Question

I would like to find the method of moments estimator for Uniform(-0, distributions. The density for Uniform(-0,0) is fu(ul0) = for - Suco 10 otherwise 28 Calculate the expected value of U. Why is it impossible to use this to estimate o? b) (7 points) Suppose that we observe n IID observations 2, with pdf 8 (170) exp 21V2 2 363 +5log 33} -> 0 for some unknown 8 and where the 8 must be positive. Find the maximum likelihood estimates of d and % (no need to check second derivative)

Answer 1

Solution or ay U um (-0,0) (U) u fuluodu 2. U. I du u 20 40 0 U is the estimations of parameter 80 not o. That's why it is impossible to estimate 'or noi C eing

by : 14,5) = 3 ex 7J* å 2,70 direconstants ý syo we fzi(9,8) 1 & TT 1/2 Celta dingi+Y?? e 51, deri 25 => - 1/2 71802 - 1 - 1 L 27 o dzi (72 no The r=18 M & P78 He know the toolf of { a } dzi 51. close from A= 1 - Y2 Cinayetlering? birra 6-vje on integrating gives and zi follows logmosomal distribution and = 18 (30) Let L² be the likelihood function to f and r. LC89) = . 1 FC8738,17 8 17! Then with M=-7 1/2 (dinzi+rga e nii

3 & an 2c&inzitt InL din Taking in on both sides; Ink + mind Zinzi * mo- 1 x 2 ; // 2 A coinority) in i = 0 ☆ Mis sonhoi)? - Y21nY; = 0. 13! dinh =0 ar => o 12x2I coinsrity) Î lainerite) =0 & Elinizi - MY=o. 잇 5! Ingri 77 YE - В n. zliseniz W = = > In (A); E substitute 'y' from <B>inens? n -012 1n7i)? + Mis m 82 | Zlinari)? (Fingri)? 82 ชา) (1-1/m) Elnai)? n? hil2 (1-1) (Zin87)" : SME (24191992 zent EUS i ne n-1

n2 (21nzrije-2 and FMLE - (SMLE) a SME &=&MLE į GoM geometrich & in n-1 Mue, ÅMLE Einari MLE N Zinzi n BAKE d2in1 d²luL dd dr >o Now, d'inL danL addy dy2 P= PAMLE thene is extreme exi'et fora inc. co and dine dr "P=VMLE Hence extreme Mle's of of and of exist, À Zini ETA extra 'MLE 2 Inari ET 7-1 n in Imeden AGM Tom implies peintes vaines drinL a82 das MLE Sphytl 14 EMLE TE Thin poi) Intiorigin and iMLE = mean ZGIMO MLE