Integrating Taylor's Approximation












2














I am reviewing Casella's All of Statistics. A particular integral comes up when reviewing non-parametric statistics in particular Histograms.



Assume that some $xin B_j$ and any other $u in B_j$ where $B_j$ is some particular bin in the histogram. I don't think the frequencies are going to be too important here...



$$int_{B_j}f(u)du approx int_{B_j}f(x) + (u-x)f^{prime}(u)$$
$$= f(x)h + hf^{prime}(x) ( h (j-frac{1}{2})-x)$$



Now I can see that the first term is just base times height.... this is a histogram after all...



But that second term is bothering me. I can't think of a simple geometrical interpretation for what it means. I figure it is some sort of compensation for the value that exists between the histogram and the true $f$. Is this accurate? And if so where does a derivation of this exist? Or did I just forget my elementary calculus?






calculus statistics







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  • 3




    this video has all the necessary info: youtube.com/watch?v=3d6DsjIBzJ4
    – tp1
    2 days ago










  • Oh nice! I love three blue one brown.
    – JetRex
    2 days ago
















2














I am reviewing Casella's All of Statistics. A particular integral comes up when reviewing non-parametric statistics in particular Histograms.



Assume that some $xin B_j$ and any other $u in B_j$ where $B_j$ is some particular bin in the histogram. I don't think the frequencies are going to be too important here...



$$int_{B_j}f(u)du approx int_{B_j}f(x) + (u-x)f^{prime}(u)$$
$$= f(x)h + hf^{prime}(x) ( h (j-frac{1}{2})-x)$$



Now I can see that the first term is just base times height.... this is a histogram after all...



But that second term is bothering me. I can't think of a simple geometrical interpretation for what it means. I figure it is some sort of compensation for the value that exists between the histogram and the true $f$. Is this accurate? And if so where does a derivation of this exist? Or did I just forget my elementary calculus?






calculus statistics







share|cite|improve this question


















  • 3




    this video has all the necessary info: youtube.com/watch?v=3d6DsjIBzJ4
    – tp1
    2 days ago










  • Oh nice! I love three blue one brown.
    – JetRex
    2 days ago














2












2








2







I am reviewing Casella's All of Statistics. A particular integral comes up when reviewing non-parametric statistics in particular Histograms.



Assume that some $xin B_j$ and any other $u in B_j$ where $B_j$ is some particular bin in the histogram. I don't think the frequencies are going to be too important here...



$$int_{B_j}f(u)du approx int_{B_j}f(x) + (u-x)f^{prime}(u)$$
$$= f(x)h + hf^{prime}(x) ( h (j-frac{1}{2})-x)$$



Now I can see that the first term is just base times height.... this is a histogram after all...



But that second term is bothering me. I can't think of a simple geometrical interpretation for what it means. I figure it is some sort of compensation for the value that exists between the histogram and the true $f$. Is this accurate? And if so where does a derivation of this exist? Or did I just forget my elementary calculus?






calculus statistics







share|cite|improve this question













I am reviewing Casella's All of Statistics. A particular integral comes up when reviewing non-parametric statistics in particular Histograms.



Assume that some $xin B_j$ and any other $u in B_j$ where $B_j$ is some particular bin in the histogram. I don't think the frequencies are going to be too important here...



$$int_{B_j}f(u)du approx int_{B_j}f(x) + (u-x)f^{prime}(u)$$
$$= f(x)h + hf^{prime}(x) ( h (j-frac{1}{2})-x)$$



Now I can see that the first term is just base times height.... this is a histogram after all...



But that second term is bothering me. I can't think of a simple geometrical interpretation for what it means. I figure it is some sort of compensation for the value that exists between the histogram and the true $f$. Is this accurate? And if so where does a derivation of this exist? Or did I just forget my elementary calculus?






calculus statistics




calculus statistics






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 2 days ago









JetRex

977




977








  • 3




    this video has all the necessary info: youtube.com/watch?v=3d6DsjIBzJ4
    – tp1
    2 days ago










  • Oh nice! I love three blue one brown.
    – JetRex
    2 days ago














  • 3




    this video has all the necessary info: youtube.com/watch?v=3d6DsjIBzJ4
    – tp1
    2 days ago










  • Oh nice! I love three blue one brown.
    – JetRex
    2 days ago








3




3




this video has all the necessary info: youtube.com/watch?v=3d6DsjIBzJ4
– tp1
2 days ago




this video has all the necessary info: youtube.com/watch?v=3d6DsjIBzJ4
– tp1
2 days ago












Oh nice! I love three blue one brown.
– JetRex
2 days ago




Oh nice! I love three blue one brown.
– JetRex
2 days ago










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