unbiased test of a nullhypothesis against an alternative. ... Let the distribution of Y1,. ,Yn be as in (i) above, and suppose we wish to test,as in -
76 Paper 1, Section II 28J Principles of StatisticsThe ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2010.pdf28 Apr 2023: How might you apply theKolmogorov–Smirnov test to test the hypothesis H0? -
77 Paper 1, Section II29J Principles of Statistics State ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2020.pdf28 Apr 2023: Θ = R (0,). Derive the likelihood ratio test statistic Λn(Θ, Θ0) for the compositehypothesis. -
84 Paper 2, Section II 26K Principles of StatisticsWe ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2017.pdf28 Apr 2023: d) For Σ = Id, we want to test the null hypothesis Θ0 = {0} (i.e. -
78 Paper 4, Section II 28J Principles of StatisticsWe ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2019.pdf28 Apr 2023: Suppose we wanted to test the nullhypothesis that our data arises from the model in part (a). -
STATISTICS Part IBExample Sheet 1 (of 3) Lent 2022 ...
www.statslab.cam.ac.uk/~lab85/resources/StatsL23a.pdf27 Apr 2023: STATISTICS Part IBExample Sheet 1 (of 3) Lent 2022. 1. Ask your supervisor to test you on the sheet of common distributions on the course website. -
28 1/I/7C Statistics Let X1, . . . ,Xn ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2007.pdf28 Apr 2023: Explain carefully how the test should be implemented. 1/II/18C Statistics. Let X1,. ... yθ1eydy and θ > 1 is unknown. Find the most powerful test of size α,. -
40 Paper 1, Section I 7E StatisticsSuppose X1, . ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2010.pdf28 Apr 2023: Derive and carry out an appropriate test at the 5% significance level of thehypothesis that the advertisement has had no effect on people’s preferences. -
38 Paper 1, Section I 7H StatisticsConsider the experiment ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2011.pdf28 Apr 2023: σ2X, σ2Y are unknown. Derive the generalised likelihood ratio test of size α of the null. -
78 Paper 1, Section II 28K Principles of StatisticsDefine ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2011.pdf28 Apr 2023: How could you use the sample correlation r to test the hypothesis ρ = 0? -
32 1/I/7D Statistics The fast-food chain McGonagles have three ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2005.pdf28 Apr 2023: It is desiredto test the hypothesis. H0 : f = f0 against H1 : f = f1,. ... You should state, but need not derive, the distribution of the test statistic.]. -
UNIVERSITY OF CAMBRIDGE Faculty of Mathematics SCHEDULES OF LECTURE…
www.statslab.cam.ac.uk/~lab85/resources/schedules2324.pdf31 Oct 2023: Comparison and ratio tests, alternating series test. [6]. ContinuityContinuity of real- and complex-valued functions defined on subsets of R and C. -
UNIVERSITY OF CAMBRIDGE Faculty of Mathematics SCHEDULES OF LECTURE…
www.statslab.cam.ac.uk/~lab85/resources/schedules1920.pdf2 May 2023: Comparison and ratio tests, alternating series test. [6]. ContinuityContinuity of real- and complex-valued functions defined on subsets of R and C. -
42 Paper 1, Section I 7H StatisticsDescribe the generalised ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2012.pdf28 Apr 2023: It is desired to test the hypothesis H0 : λ = µ against H1 : λ 6= µ. ... Let w(θ) denote the power function of this test at θ (> 1). -
48 Paper 1, Section I7H Statistics What does it ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2009.pdf28 Apr 2023: What is the critical region C of a test of the null hypothesis H0 : θ Θ0 against. ... the alternative H1 : θ Θ1? What is the size of a test with critical region C? -
41 Paper 1, Section I 7H StatisticsLet x1, . ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2013.pdf28 Apr 2023: It. is desired to test H0 : f(x) = 1 against H1 : f(x) = 2x. ... Let δ : [0, 1] {0, 1} define a test by δ(x1) = i ‘accept Hi’. -
39 Paper 1, Section I6H Statistics State the Rao-Blackwell ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2022.pdf27 Apr 2023: d) Describe a test of size α for the null hypothesis H0 : σ21 = σ. ... Suppose thereexists a likelihood ratio test of size α for H0 against H1. -
25 1/I/3D Statistics Let X1, . . . ,Xn ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2001.pdf28 Apr 2023: Define the terms size and power for a test of onesimple hypothesis against another. ... Part IB. 26. 2/II/12D Statistics. What is meant by a generalized likelihood ratio test? -
UNIVERSITY OF CAMBRIDGE Faculty of Mathematics SCHEDULES OF LECTURE…
