11 Theorem 8.2:Suppose that Q(0;Xn)is continuous in 0 and there exists a function Qo(0)such that 1.Qo(0)is uniquely maximized at 0o 2.Θis compact 3.Qo(0)is continuous in 0 4.Q(0;Xn)converges uniformly in probability to Qo(0). then (Xn)defined as the value of aee which for each Xn=n maximizes the objective function Q(;Xn)satisfies 0(Xn)00.11 Theorem 8.2: Suppose that Q(θ; Xn) is continuous in θ and there exists a function Q0(θ) such that 1. Q0(θ) is uniquely maximized at θ0 2. Θ is compact 3. Q0(θ) is continuous in θ 4. Q(θ; Xn) converges uniformly in probability to Q0(θ). then ˆ θ(Xn) defined as the value of θ ∈ Θ which for each Xn = xn maximizes the objective function Q(θ; Xn) satisfies ˆ θ(Xn) P→ θ0