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17 Jun 2019: Find the least squares estimator β̂ of β. State, without proof, the distribution of β̂and describe how you would test H0 : β = β0 against H1 : β 6= β0, where β0 is

30 Aug 2019: Consider the hypothesis that the time series y1, y2,. , yT is purely random.Describe the turning point test of this hypothesis.

17 Jun 2019: σ2X, σ2Y are unknown. Derive the generalised likelihood ratio test of size α of the null.

17 Jun 2019: iii) Determine the joint distribution of α̂, β̂ and σ̂2. (iv) Explain carefully how you would test the hypothesis H0 : α = α0 against thealternative H1 : α 6= α0.

17 Jun 2019: 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.

17 Jun 2019: Surveys suggest that 60% of the populationare already immune, but accurate tests to detect vulnerability in any individual are toocostly for mass screening. ... A simple skin test has been developed, but is not completelyreliable.

17 Jun 2019: 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.

17 Jun 2019: whether they are independent or not, we usually compute their. Pearson correlation (or a related test statistic). ... Figures: Data on the LHS to be tested for dependence, and graph of test power as a function of sample size for various dependence tests.

17 Jun 2019: Paper 3 [TURN OVER. 10. 12M Principles of Statistics. (i) Describe in detail how to perform the Wald, score and likelihood ratio tests of asimple null hypothesis H0 : θ = θ0 given ... Xn from a regular one-parameter density f(x; θ). In each case you

17 Jun 2019: 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′().