sol10 - $ m & $ yx p Q & x p m p &...

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Unformatted text preview: $ m & $ yx p Q & x p m p & x p ~ | i }{ m u { Q um z Y g p itC g p and . Now . The MSE is With this is is . By simulation, The mle has MSE about .015, substantially smaller than the nonparametric plug-in. g d g v v Tv Ty8Q l m 8 5w xw v Q rd ug h m t p ig h m p o nI Pl0 j h g e qSsird )q 8 E p v ki UWf ` $ d h q $ x X X & x ` Th y` S v BE 9 f10 $ Q x b x ` ` v $ x ` x y` v $ C PI0 HFDB@ 9 v t ( & r 8 GEC A b3 y ywY 210 Tu)$ '% s" UTS x ` x ` 3 R p h q ig feSc d b3 aY ` 3 X (1b) is the mean, which, for a is . (1c) The nonparametric plug-in estimator is Q T YBWUTS ` XV X R Q C PI0 HFDB@ 9 ( & 8 " GEC A 3 3 7654 210 )$ '% #! Homework 10 Solutions where is the smallest data point and hood is maximized by taking If then the observation contributes a factor of it makes the product 0. Hence, The likelihood function is (2) We need the Fisher information matrix for { z U { z And (1) Let . The density is 1 T TFu Y is the largest data point. The likeli- . The mle is otherwise 8 I P v Q p g E 8yQ $ { z & { Q uh z p n v H p E $ v )qb3 E p ` 3 EGp w FDB p v A E C@ @ x b 9 8 ` m6 42 0 " Q #! 7531 g ( )p | g d " #! Qm 8yQ $ & $ { | " Q m 8Q $ & g $ d { $ & d ' % #! $ & $ Q 8Q g & i e D$ d m { e $ d z m z p zX ` X & p { z X ` X & p $ { 8Q & uh z p 3 3 ` Q m m 8 { U { z $ & 8 m Q 8 $ 8 { z m { m z p h ` 8 U z p h ` An approximate The estimates standard error is The asymptotic standard error of The inverse Fisher information is Since (3) we get (4) The mle is and and confidence interval is , the maximum data point. Note that is 2 R . The mle is . The gradient of , we have . Solving for where . Hence, z %2 8 q 0p g 1 ( ( 8 q )p g 8 3 p g 8 C g ( ` ` z z H p g p ` p p z $ ` h ` T} T S Q B )p g T p g $ d h #$ p g d h E l o $ " % x E l o p g x E l o 0 ' &9 EC $ $ ! h | Q d ` b0 ` H p so . The ARE is and p g (b) Let dard error of is 95 per cent confidence interval is | | d h ` d TS ` H p g ! H )p $ g p p #C qT p C 7p ` p h " #T #T 7! #! @ @ " p 3p g TS $ p )p g $ !T WW g p 3 8 C ` g p 9 & " U . (5) so R and m Q g P I h U m m Q p m (c) has mean numbers. (d) Note that (e) By the law of large numbers, converges in probability to . So converges in probability to . The true value of is . For an arbitrary distribution , we have so the mle is inconsistent. On the other hand, is still consistent. Thus, d as (6a) . The mle is . Then, . Consistency follows from the weak law of large 3 . . The estimated stan. An approximate p g ...
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