By Bouchaud J.-P.

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The normal form of the score statistic simplifies to zS s uŽ ␲ 0 . ␫ Ž␲ 0 . 1r2 s y y n␲ 0 'n␲ 0 Ž1 y ␲ 0 . s '␲ ␲ ˆ y ␲0 0 Ž 1 y ␲ 0 . rn . 11 . Whereas the Wald statistic z W uses the standard error evaluated at ␲ ˆ , the score statistic z S uses it evaluated at ␲ 0 . The score statistic is preferable, as it uses the actual null SE rather than an estimate. Its null sampling distribution is closer to standard normal than that of the Wald statistic. 7. equals L 0 s ylog␲ 0 q Ž n y y . log Ž1 y ␲ 0 .

08, where a higher T provides more evidence against the null. a. With the usual P-value, show that the actual P Žtype I error. s 0. b. With the mid-P-value, show that the actual P Žtype I error. 08. c. Find P Žtype I error. in parts Ža. and Žb. when P ŽT s 0. 30, P ŽT s 1. 66, P ŽT s 2. 04. Note that the test with midP-value can be conservative or liberal. The exact test with ordinary P-value cannot be liberal. d. , a randomized-decision test generates a uniform random variable U from w0, 1x and rejects H0 when T s 2 and U F 58 .

C. Pearson proposed the test statistic X s 2 Ý j Ž nj y ␮j . ␮j 2 . 15 . Greater differences Ä n j y ␮ j 4 produce greater X 2 values, for fixed n. Let X o2 denote the observed value of X 2 . The P-value is the null value of P Ž X 2 G X o2 .. This equals the sum of the null multinomial probabilities of all count arrays Žhaving a sum of n. with X 2 G X o2 . For large samples, X 2 has approximately a chi-squared distribution with 2 2 df s c y 1. , where ␹cy1 denotes a chi-squared random variable with df s c y 1.

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An introduction to statistical finance by Bouchaud J.-P.

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