2. Model

Jan Kowalski and Jacek Wesołowski

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Let ${\left(X_{i\mathrm{,}j}\right)}_{i\mathrm{,}j\in \mathbb{Z}}$  be a doubly infinite matrix of random variables. Heuristically, $X_{i\mathrm{,}j}$  represents the value of variable $\mathcal{X}$  measured for the unit (rotation group) $i$  on the occasion $j.$  We assume that the expectation of $X_{i\mathrm{,}j}$  depends only on the occasion and not on the unit, that is

$$\mathbb{E}{X}_{i\mathrm{,}j}={\mu }_j\mathrm{,}\forall i\mathrm{,}j\in \mathbb{Z}\mathrm{.}$$

Moreover, we assume exponential in time correlations between $X_{i\mathrm{,}j}’\text{s}$  for the same unit and no correlations between different units (following Patterson (1950) model), that is

$$ℂ\text{ov}\left(X_{i\mathrm{,}j}\mathrm{,}{X}_{k\mathrm{,}l}\right)={\rho }^{|j-l|}{\delta }_{i\mathrm{,}k}\forall i\mathrm{,}j\mathrm{,}k\mathrm{,}l\in \mathbb{Z}\mathrm{,}$$

where $\left|\rho \right|\in \left(\mathrm{0,1}\right)$  and ${\delta }_{i\mathrm{,}k}=1$  if $i=k,$  otherwise ${\delta }_{i\mathrm{,}k}=0.$  (In practical situations often $\rho$  is in $[0,1).$  In the case $\rho =0$  observations from the past cannot improve present linear estimator of the mean, therefore we do not consider such case below.) Consequently,

$$\mathbb{V}\text{ar }{X}_{i\mathrm{,}j}=1,i\mathrm{,}j\in \mathbb{Z}\mathrm{.}$$

For any $j\in \mathbb{Z}$  we are interested in the BLUE of ${\mu }_j$  based on all available observations from occasions $i\le j.$  For a fixed positive integer $N$  denote by

$${\underset{\_}{X}}_j={\left(X_{j\mathrm{,}j}\mathrm{,}{X}_{j+\mathrm{1,}j}\mathrm{,}\dots \mathrm{,}{X}_{j+N-\mathrm{1,}j}\right)}^T$$

the maximal sample (of size $N)$  on the occasion $j\in \mathbb{Z}\mathrm{.}$  Then

$$\mathbb{E}{\underset{\_}{X}}_j={\mu }_j\underset{\_}{1}\mathrm{,}j\in \mathbb{Z}\mathrm{,}$$

where $\underset{\_}{1}={\left(\mathrm{1,1,}\dots \mathrm{,1}\right)}^T\in {ℝ}^N,$  and

$$ℂ\text{ov}\left({\underset{\_}{X}}_j\mathrm{,}{\underset{\_}{X}}_{j-k}\right)=C^k={\left[ℂ\text{ov}\left({\underset{\_}{X}}_j\mathrm{,}{\underset{\_}{X}}_{j+k}\right)\right]}^T,j\in \mathbb{Z}\mathrm{,}k\ge \mathrm{0,}$$

where $C$  is an $N\times N$  matrix of the form

$$C=\left[\begin{array}{cccc}0& \rho & & 0\\ 0& \ddots & \ddots & \\ & \ddots & \ddots & \rho \\ 0& & 0& 0\end{array}\right]\mathrm{.}$$

