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SOME GENERAL PROBLEMS 1

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Problem A certain lion has three possible states of activity each night; they are ‘very active’ (denoted by θ 1 ), ‘moderately active’ (denoted by θ 2 ), and ‘lethargic (lacking energy)’ (denoted by θ 3 ). Also, each night this lion eats people; it eats i people with probability p(i|θ), θ ϵ Θ={θ 1, θ 2, θ 3 }. Of course, the probability distribution of the number of people eaten depends on the lion’s activity state θ ϵ Θ. The numeric values are given in the following table. 2

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Problem i01234 p(i|θ 1 ) p(i|θ 2 ) p(i|θ 3 ) If we are told X=x 0 people were eaten last night, how should we estimate the lion’s activity state (θ 1, θ 2 or θ 3 )?

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Solution One reasonable method is to estimate θ as that in Θ for which p(x 0 |θ) is largest. In other words, the θ ϵ Θ that provides the largest probability of observing what we did observe. : the MLE of θ based on X (Taken from “Dudewicz and Mishra, 1988, Modern Mathematical Statistics, Wiley”) 4

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Problem Consider the Laplace distribution centered at the origin and with the shape parameter β, which for all x has the p.d.f. Find MME and MLE of β. 5

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Problem Let X 1,…,X n be independent r.v.s each with lognormal distribution, ln N( , 2 ). Find the MMEs of , 2 6

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STATISTICAL INFERENCE PART III BETTER OR BEST ESTIMATORS, FISHER INFORMATION, CRAMER- RAO LOWER BOUND (CRLB) 7

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8 RECALL: EXPONENTIAL CLASS OF PDFS If the pdf can be written in the following form then, the pdf is a member of exponential class of pdfs. (Here, k is the number of parameters)

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9 EXPONENTIAL CLASS and CSS Random Sample from Regular Exponential Class is a css for .

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10 RAO-BLACKWELL THEOREM Let X 1, X 2,…,X n have joint pdf or pmf f(x 1,x 2,…,x n ; ) and let S=(S 1,S 2,…,S k ) be a vector of jss for . If T is an UE of ( ) and (S)=E(T S), then i) (S) is an UE of ( ). ii) (S) is a fn of S, so it is free of . iii)Var( (S) ) Var(T) for all . (S) is a better unbiased estimator of ( ).

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RAO-BLACKWELL THEOREM Notes: (S)=E(T S) is at least as good as T. For finding the best UE, it is enough to consider UEs that are functions of a ss, because all such estimators are at least as good as the rest of the UEs. 11

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Example Hogg & Craig, Exercise X 1,X 2 ~Exp(θ) Find joint p.d.f. of ss Y 1 =X 1 +X 2 for θ and Y 2 =X 2. Show that Y 2 is UE of θ with variance θ². Find φ(y 1 )=E(Y 2 |Y 1 ) and variance of φ(Y 1 ). 12

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THE MINIMUM VARIANCE UNBIASED ESTIMATOR Rao-Blackwell Theorem: If T is an unbiased estimator of , and S is a ss for , then (S)=E(T S) is – an UE of , i.e.,E[ (S)]=E[E(T S)]= and – with a smaller variance than Var(T). 13

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14 LEHMANN-SCHEFFE THEOREM Let Y be a css for . If there is a function Y which is an UE of , then the function is the unique Minimum Variance Unbiased Estimator (UMVUE) of . Y css for . T(y)=fn(y) and E[T(Y)]= . T(Y) is the UMVUE of . So, it is the best unbiased estimator of .

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15 THE MINIMUM VARIANCE UNBIASED ESTIMATOR Let Y be a css for . Since Y is complete, there could be only a unique function of Y which is an UE of . Let U 1 (Y) and U 2 (Y) be two function of Y. Since they are UE’s, E( U 1 (Y) U 2 (Y) )=0 imply W(Y)=U 1 (Y) U 2 (Y)=0 for all possible values of Y. Therefore, U 1 (Y)=U 2 (Y) for all Y.

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Example Let X 1,X 2,…,X n ~Poi(μ). Find UMVUE of μ. Solution steps: – Show that is css for μ. – Find a statistics (such as S*) that is UE of μ and a function of S. – Then, S* is UMVUE of μ by Lehmann-Scheffe Thm. 16

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Note The estimator found by Rao-Blackwell Thm may not be unique. But, the estimator found by Lehmann-Scheffe Thm is unique. 17

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18 RECALL: EXPONENTIAL CLASS OF PDFS If the pdf can be written in the following form then, the pdf is a member of exponential class of pdfs. (Here, k is the number of parameters)

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19 EXPONENTIAL CLASS and CSS Random Sample from Regular Exponential Class is a css for . If Y is an UE of , Y is the UMVUE of .

