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Spring 2009
Statistics 131C
Homework 1
Due : April 10
- Problem 8.1.2 : Suppose that X 1 ,... , Xn form a random sample from a uniform distri- bution on the interval [0, θ], and that the following hypotheses are to be tested:
H 0 : θ ≥ 2 against H 1 : θ < 2.
Let Yn = max{X 1 ,... , Xn}, and consider a test procedure such that the critical region contains all the outcomes for which Yn ≤ 1 .5.
(a) Determine the power function of the test. (b) Determine the size of the test.
- Problem 8.1.6 : Suppose that a single observation X is to be taken from a uniform distribution on the interval [θ − 12 , θ + 12 ], and suppose that the following hypotheses are to be tested: H 0 : θ ≤ 3 against H 1 : θ ≥ 4.
Construct a test procedure δ for which the power function has the following values: π(θ|δ) = 0 for θ ≤ 3, and π(θ|δ) = 1 for θ ≥ 4.
- Problem 8.1.11 : Assume that X 1 ,... , Xn are i.i.d. having a Bernoulli distribution with parameter p. Suppose that we wish to test the hypotheses:
H 0 : p = 0. 4 against p 6 = 0. 4.
Let Y =
i=1 Xi. (a) Find c 1 , c 2 such that
P(Y ≤ c 1 |p = 0.4) + P(Y ≥ c 2 |p = 0.4)
is as close as possible to 0.1 without being larger than 0.1. (b) Let δ be the test that rejects H 0 if either Y ≤ c 1 or Y ≥ c 2. What is the size of the test δ? (c) Draw a graph of the power function of δ.
- Problem 8.1.14 : Let X 1 ,... , Xn be i.i.d. with an exponential distribution having param- eter θ. Suppose that we wish to test the hypotheses
H 0 : θ ≥ θ 0 against H 1 : θ < θ 0.
Let X =
∑n i=1 Xi. Let^ δc^ be the test that rejects^ H^0 if^ X^ ≥^ c. (a) Show that π(θ|δc) is a decreasing function of θ. (b) Find c in order to make δc have size α 0. (c) Let θ 0 = 2, n = 1 and α 0 = 0.1. Find the precise form of the test δc and sketch the power function.
- Problem 8.2.2 : Consider two p.d.f.’s f 0 (x) and f 1 (x) which are defined as follows:
f 0 (x) =
1 for 0 ≤ x ≤ 1 0 otherwise,
and
f 1 (x) =
2 x for 0 ≤ x ≤ 1 0 otherwise.
Suppose that a single observation X is taken from a distribution for which the p.d.f. f (x) is either f 0 (x) or f 1 (x), and the following simple hypotheses are to be tested:
H 0 : f (x) = f 0 (x) against H 1 : f (x) = f 1 (x).
(a) Describe a test procedure δfor which the value of α(δ) + 2β(δ) is minimum. (b) Determine the minimum value of α(δ) + 2β(δ) attained by that procedure.
- Problem 8.2.10 Suppose that X 1 ,... , Xn form a random sample from a Poisson distribu- tion for which the value of the mean λ is unknown. Let λ 0 and λ 1 be specified values such that λ 1 > λ 0 > 0, and suppose that it is desired to test the following simple hypotheses:
H 0 : λ = λ 0 against H 1 : λ = λ 1.
(a) Show that the value of α(δ) + β(δ) is minimized by a test procedure which rejects H 0 when Xn > c. (b) Find the value of c. (c) For λ 0 = 1/4, λ 1 = 1/2 and n = 20, determine the minimum value of α(δ) + β(δ) that can be attained.