Note that $C^n=0$  for any $n\ge N.$

The effective sample will be defined by a cascade pattern, which is a vector $\underset{\_}{\epsilon }=$   ${\left({\epsilon }_1\mathrm{,}\dots \mathrm{,}{\epsilon }_N\right)}^T\in {\left\{\mathrm{0,1}\right\}}^N$  with ${\epsilon }_1={\epsilon }_N=1.$  Let

$$n={\displaystyle \sum _{j=1}^N{\epsilon }_j}\text{ and }h=N-n\mathrm{.}$$

Let $H$  be the set of zeros in the pattern $\underset{\_}{\epsilon },$  that is $j\in H$  iff ${\epsilon }_j=0.$  Obviously, $\#H=h.$  A gap of size $m$  is a maximal set of sequential $m$  zeros, that is a set satisfying

$$\left\{j\mathrm{,}j+\mathrm{1,}\dots \mathrm{,}j+m-1\right\}\subset H\text{ and }j-\mathrm{1,}j+m\cancel{\in }H\mathrm{.}$$

Consequently, $H$  is a union of, say, $s$  gaps of sizes $m_r,r=1,2,\dots \mathrm{,}s,$  and ${\displaystyle {\sum }_{r=1}^s{m}_r}=h.$

The coverage $p$  of the pattern (see Kowalski 2009 for equivalent definition) is the size of the largest gap increased by one:

$$p=1+\underset{1\le r\le s}{\max }{m}_r\mathrm{.}$$

On each occasion $j\in \mathbb{Z}$  we may not observe the maximal sample ${\underset{\_}{X}}_j$  but the effective sample of size $n$  defined by the cascade pattern $\underset{\_}{\epsilon },$  that is the vector

$${\underset{\_}{Y}}_j={\left(X_{j+k-\mathrm{1,}j}\mathrm{,}k\in \left\{\mathrm{1,}\dots \mathrm{,}N\right\}\backslash H\right)}^T,$$

that is values of $X_{i\mathrm{,}j}’\text{s}$  represented by zeros (gaps) in the cascade pattern $\epsilon$  are removed from the sample.

We consider BLUE ${\widehat{\mu }}_t$  of the mean ${\mu }_t$  on the occasion $t\in \mathbb{Z}$  which is based on observations ${\underset{\_}{Y}}_j,j\le t.$  That is

$${\widehat{\mu }}_t={\displaystyle \sum _{i=0}^{\infty }{\widetilde{\underset{\_}{w}}}_i^T{\underset{\_}{Y}}_{t-i}}$$

with ${\underset{\_}{\tilde{w}}}_i\in {ℝ}^n,i\ge 0,$  which minimize $\mathbb{V}\text{ar}{\widehat{\mu }}_t$  under the unbiasedness constraints

$${\widetilde{\underset{\_}{w}}}_0^T\underset{\_}{1}=1\text{ and }{\widetilde{\underset{\_}{w}}}_i^T\underset{\_}{1}=0,i\ge 1.$$

It is both obvious and crucial for our approach that, equivalently,

$${\widehat{\mu }}_t={\displaystyle \sum _{i=0}^{\infty }{\underset{\_}{w}}_i^T{\underset{\_}{X}}_{t-i}}(2.1)$$

with ${\underset{\_}{w}}_i\in {ℝ}^N,i\ge 0,$  minimizing $\mathbb{V}\text{ar}{\widehat{\mu }}_t$  under unbiasedness constraints

$${\underset{\_}{w}}_0^T\underset{\_}{1}=1,{\underset{\_}{w}}_i^T\underset{\_}{1}=0,i\ge \mathrm{1,}(2.2)$$

and cascade pattern constraints

$${\underset{\_}{w}}_i^T{\underset{\_}{e}}_j=0\forall i\ge \mathrm{0,}\forall j\in H\mathrm{,}(2.3)$$

where ${\underset{\_}{e}}_j={\left(\mathrm{0,}\dots \mathrm{,0,1,0,}\dots \mathrm{,0}\right)}^T$  (with 1 at $j^{\text{th}}$  position) is $j^{\text{th}}$  vector of the canonical basis in ${ℝ}^N,j\in H.$  Note that the constraint (2.3) actually says that $j^{\text{th}}$  entries $\left(j\in H\right)$  of vectors ${\underset{\_}{w}}_i,i\ge 0,$  are all zeros.

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