Many researchers used time series (rainfall, temperature etc.) with varying length of records (say 30, 40, 50, 75, 100, 135 years) for analysing trends using well known Mann-Kendall test. Is it appropriate to use any length or is there any metric that can help in deciding optimal length of the time series for MK test.
You can go through the articles/paper
Trend analysis of annual and seasonal rainfall time series
in the Mediterranean area
Using Mann-Kendall and Time Series Techniques for Statistical
Analysis of Long-Term Precipitation in Gorgan Weather Station
Trend Analysis
Precipitation Trends over Time Using Mann-Kendall and Spearman’s rho Tests in Swat River Basin, Pakistan
Based on the calculation principle of MK test, I think the core step of the test is the hypothetical test of the rank. And the hypothetical test of the rank must obey the Law of Large Number. Accordithng to the Law of Large Number, the larger the sample size is, the better the hypothetical test is. And in hydrology and water resources, the sample size should be larger than 30. So I think the length of the time series shoule be longer than 30.
The empirical studies have proved the reliability of samples over 30-40 years. Actually there is not any formula, it is an experimental fact.
One should be aware that a trend analysis is implicit in many econometric regressions. E.g. the constant in a dln-regression shows you the annual unexplained change of the endogenuous variable. Therefore, I would not be so strict what concerns the minimum length of a time series. The difference lies mainly in the interpretation of the trend. If it is based on a rather short time span, there is a higher probability of a trend break in the short-run, but one could also never exclude such a change (in this case slower, but undetected for a longer time) for trends calculated on long-term series. The problem is that a trend is no expanation but a technical assumption (which in many cases cannot be avoided).
Geneally, in water resources hydrologists usually go with 30 years as length of a time series. But I believe that the larger the time series the better the outcomes. However if you have a record of more than 100 years, it could be a beneficial to explore the trend variartion ( say for each 50 years) and hence you will be able to understand which period is responsible for the variation.
I think time series is concerned with the intervals of observations where at least fifty observations may work
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