We have a sequence $\lbrace b \rbrace$ which is defined in a relation $b_i = b_{i-1} * 0.9 + t_i * 0.1$. If we have observations of the full sequence of $\lbrace b \rbrace$, it would be quite simple to calculate the values of $\lbrace t \rbrace$. However, the case is that we only have around $50\%$ to $80\%$ observations in $\lbrace b \rbrace$ and we still want to calculate the values of $\lbrace t \rbrace$ as accurately as possible. Are there any well-known mathematical methods to solve this problem? Thanks!
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Between $b_i$ and $b_j$ you have $$b_j=0.9^{j-i}b_i +0.1 t_{j} +0.09 t_{j-1} + \cdots + 0.1 \times 0.9^{j-i+1} t_{i+1}$$ If you have no other information for this gap, one approach would be to make all the values of $t$ in this gap equal (though, as whuber says, there are other approaches). If so, you get $$ t = \frac{b_j -0.9^{j-i}b_i}{1-0.9^{j-i}}. $$ This approach minimises the variance of $t$ and so may underestimate variations in the values of $t$. It may be unsuitable if you think there are patterns in $t$, such as a trend. |
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