www.statslab.cam.ac.uk/~lab85/resources/schedules1819.pdf2 May 2023: Comparison and ratio tests, alternating series test. [6]. ContinuityContinuity of real- and complex-valued functions defined on subsets of R and C. -
46 A1/12 B1/15 Principles of Statistics (i) A public ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2003.pdf28 Apr 2023: How might the parametric bootstrapbe used to obtain a test of H0 of exact size α? ... Your account should include a discussion of natural statistics andtheir properties and of various conditional tests on natural parameters. -
29 1/I/7C Statistics A random sample X1, . . ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2006.pdf28 Apr 2023: It is desired to test the hypothesis H0 : θ 0 against thealternative H1 : θ > 0. ... Show that there is a uniformly most powerful test of size α = 1/20and identify a critical region for such a test in the case n = 9. -
45 Paper 1, Section I 7H StatisticsLet X1, . ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2016.pdf28 Apr 2023: Find the criticalregion of the most powerful test of size α for the hypotheses H0 : λ = 1 versus H1 : λ = 2.Determine whether or not this test is uniformly most powerful ... Prove that the power function of this test is of theform w(α,β,σ2) = -
57 1/II/27I Principles of Statistics Suppose that X has ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2007.pdf28 Apr 2023: If it is, how (according to the conditionalityprinciple) do we test hypotheses on ψ? ... Explain when this test may be used. (ii) Independent identically-distributed observations X1,. -
31 1/I/3H Statistics Derive the least squares estimators α̂ ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2003.pdf28 Apr 2023: Derive carefully the test of homogeneity and test the hypothesis that the distributions ofscores among students in the two cities are the same. ... Conduct a goodness-of-fit test at the 0.05 level for the null hypothesis that the coins areall fair. -
UNIVERSITY OF CAMBRIDGE Faculty of Mathematics SCHEDULES OF LECTURE…
www.statslab.cam.ac.uk/~lab85/resources/schedules2021.pdf2 May 2023: Comparison and ratio tests, alternating series test. [6]. ContinuityContinuity of real- and complex-valued functions defined on subsets of R and C. -
41 Paper 1, Section I 7H StatisticsSuppose that X1, ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2015.pdf28 Apr 2023: For 0 < α < 1, derive the test that is most powerful among all tests of size atmost α. ... x6. Explain how to obtain anapproximate size 0.05 test using the value of T. -
83 Paper 1, Section II29J Principles of Statistics Let ...
www.statslab.cam.ac.uk/~lab85/resources/PoS2021.pdf28 Apr 2023: b) Suppose you do not have access to the quantiles of the sampling distribution ofthe Kolmogorov–Smirnov test statistic. ... Describe a test of H0 : F = F0 with size exactly 1/(m 1). -
35 1/I/7H Statistics A Bayesian statistician observes a random ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2008.pdf28 Apr 2023: ii) Find the form of the size-α likelihood–ratio test of the hypothesis. ... The t-test of the null hypothesis that α = 0 against an unrestricted alternativedoes not reject the null hypothesis. -
45 Paper 1, Section I 6H StatisticsLet X1, . ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2021.pdf27 Apr 2023: Conduct a 5% significance level test of the null hypothesis that the medicine andplacebo have the same effect, against the alternative that their effects differ. ... Find a uniformly most powerful test of size α of H0 : θ = 0 againstH1 : θ > 0. -
UNIVERSITY OF CAMBRIDGE Faculty of Mathematics SCHEDULES OF LECTURE…
www.statslab.cam.ac.uk/~lab85/resources/schedules2122.pdf2 May 2023: Comparison and ratio tests, alternating series test. [6]. ContinuityContinuity of real- and complex-valued functions defined on subsets of R and C. -
UNIVERSITY OF CAMBRIDGE Faculty of Mathematics SCHEDULES OF LECTURE…
www.statslab.cam.ac.uk/~lab85/resources/schedules2223.pdf27 Apr 2023: Comparison and ratio tests, alternating series test. [6]. ContinuityContinuity of real- and complex-valued functions defined on subsets of R and C. -
Modern Statistical MethodsRajen D. Shah r.shah@statslab.cam.ac.uk…
www.statslab.cam.ac.uk/~rds37/teaching/modern_stat_methods/notes2.pdf15 Jan 2023: Inthe final chapter we will tackle the problem of how to handle performing thousands ofhypothesis tests at the same time and more generally the task of quantifying uncertaintyin high-dimensional settings.
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