Just a short notice here: some of the above methods do not take into account long-term persistence (aka long-range dependence). See for example:

Article Trend Detection in Hydrologic Data: The Mann-Kendall Trend T...

The MK test is the most reliable non-parametric method for trend estimation. For the second question, I recommend a book in which you can find your answers:

- Wilks D., 2011, Statistical Methods in the Atmospheric Sciences

Mann whitney

Note also that the Mann Kendall trend test assumes no serial correlation -- depending on what climate variables you are working with (and at what timescale), this assumption may not be valid. You can use block bootstrapping to deal with this issue (check https://www.researchgate.net/publication/258648184_Block_bootstrap_for_Mann-Kendall_trend_test_of_serially_dependent_data for example)

Furthermore, if your data is collected seasonally, check for the seasonal Kendall test.

I suggest checking chapter 12 of https://www.researchgate.net/publication/44429794_Statistical_Methods_in_Water_Resources and also reading https://www.researchgate.net/publication/224839899_Non-Parametric_Trend_Test_for_Seasonal_Data_With_Serial_Dependence

All the best,

David

The main answers to your questions are already given. However, I would like to point also to the Chow-Test, which allows you to identify structural breaks in your series. Fitting a linear trend, the selection of the period always matters ...

Just a short notice here: some of the above methods do not take into account long-term persistence (aka long-range dependence). See for example:

Article Trend Detection in Hydrologic Data: The Mann-Kendall Trend T...

You can use modified Mann Kendall method take take into account long term persistent.This will give you more realistic results. You can check this article in which a comparison was shown between mk and mmk.

Article Unidirectional trends in annual and seasonal climate and ext...

In addition to Hamed's paper suggested by Yannis Markoins above, which is a must-read, I am listing three papers in chronological order. The first one (2003) studied the problem and proposed a practical solution based on Monte Carlo simulation (my students still prefer this method). Further, I would highlight the following statement from the second paper (Cohn and Lins, 2005, "Nature’s style: Naturally trendy"): "From a practical standpoint, however, it may be preferable to acknowledge that the concept of statistical significance is meaningless when discussing poorly understood systems."

Article Climate Change, the Hurst Phenomenon, and Hydrological Statistics

Article Nature's style: Naturally trendy

Article Statistical Analysis of Hydroclimatic Time Series: Uncertain...

The Spear Mann correlation test also the non-parametric rank based method for testing linear trend in time series.

We don't have a conceptual model for a time series, as clear understanding about a time as the property or characteristic of reality.

"Any model is a reflection on the researcher interaction with a reality" - this is by memory reference from The Philosophy of Statistics by Dennis V. Lindley Journal of the Royal Statistical Society. Series D (The Statistician), Vol. 49, No. 3 (2000), pp. 293-337

Natural system is much more complex that applied engineering model. Researcher of natural system has to start looking at:

what system is under consideration,

the scale of measurements were done

the dimensions involved in the model for measurement

and something more

there is spearman rho test which can be used. however, use of modified MK or MK with prewhitening may be preferred.

you may look at publication titled

when to prewhiten and when not to prewhiten?

Please check the link below, might be helpful for you.

https://cran.r-project.org/web/packages/trend/vignettes/trend.pdf

Singular spectrum analysis