3.2 Verify the formula var {n i=1 ai XI(ti ) } = n i,j=1 ai aj ????(ti...
Question:
3.2 Verify the formula var {∑n i=1 ai XI(ti
)
}
= −∑n i,j=1 ai aj
????(ti − tj
)
for an intrinsic random field {XI(t)} of order 0 with semivariogram ????(h), where a(n × 1) is an increment vector of order 0, i.e. ∑ ai = 0.
Hint: Write ∑ ai XI(ti
) = ∑ ai
[XI(ti
) − XI(t0)] for any other site t0, and using
(3.3) show that var {∑ai XI(ti
)
}
= var {∑ai
[XI(ti
) − XI(t0)]}
= ∑
i,j ai aj cov{[XI(ti
) − XI(t0)], [XI(tj
) − XI(t0)]}
= ∑ai aj
{????(ti − t0) + ????(tj − t0) − ????(0) − ????(ti − tj
)}
= −∑ai aj
{????(ti − tj
)}.
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