Inference for the Analysis of Ordinal Data with Spatio-Temporal Models

Let s m , l ∈ R m and t ∈ T such that t  ≥ 1. Besides, let f ( ⋅ ; β ) denote the conditional probability mass function of Z ( s m , l , t ) | Z ( s m , l , t − 1 ) , Z ( A ( s m , l ) , t − 1 ) . Then, for j = 1, 2, we see that

Proof.

Since the sums are over a finite support, the sum and the derivative commute. Also, since f ( ⋅ ; β ) is a density with its support contained in S = {1,…, E}, the sum on the second expression is always 1. Therefore, as the derivative of a constant vanishes, we have concluded this proof.   ■

Consider the log-likelihood given by (16) and suppose that the true value of the parameter β is β 0 . For each m ∈ N and k = 1 , 2 , … , ( τ + 1 ) | R m | , let M m , k = ∑ i = 1 k b ( m ) ( s m , i ∗ ; β 0 ) and W m , k = ∑ i = 1 k A ( m ) ( s m , i ∗ ; β 0 ) − E ( A ( m ) ( s m , i ∗ ; β ) ∣ F n , i − 1 ) , where s m , i ∗ ∈ Δ m . Then, { M m , k , F m , k } and { W m , k , F m , k } are triangular martingale arrays

Consider the log-likelihood function given by (16) and suppose the true value of the parameter β is β 0 . Assume that there exists ε₀  > 0  such that

and that X m , t − 1 , e X m , t − 1 , e T is positive definite for any m ∈ N , t = 1,…, τ and e = 1 , . . . E − 1 . Then, the following holds:

1. sup | Δ m | ∥ β − β 0 ∥ ≤ 1 1 | Δ m | ∑ i = 1 | Δ m | A ( m ) ( s m , i ∗ ; β ) − A ( m ) ( s m , i ∗ ; β 0 ) → P 0 , as m → ∞ .

2. sup m , i , j E b j ( m ) ( s m , i ∗ ; β 0 ) q < ∞ , for some q > 1.

3. P − 1 | Δ m | ∑ i = 1 | Δ m | c T A ( m ) ( s m , i ∗ ; β 0 ) c > ϵ → 1 , as m → ∞ , for any vector c ∈ R ( E − 1 ) × ( E + 1 ) with ∥ c ∥ = 1 , and for some ε > 0.

4. P − 1 | Δ m | ∑ i = 1 | Δ m | A ( m ) ( s m , i ∗ ; β ) i s p o s i t i v e − d e f i n i t e f o r a l l β → 1 , as m → ∞ .

Proof.

(i). For e = 1,…, n – 1, let A e ( m ) ( s m , i ∗ ; β ) be given by

that is, A e ( m ) ( s m , i ∗ ; β ) correspond to diagonal matrices on A ( m ) ( s m , i ∗ ; β ) . For e = 1, …, E – 1, direct computations imply that

where we assumed that s m , i ∗ ≡ s m , l , t and u ≡ x s m , l , t − 1 . Observe that β 0 can be decomposed as β 0 = ( β 1 0 , β 2 0 , … , β E − 1 0 ) , where β i 0 is a vector of dimension E + 1; then, we can write

Since G ^” is bounded, using the Mean Value Theorem and the Cauchy-Schwarz inequality, we deduce that there exists C > 0 such that

Besides, since u ≡ x s m , l , t − 1 counts the number of adjacent sites in the previous time that are in each state, there exists M > 0 with 0 < [ u ] j , [ u T u ] i j < M . Therefore, there exists a constant K > 0 such that

which implies

For c, e = 1…, E – 1 such that e ≠ c, we see that

Therefore, from eq. (33) it follows that

which implies the desired result (i).

(ii) Computing b j ( m ) ( s m , i ∗ ; β ) explicitly and repeating some arguments from above, it is easy to check that b j ( m ) ( s m , i ∗ ; β ) is bounded by a constant that is independent of the parameters m, i, j and β . This implies

for any q > 1.

(iii) From eq. (34), it follows that

Then, if c = ( c 1 , … , c E − 1 ) for c i ∈ R E + 1 , we obtain that

where we assumed that s m , k ∗ ≡ s m , l , t and u ≡ x s m , l , t − 1 .

If the random field Z is at state e at time t, in the following time it can only transition to the next state e + 1 or stay in the same state. Thus,

Since G’> 0 and [ x s m , i ∗ , t − 1 ] i , j is bounded, there exists a constant g ≡ g ( β 0 ) > 0 , independent of m, such that

Besides, X m , t − 1 , e X m , t − 1 , e T is positive definite for m ∈ N , t = 1,…, τ and e = 1 , . . . E − 1 ; which implies

Therefore, if ∥ c ∥ = 1 , we see that

Thus, from eq. (32) it follows that

for any vector c ∈ R ( E − 1 ) × ( E + 1 ) with ∥ c ∥ = 1 .

(iv) Let g ( β ) > 0 be a positive constant such that G ′ ( x s m , i ∗ , t − 1 T β ) > g ( β ) . Notice that the bounds we found in the proof of (iii) can be applied for any β , namely

for any vector c ∈ R ( E − 1 ) × ( E + 1 ) with c ≠ 0 . This implies the desired result.   ■

Theorem 10.

Consider the log-likelihood function l m ( β ) given by (16) and suppose the true value of the parameter β is β 0 . Assume that there exists ε₀ > 0 such that

and that X m , t − 1 , e X m , t − 1 , e T is positive definite for any m ∈ N , t = 1,…, τ and e = 1 , . . . E − 1 . Let β ˆ m be the MLE estimator of l m ( β ) . Then, β ˆ m exists and is consistent; that is, β ˆ m satisfies β ˆ m → P β 0 .

Remark 11.

Under the conditions of Lemma 9, in the proof of such Lemma, we can observe that − 1 | Δ m | ∑ i = 1 | Δ m | A ( m ) ( s m , i ∗ ; β ) is positive definite for all β and m ∈ N . Then, we can establish the existence of the MLE β ˆ m for any m. Under the conditions of Theorem 1 in Huang and Cressie [19], it is only possible to guarantee the existence of the MLE asymptotically.

Theorem 12.

Assume the conditions of Lemma 9 hold. Define

m ∈ N . Assume that, for all ∥ c ∥ = 1 ,

as m → ∞ . Then, as m → ∞ , B m 1 / 2 ( β ˆ m − β 0 ) → d N ( 0 , I ) .

Remark 13.

In Huang and Cressie [19], they ask that condition (ii) in Lemma 9 be valid for q > 2. In the proof of Lemma 9, we checked that, for our model, it is valid for any q > 1.