Question

Let X1,... , Xn be a random sample from the Pareto distribution with pdf Ox (0+1), x > 1, f(z0) where 0>0 is unknown (a) Find a confidence interval for 0 with confidence coefficient 1-a by pivoting a ran- dom variable based on T = T log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval (b) Find a confidence interval for 0 with confidence coefficient 1 - a by pivoting the cdf of XminX1,... . X,}. (Use equal-tail confidence interval)

Answer 1

2 Log 20T -RPTa20T 2 PC29T - n; S(20TEYa) % YaXon PCin, 201 220, = 1- P 2n;1- 2 2 T 2T Xenjth on1h 2T 2T b) C.d.p of XaFxi) P( )= -PCX 1-(PCx,x)) anned w CamScanne

Let T- 2ne in X)FrCt) PCTst) PCane n a 2no - 1-e ene n) 1-eth PYaTEY) uhere p CTV amd PCTEa)- P(TEY n() 2 PCT EVa) a-eln(-) 1- PC-2 An (1-) 2ne ln(x n4 = P 7 07 In(Xen) n C.T n (Xcv) Cs Scanned with

Define pdf of y- fyy)eey N BX let x 2z P2P d-) e 2 2 X 2 2 Xx 2