Explicit expressions for the variogram of first--order intrinsic autoregressions

Exact and explicit expressions for the variogram of first--order intrinsic autoregressions have not been known. Various asymptotic expansions and approximations have been used to compute the variogram. In this note, an exact and explicit expression applicable for all parameter values is derived. The expression involves Appell's hypergeometric function of the fourth kind. Various particular cases of the expression are also derived.


Introduction
Let {X u,v : u, v ∈ Z} be a homogeneous first-order intrinsic autoregression on the two-dimensional rectangular lattice Z 2 [11], [3] with generalized spectral density function for w ∈ (−π, π), η ∈ (−π, π) and the conditional expectation structure where a > 0, b > 0, a We can assume without loss of generality that κ = 1 and that {X u,v } is Gaussian, [4].It follows that the difference X u,v − X u+s,v+t has a well defined distribution with zero mean and lag (s, t) variogram The computation of (2), both analytically and numerically, has been a subject of considerable interest.The symmetric case a = b = 1/4 was considered by McCrea and Whipple [13], Spitzer [18] and Besag and Kooperberg [3].Besag and Mondal [4] derived explicit expressions for ν s,s and ν s,0 for the asymmetric case with the latter expressed as a finite sum of incomplete beta functions.Duffin and Shaffer [9, Theorem 4] and Besag and Mondal [4, Theorem 2] provided asymptotic expansions for ν s,t in terms of r = √ 4bs 2 + 4at 2 for the symmetric and asymmetric cases.Their approach was to exploit the recurrence equation with respect to integer time variables s and t [4, Eq. ( 4)].See also [2] and [11].
The aim of this note is to derive an explicit expression for (2) for general s, t, a and b.Our approach is quite different -we attack the problem varying a and b to find the explicit formula for ν st (a, b).
A transformation formula between F 4 and the Appell's hypergeometric function F 2 defined by This formula has been proved earlier (in equivalent form) in [1, p. 11 Proof.By equation 3.915(2) in [10], we can write where i = √ −1 and J s (x) denotes the Bessel function of the first kind of order s.Setting we can rewrite (5) as Consider now for instance the Laplace-transform formula 3.12.15(20) in [16], when The result of the theorem follows from ( 6) and ( 8).
Remark 1.The explicit expression in (4) has some applicability.Although one assumes that |a| + |b| = 1 2 in (1) the case |a| + |b| < 1 2 has been of interest.For instance, see [3,Example 3.3] quoting a result by Kempton & Howes on a four-neighbour auto-normal scheme with the maximum likelihood estimates a = 0.4848 and b = 0.0132, i.e. a + b = 0.4980 < 1 2 .Theorem 2. For all a ∈ 0, 1  2 we have Proof.Consider the Laplace transform (7) for p = (1 − θ) −1/2 and λ = 0.The resulting Appell's function F 4 obviously converges on the edge a + b = 1 2 of the convergence region.Considering that ν st is a difference of two F 4 terms, the result follows by Abel's summation method.
Remark 2. The limit in (9) gives an approximation formula, viz where the quality of the approximation is controlled by suitably chosen small θ > 0.

The symmetric case a
Here, we present technical details to calculate ν s,t . Using Burchnallformula [21, §9.4,Eq. ( 149)], we can transform The asymptotics of generalized hypergeometric series p+1 F p , p ≥ 3 have been studied by Bühring and Srivastava [6], Saigo and Srivastava [19] and by A.K. Srivastava [20].Since we are faced with we have a hypergeometric term in [6] for p = 3: valid for θ → 0+.Here and where γ = −Ψ(1) denotes the Euler-Mascheroni constant and Ψ(x) = d dx ln Γ(x) denotes the digamma function.The B in ( 12) can be represented by various forms, see (4.6), (4.7), and (4.12) in [6].An example is [6, Eq. (4.7)]: An advantage of this formula is that it is a single infinite series of hypergeometric terms and the series for each hypergeometric term is finite because of the −k in the numerator.Combining ( 11), ( 13) and ( 14), we can write L st := L, B st := B and Γ st := Γ as and for all (s, t) ∈ N 2 0 .Theorem 3.For all s, t ∈ N 0 we have Proof.In (17) assume that s + t is even.So, applying properties of digamma function Ψ to expression (17), we conclude Ψ 1 2 (s + t + 1) + Ψ For odd s + t repeat this procedure.