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by: Celestino Bergnaum


Marketplace > Texas A&M University > Statistics > STAT 611 > THEORY OF INFERENCE
Celestino Bergnaum
Texas A&M
GPA 3.5

Jeffrey Hart

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About this Document

Jeffrey Hart
Class Notes
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This 3 page Class Notes was uploaded by Celestino Bergnaum on Wednesday October 21, 2015. The Class Notes belongs to STAT 611 at Texas A&M University taught by Jeffrey Hart in Fall. Since its upload, it has received 55 views. For similar materials see /class/225753/stat-611-texas-a-m-university in Statistics at Texas A&M University.




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Date Created: 10/21/15
Formula Sheet for Final Exam in STAT 611 610 stuff Theorem of Total Probabilities Let the events B17B27 be mutually exclusive and such that S U2 Bi Let Z be the set of indices 239 such that PBi gt 0 Then PA ZPAlBiPBi ieZ Multiplication Rule PA0A1An PA0PA1A0PA2A0A1PAnA0A1An1 Density of Monotone Transformation of X My fxg 1y 9 1y Density of One to One Transformation of X7 Y Let u 91Ly 1 92Ly be one to one with inverse transformation z hluv7 y h2ui Let J 3737 7 Then the joint pdf of U7 V is fLCVWv fxy h1u7v7 h2u l Ul Moment Generating Function of X Mm E 5quot Density of order statistic Let X1 lt lt XW denote the order statistics for a random sample from pdf f7 which has cdf F The density of X is W X fxlFIlquot 11FWWH j i 1W i j Math stuff Binomial Sums i ltZgtakb k Mb iltn271gtzk m lt1 k0 k 0 Exponential Series ooak k0 k 5a Geometric Series Let n 1 in the second binomial sum 00 l Zak lallt1 k0 17a Gamma Function The gamma function satis es No 1 OLPOL 04 gt 07 and for every nonnegative integer n7 Pn 1 71 Incomplete Gamma Function Nam uu 7 leiudii7 04 gt 07 z gt 0 0 Useful Limits n lim ze m 0 lim 1 3 e mace Hoe n Estimation stuff Cram r Rao lower bound d 2 macaw Var 2 2 9 a am xm If the data are a random sample and dinge ltlogfX10gt 10gfz0 fx9 then the denominator in the CRLB is dx 7 62 nEQ w logfX10 Rao Blackwell Theorem If W is unbiased for 70 and T is suf cient7 then is unbiased for 70 and Var9EWT S Var9W for all 0 Lehmann Sche Theorem An unbiased estimator that depends on the data only through a complete7 suf cient statistic is the unique UMVUE Asymptotic normality of a function of an asymptotically normal statistic If WTn 7 0 converges in distribution to N0UZ7 then gTn 7 90 converges in distribution to N07 72g 027 so long as g has a nonzero derivative Testing stuff Type I and H errors A type I error is rejecting H0 when H0 is true A type II error is accepting H0 when H1 is true Likelihood ratio statistic L 6 su x We pm lt w gt supeee 19 95 Monotone likelihood ratio The family of distributions fw0 6 E 9 is said to have the monotone likelihood ratio property in the statistic TX if for all 91 lt 02 the ratio fw 02fw 01 is a nondecreasing function of Tw on w fw 01 gt 0 or fw 02 gt 0 Unbiased tests An unbiased test b is one satisfying infc E9 X 2 sup E9 X 9660 9680 Interval estimation stuff Pivotal quantity Given data X7 QX7 9 is a pivotal quantity if the distribution of QX0 is completely free of all unknown parameters Bonferroni inequality PA B 2 17 PAc 7 PB0 HPD regions An HPD region is a Bayes credible set of the form 6 WM 2 c where c is a constant


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