Question

2. Suppose that we have a random sample of normally distributed random variables: X;2.2.4. N (u,02) for i = 1...n Derive the maximum likelihood estimators of u and o2.

Answer 1

Estimation of the parameters of a Normal distribution by the method of maximum libelihood Consider a univariate normal disbn with parameters M and o? If anyone of the two or both remain unknown, we can apply the method of monem, & likelihood to estimate at (them). In Suppose that a simple random sample has been drawn from the normal population," consisting of the values X13X2,..., ben. The goint pdf of these values, for fireed M and p(2.,;. & 2; - - ...,20) 14,03) cs f (xyl M,02) - (82|4,0")... f(x01 M, 02) Cam's canner

(20-02 - My ! (>=M.) 20 ) lie 2o 1 e. . 02 Ova = i. . Como conse E this function can be treated If any d the parameters op unknown, then, this function can be as the likelihood function of the S Case I e unknown, o? known Suppose that o' 02 (given.) Then, the likelihood function of u is given by LEV-M) . :.4(4) Marathi ė 2007 We shall maximise this function of its log transformation for finding the MLE of M. Now on Ini LCM) = - ņ In (53) - ņ (27) - 3712 The foc for maximisation is e d en L(a) --0 0:- 2.2(xi-4). (-) = 0 du 2 1 -M 10. in E (Xp-) = > nga nazo = nulna iyid = JT 4/6). 4-2014] Scanned with CamScanner

d² LR (M) dM² & The long likelihood and hence the likelihood of function is moreimised at u=X... Les This implies that the MLE of all is ū= 5, the sambleanthean. 0 5

Case II M known, or unknown let, the known value of u be do. Then, or the basis of the sample values we form the Ukellhood function L102= 1 -Mo)? 8.0. In (21)= 36 Thus, in 2004)= - R In )- We maxpmise this wit o? - The FOC is Id(02)22-2620 a, o- I. CA - Albert edhe 3042-70* (** 64 do The Lpkelihood functh attains its maximum cformed with CamScanner

02= E(X7-Mo)? Elence, the MLE of one - use II ell einknown, or unknown The likelihood function of u and oz takes the form Paulo Hot 2(x-4) L(11.03) - 1 e20 2 00, (nda nk)=0 P 02- M -0-0- au Thues, In L(2,64) = _n_ Inco ) - R 1n(21) - 2012 We shall maximise this log opkelihood writ . M and o2 The foes are Olet = -0 -0 - 22(Pi-M) (-) = 06 SELALU 202 - 0, 90, do 2 = - - 04 2¢17-432 E-1) 202 203 204 From the first condition we get ninu op, M = X substituting this into the seond condition, -- + z(xp-x} Zo 202904 mp , cxi-)? Scanned 20- 2041 CS CamScanner

er o? I = (xp-)' = 5?, the sample verfance.. second order fortfal derivatives are, 22n) au² = -2 2 2 = -2 at 8²-5² ( 97-04)- (m_ntaze 22ln L andora 210 362)2 = pi + ) (-302 at M=T 206 n NS2 at o² S² 54" second order partial .The determinant of derivatives is 1-2 .0 12 )0.- Blast Scanned with CamScanner

on The matrex of second onder Badial dorvatives som det hele e r det negative deferate. 22 112 334 L i SI odoon a(0)2 ] L 8. The SoCs for maximisation are satisfied. Hence, the MLE'S R M and ozare jij û zł CamScanner lubin the