Question

If the modom Variable Y denotes an individual's income, Pareto's law claims that P(Yay)-(W) wherek is the entire population's minimum income It follows that 540)**** *** yak 21. ample of The income information has been collected on a ra. individuals: Y.Y2....,Y To answer this question, enter your answer as a formula. In addition to the usual guidelines, a few more instructions for this problem: write y.) as single variable p and y asm. These can be used as the input of functions as though they were usual variables e.g log(P), m^2, exp(m) etc... Remember p represent product of I s only..not the product of any functions of them. The order statistic miny,=Y (1) can be entered as min(y). Same for max(y). For all parts below, note that you will need to make your responses an expression in terms of p, m, min(y) or max(y). sum() is not a recognized function, and "x1", "x2", etc... are not recognized variables. For part (1) and (2) assume k is known. Find the maximum likelihood estimator of 8 1/(log(P) -log(k)) Submit Answer Incorrect. Tries 3/5 Previous Tries 2. Find the Method of moments estimator of 8. m/m-k) You are correct. Your receipt no. is 159-9689 Previous Tries 3. For parts (3) and (4) assume 8 is known. Find the Maximum likelihood estimator of k. min(y) You are correct. Your receipt no. is 159-5892 Previous Tries 4. Find the Method of moments estimator of k. Use "t" as symbol for 8 in the response. (m(-+ + 1)y-t Submit Answer Incorrect. Tries 1/2 Previous Tries Post Discussion Send Feedback • Type here to search o f @ w

Answer 1

sample values in of random fy (Y) = oko ( 4 ) et ; y 7,K ,871 likelihood function of random sam 9, 92, ..., Yn is given as LIK, 0)4,) = 4,00) 1 OK0.15.)°F) = (K0)". ( 19. 1041 since, we have given that ri = b .LCK, 04,..,Y) = (ekº)". 1)0+) Now, taking log of both sides, Dog Lik,019.-)) = nloge + onlogk H0+i) dog (1) To get the MLE of o, for known K, we equate 2 (log 2 (K, 0141. Yn) = 0 and 2 / dog 1/4, 0) o differentiating is wort o, we get e Dog L (K,0) = n + nilagk + log 1) Now

2 log L (8,0) = 0 2 n +nlogk - logp = 0 = logp - nlogk = logo = 1oo loga 1 = loga- logo (1.) gog ten : lôme = mLE of o = MLE = MLE of o = n when log (b) / kis known [ Beat L(t,s) - - < 0 ! . (2.) To find the method of moment estimates of o - we equate E(X) = mg ; 8=1,2,... where, Mr = t Exit we have given that, (m - m = 2 = 5 NOW, EU) = 48.6KO (4" by - Oko. Jy - ody my FW.I s 0 к

-041 -ot y-of 900 => ELY) = OK0. ( y-04) - OK (O K ) (2 071) list - Ely) = OKO KI-O -(1-0) = -178 Now, for known k, to get mme of o, we solve. E(Y) = m - Ko = m Ko =-m+mo olk-m) --M > 0= m u omme = Mme of 0 = m 7 for known m-K E (3) Now, suppose, o is known, we have to find MLE of K. → LIK, 014.,Yn) = (020)" (1) ®+ ksy let Yew, 9(2), ... Sens be the ordered statistic of 91, 92, ... Yh. Kuyu Yes .. <you Dy K You

likelihood would whood would be marimum value of a lifo is known ). for the monimum le. L(koly, ...gr (K, 019,--yo) - (OKO)" (+0+1 u be marimum for maximum wedlue of K and the maximum value of K = 4 ) = min se up KMEE = MLE of K = Yes = minyi) when non lo is known (4). we have to find mme af K 10 is known) . Ely) = ko , m = ko o-1 m = kt ( we have given that mt-m=kt K = m(t-1) Rmme = mme of K = m(t-17 when out Emme = m(t-1) / when 7 when ot is known