In der Praxis ist die Verwendung eines Standard-T-Tests zur Überprüfung der Signifikanz eines linearen Regressionskoeffizienten gängige Praxis. Die Mechanik der Berechnung macht für mich Sinn.
Warum kann die T-Verteilung verwendet werden, um die Standardteststatistik zu modellieren, die beim Testen von linearen Regressionshypothesen verwendet wird? Standardteststatistik, auf die ich mich hier beziehe:
regression
hypothesis-testing
linear-model
t-distribution
Nate Parke
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Antworten:
To understand why we use the t-distribution, you need to know what is the underlying distribution ofβˆ and of the Residual sum of squares (RSS ) as these two put together will give you the t-distribution.
The easier part is the distribution ofβˆ which is a normal distribution - to see this note that βˆ =(XTX)−1XTY so it is a linear function of Y where Y∼N(Xβ,σ2In) . As a result it is also normally distributed, βˆ∼N(β,σ2(XTX)−1) - let me know if you need help deriving the distribution of βˆ .
Additionally,RSS∼σ2χ2n−p , where n is the number of observations and p is the number of parameters used in your regression. The proof of this is a bit more involved, but also straightforward to derive (see proof here Why is RSS distributed chi square times n-p?).
Up until this point I have considered everything in matrix/vector notation, but let's for simplicity useβˆi and use its normal distribution which will give us:
Additionally, from the chi-squared distribution ofRSS we have that:
This was simply a rearrangement of the first chi-squared expression and is independent of theN(0,1) . Additionally, we define s2=RSSn−p , which is an unbiased estimator for σ2 . By the definition of the tn−p definition that dividing a normal distribution by an independent chi-squared (over its degrees of freedom) gives you a t-distribution (for the proof see: A normal divided by the χ2(s)/s−−−−−−√ gives you a t-distribution -- proof) you get that:
Wheres(XTX)−1ii−−−−−−−−√=SE(βˆi) .
Let me know if it makes sense.
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The answer is actually very simple: you use t-distribution because it was pretty much designed specifically for this purpose.
Ok, the nuance here is that it wasn't designed specifically for the linear regression. Gosset came up with distribution of sample that was drawn from the population. For instance, you draw a samplex1,x2,…,xn , and calculate its mean x¯=∑ni=1xi/n . What is the distribution of a sample mean x¯ ?
If you knew the true (population) standard deviationσ , then you'd say that the variable ξ=(x¯−μ)n−−√/σ is from the standard normal distribution N(0,1) . The trouble's that you usually do not know σ , and can only estimate it σ^ . So, Gosset figured out the distribution when you substitute σ with σ^ in the denominator, and the distribution is now called after his pseduonym "Student t".
The technicalities of linear regression lead to a situation where we can estimate the standard errorσ^β of the coefficient estimate β^ , but we do not know the true σ , therefore Student t distribution is applied here too.
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