ative. This part can be shown by using a Cauchy-Schwarz inequality. N ∑ i =1 (√ θ$_{i}$e$^{ω}$ i µ ) 2 N ∑ i =1 (√ θ$_{i}$ω 2 i e$^{ω}$ i µ ) 2 ≥ [ N ∑ i =1 √ θ$_{i}$e$^{ω}$ i µ · √ θ$_{i}$ω 2 i e$^{ω}$ i µ ] 2 (80) ∴ ∂ $^{2}$G ∂µ 2 is always nonnegative. Second, we show that det ∇ $^{2}$G ≥ 0 . From (77), we calculate ∂ $^{2}$G ∂ω$_{k}$∂µ where k ∈ { 1 , 2 , ..., N } . Similarly, 46 ∂ $^{2}$G ∂ω$_{k}$∂µ = ∂ ∂ω$_{k}$ [ ln ( N ∑ i =1 θ$_{i}$e$^{ω}$ i µ ) − 1 µ ∑ N i $_{=1}$θ$_{i}$ ω$_{i}$e$^{ω}$ i µ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ] = θ$_{k}$ 1 $_{µ}$e$^{ω}$ k µ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ − 1 µ ( ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ) · ∂ ∂ω$_{k}$ θ$_{k}$ω$_{k}$e$^{ω}$ k µ − θ$_{k}$ µ e$^{ω}$ k µ ( ∑ N i $_{=1}$θ$_{i}$ ω$_{i}$e$^{ω}$ i µ ) [ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ] 2 = θ$_{k}$ 1 $_{µ}$e$^{ω}$ k µ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ − 1 µ ( ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ) ( θ$_{k}$e$^{ω}$ k µ + θ$_{k}$ω$_{k}$ µ e$^{ω}$ k µ ) − θ$_{k}$ µ e$^{ω}$ k µ ( ∑ N i $_{=1}$θ$_{i}$ ω$_{i}$e$^{ω}$ i µ ) [ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ] 2 = θ$_{k}$ 1 $_{µ}$e$^{ω}$ k µ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ − θ$_{k}$e$^{ω}$ k µ + θ$_{k}$ω$_{k}$ µ e$^{ω}$ k µ µ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ + θ$_{k}$e$^{ω}$ k µ ( ∑ N i $_{=1}$θ$_{i}$ ω$_{i}$e$^{ω}$ i µ ) µ 2 [ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ] 2 = − θ$_{k}$ω$_{k}$e$^{ω}$ k µ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ µ 2 [ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ] 2 + θ$_{k}$e$^{ω}$ k µ ( ∑ N i $_{=1}$θ$_{i}$ ω$_{i}$e$^{ω}$ i µ ) µ 2 [ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ] 2 = θ$_{k}$e$^{ω}$ k µ ( N i $_{=1}$θ$_{i}$ ω$_{i}$e$^{ω}$ i µ − ω$_{k}$ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ) µ 2 [ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ] 2 = θ$_{k}$e$^{ω}$ k µ ∑ N i $_{=1}$θ$_{i}$ ( ω$_{i}$ − ω$_{k}$ ) e$^{ω}$ i µ µ 2 [ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ] 2 (81) ∂G ∂ω$_{k}$ = ∂ ∂ω$_{k}$ µ ln ( N ∑ i =1 θ$_{i}$e$^{ω}$ i µ ) = µ θ$_{k}$ 1 $_{µ}$e$^{ω}$ k µ ∑ N i $_{=1}$¯ θ$_{i}$e$^{ω}$ i µ = θ$_{k}$e$^{ω}$ k µ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