According to the identifiability results in theorems and the asymptotic property of the M -estimator (Chapter 5.1 of [32]), (ˆ p , ˆ π ) converges weakly to the true parameter. Furthermore, according to chapter 5.3 [32], the MLE is asymptotically normally distributed. Thus, for each j , α , and y , we have We say a random sequence a$_{n}$ = O$_{p}$ ( n − 1 / $^{2}$) if √$_{na}$$_{n}$ is tight. Notice that the identifiability is subject to a permutation of the latent class labels. To simplify notation, we assume that the class labels of ˆ p y j α have been arranged in an appropriate order. Otherwise, we need to write ˆ p y j α − p y j α = O$_{p}$ ( n − 1 / $^{2}$). Thus, we proceed assuming that the permutation λ is identity. We now construct an estimator of the partial information for each item. The basic idea is that if [ α $_{1}$]$_{j}$ = [ α $_{2}$]$_{j}$ , then p y j $^{α}$1 = p y j $^{α}$2 for all y . Together with (13), we have that Based on this fact, we define an equivalent class such that Based on (14), we have that as n → ∞ . If [ α $_{1}$]$_{j}$ = [ α $_{2}$]$_{j}$ , then there exists an ε and y such that (ˆ p y jλ ( α $_{1}$) − ˆ p y jλ ( α $_{$_{2}$)}$) 2 > ε and thus as n → ∞ . Let “ 〈 〉 $_{j}$” be the canonical map of the estimated equivalence class as in (15). Based ˆ p y j α − p y j α = O$_{p}$ ( n − 1 / $^{2}$) . (13) d$_{j}$ ( α $_{1}$, α $_{2}$) = k$_{j}$ ∑ y =1 (ˆ p y jλ ( α $_{1}$) − ˆ p y jλ ( α $_{$_{2}$)}$) 2 = O$_{p}$ ( n − $^{1}$) . (14) α 1 j = α 2 if d$_{j}$ ( α $_{1}$, α $_{2}$) ≤ n − 1 / $^{2}$. (15) P ( α 1 j = α $_{2}$) → 1 P ( α 1 j = α $_{2}$) → 1 40