# Mathematical definition of Infill Asymptotics

I am writing a paper that uses infill asymptotics and one of my reviewers has asked me to please provide a rigorous mathematical definition of what infill asymptotics is (i.e., with math symbols and notation).

I can't seem to find any in the literature and was hoping someone could either point me in the direction of some or provide me with a self-written definition.

If you are unfamiliar with infill asymptotics (also called fixed domain asymptotics) they are the following: Infill asymptotics are based on observations that get increasingly dense in some fixed and bounded region as their number increases.

Stated otherwise, infill asymptotics is where more data is collected by sampling more densely in a fixed domain.

I've already looked at Stein 1999 and Cressie 1993 but nothing "mathematically" rigorous there.

Here is the quoted passage from my paper.

Therefore, it is important to recognize the kind of asymptotics we are dealing with. In our case, the asymptotics we deal with are based on observations that get increasingly dense in some fixed and bounded region as their number increases. These types of asymptotics are known as fixed-domain asymptotics (Stein, 1999) or infill asymptotics (Cressie, 1993). Infill asymptotics, where more data are collected by sampling more densely in a fixed domain, will play a key role in helping us develop an argument for...

Impotrant to note, I am sampling my observations using Latin hypercube sampling.

Here is what Cressie's book has to say about infill asymptotics.

• Section 5.8, Infill Asymptotics, of the first (1991) edition of Cressie's book is clear. Although it does not provide a definition in mathematical notation, an example (of asymptotics that are "more delicate than infill") is explicitly given two pages later using mathematical notation. Could you perhaps quote your own paper's description of "infill asymptotics"? – whuber Dec 11 '15 at 15:15
• @whuber I added the quote to the original question – user95564 Dec 11 '15 at 15:56
• Thank you. That quotation does not seem to be sufficiently specific. How, exactly, do you go about sampling the fixed domain? An example (offered by Cressie) is that you sample one point and then, forever after, sample in a cluster around a different point. That would likely have different asymptotic behavior than sampling with a homogeneous Poisson process, for instance. – whuber Dec 11 '15 at 16:58
• @whuber I am using Latin hypercube samples. – user95564 Dec 11 '15 at 17:00
• Please include that information in your question, because it's crucial for the answer. – whuber Dec 11 '15 at 17:04

The definition of infill asymptotics is not particularly useful (technically, if the domain stays fixed and sample size rises, that is infill asymptotics. But consider the case where you sample on a transect from 0 to 1, taking one sample in 0,1/2, another sample in 1/2,3/4, another in the interval 3/4, 7/8, etc. You will be able to say a lot about the values at 1, but won't be able to say much else.)

To get typical result in infill asymptotics, you need a design with properties such as: for all subregions of area $\epsilon$, for any $\epsilon>0$, the probability of a sample occurring in the subregion approaches 1 as $n\rightarrow\infty$. Such a sample is dense in the domain.

Sometimes the infill is not given explicitly, only a design is given. For instance, in the paper by Lahiri (On Inconsistency of Estimators Based on Spatial Data under Infill Asymptotics), he describes a design that is essentially a 'jittered' grid (some randomness as the small level, but generally based on sampling in hyper rectangular subregions) that is asymptotically dense in the fixed domain. He obtains the result (common for infill problems) that most variogram parameters are estimated inconsistently.

Lahiri, Lee and Cressie (On asymptotic distribution and asymptotic efficiency of least squared estimators of spatial variogram parameters, J.StatPlanInf 2002, vol. 103, pp. 65-85) similarly consider infill grids that become systematically more closely spaced, again, yielding a dense sample.

(The general result for dense samples is that since infill asymptotics really is a single realization of a spatial process, the only parameter of the (super population) true variogram that can be consistently estimated is the slope at zero, but predictions are increasingly good.)

• Do you know how to prove this statement? "for all subregions of area ϵ, for any ϵ>0, the probability of a sample occurring in the subregion approaches 1 as n→∞. Such a sample is dense in the domain." – user95564 Dec 11 '15 at 21:55
• It's almost the definition of 'dense'. A dense sequence has a limit point at all locations. So if you pick any location in the region and any disc of radius $\epsilon$ around it, since this disc is a subregion, you'll eventually get an observation in it. (NOTE: I'm assuming the domain is some reasonable polygon and not some pathological topological oddity...) Note that lots of processes yield dense sequences, including Poisson spatial processes (with intensity > 0 everywhere), ergodic sequences, simple random samples and grids. – AlaskaRon Dec 11 '15 at 23:00
• Do you know of any papers that say that Latin Hypercubes are asymptotically dense? – user95564 Dec 11 '15 at 23:24

Let's start with a definition of Latin Hypercube sampling, just to make things perfectly clear and establish a notation. Then we can define infill asymptotics.

## LHS

Latin Hypercube Sampling of a box $\mathcal{B}=[l_1,u_1)\times [l_2,u_2)\times \cdots [l_d,u_d) \subset \mathbb{R}^d$ proceeds by dividing each dimension into $N \ge 1$ parts of equal lengths $\delta_i(N) = (u_i-l_i)/N$, thereby partitioning it into $N^d$ cells

$$c_N(i_1,i_2,\ldots, i_d) = [l_1 + i_1\delta_1(N), l_1 + (i_1+1)\delta_1(N))\times \cdots [l_d + i_d\delta_d(N), l_d + (i_d+1)\delta_d(N)),$$

where $0 \le i_j \lt N$ for each index $j$.

Sampling occurs by first selecting $N$ such cells $S=\{c_N(i_1^1, \ldots, i_d^1), \ldots, c_N(i_1^N, \ldots, i_d^N)\}$ uniformly, independently, and without replacement from the collection of all such cells in such a way that

$$\{i_j^1, i_j^2, \ldots, i_j^N\}=\{1, 2, \ldots, N\},\ j=1, 2, \ldots, d.$$

(This is the $d$-dimensional generalization of the $2$-dimensional situation where "there is only one sample in each row and each column.") Each of the $N$ cells in $S$ is then sampled at a location chosen uniformly and independently among all points in the cell, producing a set of $N$ ordered pairs $$X(N)=\{(Z_1^N,Y_1^N), \ldots, (Z_N^N,Y_N^N)\}$$ of (location, observation) values.

## Infill Asymptotics

Presumably, some procedure $t_N$ is applied to each Latin Hypercube sample $X(N)$ of size $N$ of a fixed box $\mathcal{B}$, yielding an estimate $t_N(X(N))$ for each $N$. This results in a sequence

$$t_1(X(1)), t_2(X(2)), \ldots, t_N(X(N)), \ldots$$

of random variables. Infill asymptotics refers to the behavior of this sequence as $N$ grows without bound.