Question

Let d ln(B)e where the unknown 3 E R, k = 1,..., M and ek are dent and identically distributed according to the (standard) Cauchy distribution, that is mutually indepen- 1 1 т (еx) k (a) Construct the likelihood T(d|B) and the negative log-likelihood (b) In this case, you cannot find an likelihood estimate BML But do this: derive the nonlinear equa- explicit form for the maximum tion of the form (- In т(d|B)) — F(83; d,, .., dм) dp 0 The ML estimate could then be solved numerically from this equa- tion, for example, using the Newton iteration 3 2.5 + 2 + + 1.5 1 0.5 0- -0.5 0.2 0.4 0.1 0.3 0.5 0.6 0.7 0.8 0.9 Figure 1: Data for Q5.

Answer 1

Solution Given that dKnCA)teK ek is a standand Cauchy distonibu tion CPolt) a )Tartan(p) t% odt PCde d)P nCRH< d/) Then F Tarntan -In CA))2 )Tatand-toca))t The cdf d d is dengity bunchon dde is Than the aThen the likelihood is Cde) にこ CS Scanned with CamScanner

m TTE CdIn CA) J (+llDoy- Lo Ce-ncalt,) F C dm) /e Je-iaI} 2 CA)-dk Csanned with wl CamScanner