Question

Let Yı,Y2, ..., Yn be iid from a population following the shifted exponential distribution with scale parameter B = 1. The pdf of the population distribution is given by fy(y\0) = y-0) = e x I(y > 0). The "shift" @ > 0 is the only unknown parameter. (a) Find L(@ly), the likelihood function of 0. (b) Find a sufficient statistic for 0 using the Factorization Theorem. (Hint: O is bounded above by y(1) min{Y1, 42, ..., .., Yn}.) (c) Find the MLE of 0. =

Answer 1

Question - a b.did fly)= e(4-0) ; y so n Likelihood function, 4 (4-0) Å flyi). no- y iz re L=e Auswer b) LY (6) (Yin) Arrange the sample obsez vatous air ascending order of magnitude, Yo <Y (2) < noyi ; ocY (1) <Y (2) < define Indicatos variable as, I (40), 0) = 0; otherwise We j OLY (1) if so

Hence, L can be written as, noyi L=e ot no hi L = e I/, 0] le no L= e 2 3 I lyci, o]. Comparing it with Neyman factorization Theorem Lea (1, ). H(41, 42, -- yn) Comparnig ② and ③, we have GCTIO) eno I 40), 0] [where T= you) and Hlys Y ..Yon) Thus T= y0) is sufficient statistic for a Yi Auswer

© From D L=enomy n dog L= no hay alogh ; Hence method of diferentiation fail hese For M. L.E. of o, we must find value of o for which L (or log() is maximum We can weite (Yon) no- y L=e OcY,< Y2) < It is evident that Lis maximum if o is maximian, but oLY (1) re Omaximums Y(1) ie M. L.E. of B = ô = you = mint yo, yon --- yn} Answer