The main assumption is that T(t) as a function of time (t) is a sum of a zero-mean unknown "noise" U(t) and a linear trend a+b*t. We are estimating the trend coefficient "b". For simplicity, we may also assume that the "noise" is a Gaussian random process with zero mean and some fixed, stationary auto-correlation function.
The least-squares estimator for the trend coefficient "b" is a linear function of T(t). So, the trend estimator is then also a Gaussian distributed value with some (possibly nonzero) mean and some standard deviation. Our goal is to compute the mean and the standard deviation of the estimator for "b". This was the main goal in my Fourier-based analysis that I pointed out before.
The assumption of Gaussian or normal distributions is not at all important. We will conclude that the trend is nonzero not because the distribution is not normal or not Gaussian, but because the mean value is far enough from zero so that it can't be just a natural fluctuation. The mean value and the standard deviation are defined just as well for non-Gaussian distributions.
Even if the distribution of noise is not Gaussian, we still assume that it has zero mean (by definition, it is "noise" and has no trend). So, we can still assume that U(t) has some fixed auto-correlation function. The estimator for "b" is still a linear function of T(t), so we can compute the standard deviation of b.
The usual procedure is to require that some value is beyond 2 sigma away from zero. The null hypothesis is that there is zero trend. We can refute the null hypothesis at high confidence if we show that the mean of "b" is larger than 2 sigma.
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timeasmymeasure
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Date: 2020-01-08 02:41 pm (UTC)The least-squares estimator for the trend coefficient "b" is a linear function of T(t). So, the trend estimator is then also a Gaussian distributed value with some (possibly nonzero) mean and some standard deviation. Our goal is to compute the mean and the standard deviation of the estimator for "b". This was the main goal in my Fourier-based analysis that I pointed out before.
The assumption of Gaussian or normal distributions is not at all important. We will conclude that the trend is nonzero not because the distribution is not normal or not Gaussian, but because the mean value is far enough from zero so that it can't be just a natural fluctuation. The mean value and the standard deviation are defined just as well for non-Gaussian distributions.
Even if the distribution of noise is not Gaussian, we still assume that it has zero mean (by definition, it is "noise" and has no trend). So, we can still assume that U(t) has some fixed auto-correlation function. The estimator for "b" is still a linear function of T(t), so we can compute the standard deviation of b.
The usual procedure is to require that some value is beyond 2 sigma away from zero. The null hypothesis is that there is zero trend. We can refute the null hypothesis at high confidence if we show that the mean of "b" is larger than 2 sigma.