Likelihood Ratio Tests Flashcards
Let Y1 , Y2 , . . . , Yn denote a random sample from a normal distribution with mean μ (unknown) and variance σ^2. For testing H0 :σ^2 = σ0^2 against H :σ^2 > σ0^2, show that the likelihood ratio test is equivalent to the χ2 test given in Section 10.9.
Answer
A survey of voter sentiment was conducted in four midcity political wards to compare the fraction of voters favoring candidate A. Random samples of 200 voters were polled in each of the four wards, with the results as shown in the accompanying table. The numbers of voters favoring A in the four samples can be regarded as four independent binomial random variables.
Construct a likelihood ratio test of the hypothesis that the fractions of voters favoring candidate
A are the same in all four wards. Use α = .05.
Opinion 1 2 3 4 Total
Favor A 76 53 59 48 236
Do not favor A 124 147 141 152 564
Total 200 200 200 200 800
Answer
Let S1^2 and S2^2 denote, respectively, the variances of independent random samples of sizes n and m selected from normal distributions with means μ1 and μ2 and common variance σ^2. If μ1 and μ2 are unknown, construct a likelihood ratio test of H0 :σ^2 = σ0^2 against Ha :σ^2 = σa^2, assuming that σa^2 > σ0^2.
Answer
Supposethat X1 ,X2 ,…,Xn1 ,Y1 ,Y2 ,…,Yn2 ,and W1 ,W2 ,…,Wn2 are independent random samples from normal distributions with respective unknown means μ1, μ2, and μ3 and variances σ1^2, σ2^2, and σ3^2.
a) Find the likelihood ratio test for H0 : σ1^2 = σ2^2 = σ3^2 against the alternative of at least one inequality.
b) Find an approximate critical region for the test in part (a) if n1, n2, and n3 are large and α = .05.
Answer
Let X1, X2, . . . , Xm denote a random sample from the exponential density with mean θ1 and let Y1 , Y2 , . . . , Yn denote an independent random sample from an exponential density with mean θ2.
a Find the likelihood ratio criterion for testing H0 : θ1 = θ2 versus Ha : θ1 ≠ θ2 .
b Show that the test in part (a) is equivalent to an exact F test [Hint: Transform ΣXi and ΣYj to χ2 random variables.]
Answer
Suppose that independent random samples of sizes n1 and n2 are to be selected from normal populations with means μ1 and μ2 , respectively, and common variance σ^2 . For testing H0 : μ1 = μ2 versus Ha : μ1 − μ2 > 0 (σ^2 unknown), show that the likelihood ratio test reduces to the two-sample t test presented in Section 10.8.
Answer
