SCORE-TYPE TESTS FOR NORMAL MIXTURES AMENGUAL BEI CARRASCO SENTANA 2023s-02

Second, we can easily check that jwLR M11n j > and jwLR1n j > jwLR 2n j because LR 2 LR LR 2 1 LR LR 2 1 n 1 2 2n n n = n 1 n n n 1 LR LR 2n n 8wLR2n 2 2 1 4= + [1 LR n + LR n ]n 2 (1 LR) LR (24) LRn 9 n n 1 2n (1 LR) LR2 n n 1 LR 1 LR LR 4 1 16 w + n (1 ) n (1 LR) LR 2 22n 6 n n 2 n2 n (25)8 1 n 1 LR LR 1 n 1 LR 2 LR 6 n n ; (26) 6 18 n n 92 where (24) follo. [...] To do so, 27 notice that n 1 LR 2 LRLR 2 1 LR 2 1 4 1 2 n LR LR2 2n n n n n n 2 1 n (32)1 n 2 1 LR LR 2 36n n (33) (1 2n + n)1p LR n 1 LR LR 2 w2n 1 8 n n n 2 1 LR LR 2n 2 1 n 1p n 36 (34) n 1 LR LR 2 n n 1 2jwLR 8 2n j 2 LR 2 LR 4 14n 1 n n 2 ; (35) where (32) follows from the denition of Cn, (33) follows from the denition of wLR2n , (34) follows from the bound of LRn. [...] Step 4 We now show that LR (LRn n ) = LM a n( LM n ) + op(1), that is, that 81 > 0,82 > 0, there exists N such that for all n > N , P LR (LR) LMa(LM n n n n ) < 1 > 1 2: Letting n 1 4 1 G = n 1 LR LR ; jn 1 LR 1 2 n 2 2n n n LRLRn n j; n 2 1 LRn LRn1 ;o n 1 LR LM 4 1 1 2 2 2 n n ; jn 1 LRn LMLMn n j; n 2 1 LR LMn n ; we know that max fGng = Op(1), so that for 2 > 0 there exists M. [...] Therefore, 1 h i2 21 Gdn(1;m1)2 G d n(2;m2) pH4 1 1 4 1 2 1 1 1 2 1 4 1 2 (79)n 36 8 1 2 36 2 8 22 2 + pH3 1 1 1 11 1 1 2 22Xn 2 2 (80) 2 1 p1 @ 6Ln (0; 0; 1) n o 2+ 1ij 1ij (81) i!j! n 1 2X @@ i@j 1 1 2 2 4 i+j 3;j 1 6 21 p1 @ Ln (; ; 1) n o + sup 22i2j + 22i2j i!j! n @@i@j 1 1 1 2 2 2 ; (82) i+j=5 jj;jj where 1 = (1;m1), 1 = (1;m1), 2 and 2 are dened in the same way. [...] To be more specic, for ( ;m) 2 B 1, we have 1 1 4 1 2 1 11 1 1 1 2 4 2 22 = 1 1 = 036 8 36 8 1 1 1 11 11 1 2 22 = 8(m< 1 m2) 2 2 2 = m i1j1 m i1 j1 j j : 1 1 1 2 2 2 if i 1 1i 1 i1 1 1 2 22 = ;2= 11 j1 1 j 2 2 sup (1 + 2) if i = 0 46 and the same applies to 22i2j" 1 1 1.