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20 EXAMPLES Let X 1,X 2,…~Bin(1,p), i.e., Ber(p). This family is a member of exponential family of distributions. is a CSS for p. is UE of p and a function of CSS. is UMVUE of p.

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21 EXAMPLES X~N( , 2 ) where both and 2 is unknown. Find a css for and 2.

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22 FISHER INFORMATION AND INFORMATION CRITERIA X, f(x; ), , x A (not depend on ). Definitions and notations:

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23 FISHER INFORMATION AND INFORMATION CRITERIA The Fisher Information in a random variable X : The Fisher Information in the random sample: Let’s prove the equalities above.

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24 FISHER INFORMATION AND INFORMATION CRITERIA

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25 FISHER INFORMATION AND INFORMATION CRITERIA

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26 FISHER INFORMATION AND INFORMATION CRITERIA The Fisher Information in a random variable X : The Fisher Information in the random sample: Proof of the last equality is available on Casella & Berger (1990), pg

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27 CRAMER-RAO LOWER BOUND (CRLB) Let X 1,X 2,…,X n be sample random variables. Range of X does not depend on . Y=U(X 1,X 2,…,X n ) : a statistic; does’nt contain . Let E(Y)=m( ). Let prove this!

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28 CRAMER-RAO LOWER BOUND (CRLB) -1 Corr(Y,Z) 1 0 Corr(Y,Z) 2 1 Take Z= ′(x 1,x 2,…,x n ; ) Then, E(Z)=0 and V(Z)=I n ( ) (from previous slides).

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29 CRAMER-RAO LOWER BOUND (CRLB) Cov(Y,Z)=E(YZ)-E(Y)E(Z)=E(YZ)

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30 CRAMER-RAO LOWER BOUND (CRLB) E(Y.Z)=m ʹ ( ), Cov(Y,Z)=m ʹ ( ), V(Z)=I n ( ) The Cramer-Rao Inequality (Information Inequality)

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31 CRAMER-RAO LOWER BOUND (CRLB) CRLB is the lower bound for the variance of an unbiased estimator of m( ). When V(Y)=CRLB, Y is the MVUE of m( ). For a r.s., remember that I n ( )=n I( ), so,

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32 ASYMPTOTIC DISTRIBUTION OF MLEs : MLE of X 1,X 2,…,X n is a random sample.

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33 EFFICIENT ESTIMATOR T is an efficient estimator (EE) of if – T is UE of , and, – Var(T)=CRLB T is an efficient estimator (EE) of its expectation, m( ), if its variance reaches the CRLB. An EE of m( ) may not exist. The EE of m( ), if exists, is unique. The EE of m( ) is the unique MVUE of m( ).

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34 ASYMPTOTIC EFFICIENT ESTIMATOR Y is an asymptotic EE of m( ) if

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35 EXAMPLES A r.s. of size n from X~Poi(θ). a)Find CRLB for any UE of θ. b)Find UMVUE of θ. c)Find an EE for θ. d)Find CRLB for any UE of exp{-2 θ}. Assume n=1, and show that is UMVUE of exp{- 2 θ}. Is this a reasonable estimator?

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36 EXAMPLE A r.s. of size n from X~Exp( ). Find UMVUE of , if exists.

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Summary We covered 3 methods for finding good estimators (possibly UMVUE): – Rao-Blackwell Theorem (Use a ss T, an UE U, and create a new statistic by E(U|T)) – Lehmann-Scheffe Theorem (Use a css T which is also UE) – Cramer-Rao Lower Bound (Find an UE with variance=CRLB) 37

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Problems Let be a random sample from gamma distribution, X i ~Gamma(2,θ). The p.d.f. of X 1 is given by: a) Find a complete and sufficient statistic for θ. b) Find a minimal sufficient statistic for θ. c) Find CRLB for the variance of an unbiased estimator of θ. d) Find a UMVUE of θ. 38

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Problems Suppose X 1,…,X n are independent with density for θ>0 a) Find a complete sufficient statistic. b) Find the CRLB for the variance of unbiased estimators of 1/θ. c) Find the UMVUE of 1/θ if there is one. 39

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