test1 - 18.443 Practice test 1 Consider the family of...

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± 18.443. Practice test 1. Consider the family of distributions with p.d.f. f ( x | ) = x 1 , for 0 < x < 1 , and > 0 . (1) Consider an i.i.d. sample X 1 , . . . , X n from this distribution. As always, the underlying parameter for this sample is unknown. In problems (1) and (2) the distribution is given by equation (1) above. (1) Find the MLE ˆ of . (2) Compute Fisher information I ( ) and state asymptotic normality of MLE ˆ . If n = 100 , ±nd c such that Pr( c ˆ c ) ± 0 . 95 . (3) Suppose that a covariance matrix ² ² ² 1 1 1 1 0 0 1 0 0 2 2 2 2 1 1 1 1 0 0 2 0 0 = . ± ± 2 2 2 2 0 0 1 0 0 0 0 0 1 Distribution N (0 , ) has what density in what basis? (4) Suppose that a sample X 1 , . . . , X 15 ² N ( µ x , α x 2 ), where µ x and α x 2 are unknown, has sample mean and sample variance µ ˆ x = X ¯ = 2 . 4 , α ˆ x 2 = X ¯ 2 X ¯ 2 = 0 . 55 . Find 95% con±dence intervals for µ and α 2 . (5) In addition to the sample from problem (4) suppose that we are given a sample Y 1 , . . . , Y 10 ² N ( µ y , α x 2 ) from a distribution with the same variance as X s, i.e. α x 2 = α y 2 , but possibly diﬀerent mean µ y . Suppose that
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