What did Silverman(1981) mean by 'critical bandwidth'?












2












$begingroup$


In the selection of a bandwidth for a Kernel Density Estimator, critical bandwidth according to my understanding is:



"For every integer k, where 1<k<n, we can find the minimum width h(k) such that the kernel density estimate has at most k maxima. Silverman calls these h(k) values “critical widths.”



I don't intuitively understand this concept. Any help would be appreciated.



Thank you!






econometrics kernel-smoothing







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$endgroup$

















    2












    $begingroup$


    In the selection of a bandwidth for a Kernel Density Estimator, critical bandwidth according to my understanding is:



    "For every integer k, where 1<k<n, we can find the minimum width h(k) such that the kernel density estimate has at most k maxima. Silverman calls these h(k) values “critical widths.”



    I don't intuitively understand this concept. Any help would be appreciated.



    Thank you!






    econometrics kernel-smoothing







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      In the selection of a bandwidth for a Kernel Density Estimator, critical bandwidth according to my understanding is:



      "For every integer k, where 1<k<n, we can find the minimum width h(k) such that the kernel density estimate has at most k maxima. Silverman calls these h(k) values “critical widths.”



      I don't intuitively understand this concept. Any help would be appreciated.



      Thank you!






      econometrics kernel-smoothing







      share|cite|improve this question









      $endgroup$




      In the selection of a bandwidth for a Kernel Density Estimator, critical bandwidth according to my understanding is:



      "For every integer k, where 1<k<n, we can find the minimum width h(k) such that the kernel density estimate has at most k maxima. Silverman calls these h(k) values “critical widths.”



      I don't intuitively understand this concept. Any help would be appreciated.



      Thank you!






      econometrics kernel-smoothing




      econometrics kernel-smoothing






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      share|cite|improve this question











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      asked 2 hours ago









      Miles DavisMiles Davis

      133




      133






















          2 Answers
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          $begingroup$

          If you have a really wide bandwidth, you'll get one peak in your KDE. If you reduce it a bit, its still one peak. Keep reducing it until you get to the switchover point of adding a second peak. That bandwidth is $h(1)$.



          Now make it smaller still, until you get to the switchover between two peaks and three. That's $h(2)$.



          And so forth.



          series of KDEs showing smaller bandwidths just before and after adding each new peak (i.e. at the critical bandwidths) for the 2nd, 3rd and 4th peak



          At any bandwidth between $h(i)$ and $h(i+1)$ you will have $i$ peaks in your KDE.



          Silverman wanted a name for that set of $h$-values; he called them critical bandwidths.



          This comes up in his test for multimodality, for example.






          share|cite|improve this answer











          $endgroup$





















            1












            $begingroup$

            I hate animations in Web pages, but this question begs for an animated answer:



            Figure



            These are KDEs for a set of three values (near -2.5, 0.5, and 2.5). Their bandwidths continually vary, growing from small to large. Watch as three peaks become two and ultimately one.




            A KDE puts a pile of "probability" at each data point. As the bandwidth widens, the pile "slumps." When you start with tiny bandwidths, each data value contributes its own discrete pile. As the bandwidths grow, the piles slump and merge and accumulate on top of each other (the thick blue line), ultimately becoming one single pile. Along the way, the maxima change discontinuously from the starting value of $n$ (assuming the kernel has a single maximum, which is almost always the case) to $1.$ The critical width for $k$ maxima is the first (smallest) width that reduces the KDE to a curve with no more than $k$ maxima.






            share|cite|improve this answer











            $endgroup$













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              2 Answers
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              active

              oldest

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              2 Answers
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              active

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              2












              $begingroup$

              If you have a really wide bandwidth, you'll get one peak in your KDE. If you reduce it a bit, its still one peak. Keep reducing it until you get to the switchover point of adding a second peak. That bandwidth is $h(1)$.



              Now make it smaller still, until you get to the switchover between two peaks and three. That's $h(2)$.



              And so forth.



              series of KDEs showing smaller bandwidths just before and after adding each new peak (i.e. at the critical bandwidths) for the 2nd, 3rd and 4th peak



              At any bandwidth between $h(i)$ and $h(i+1)$ you will have $i$ peaks in your KDE.



              Silverman wanted a name for that set of $h$-values; he called them critical bandwidths.



              This comes up in his test for multimodality, for example.






              share|cite|improve this answer











              $endgroup$


















                2












                $begingroup$

                If you have a really wide bandwidth, you'll get one peak in your KDE. If you reduce it a bit, its still one peak. Keep reducing it until you get to the switchover point of adding a second peak. That bandwidth is $h(1)$.



                Now make it smaller still, until you get to the switchover between two peaks and three. That's $h(2)$.



                And so forth.



                series of KDEs showing smaller bandwidths just before and after adding each new peak (i.e. at the critical bandwidths) for the 2nd, 3rd and 4th peak



                At any bandwidth between $h(i)$ and $h(i+1)$ you will have $i$ peaks in your KDE.



                Silverman wanted a name for that set of $h$-values; he called them critical bandwidths.



                This comes up in his test for multimodality, for example.






                share|cite|improve this answer











                $endgroup$
















                  2












                  2








                  2





                  $begingroup$

                  If you have a really wide bandwidth, you'll get one peak in your KDE. If you reduce it a bit, its still one peak. Keep reducing it until you get to the switchover point of adding a second peak. That bandwidth is $h(1)$.



                  Now make it smaller still, until you get to the switchover between two peaks and three. That's $h(2)$.



                  And so forth.



                  series of KDEs showing smaller bandwidths just before and after adding each new peak (i.e. at the critical bandwidths) for the 2nd, 3rd and 4th peak



                  At any bandwidth between $h(i)$ and $h(i+1)$ you will have $i$ peaks in your KDE.



                  Silverman wanted a name for that set of $h$-values; he called them critical bandwidths.



                  This comes up in his test for multimodality, for example.






                  share|cite|improve this answer











                  $endgroup$



                  If you have a really wide bandwidth, you'll get one peak in your KDE. If you reduce it a bit, its still one peak. Keep reducing it until you get to the switchover point of adding a second peak. That bandwidth is $h(1)$.



                  Now make it smaller still, until you get to the switchover between two peaks and three. That's $h(2)$.



                  And so forth.



                  series of KDEs showing smaller bandwidths just before and after adding each new peak (i.e. at the critical bandwidths) for the 2nd, 3rd and 4th peak



                  At any bandwidth between $h(i)$ and $h(i+1)$ you will have $i$ peaks in your KDE.



                  Silverman wanted a name for that set of $h$-values; he called them critical bandwidths.



                  This comes up in his test for multimodality, for example.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited 2 hours ago

























                  answered 2 hours ago









                  Glen_b♦Glen_b

                  209k22398741




                  209k22398741

























                      1












                      $begingroup$

                      I hate animations in Web pages, but this question begs for an animated answer:



                      Figure



                      These are KDEs for a set of three values (near -2.5, 0.5, and 2.5). Their bandwidths continually vary, growing from small to large. Watch as three peaks become two and ultimately one.




                      A KDE puts a pile of "probability" at each data point. As the bandwidth widens, the pile "slumps." When you start with tiny bandwidths, each data value contributes its own discrete pile. As the bandwidths grow, the piles slump and merge and accumulate on top of each other (the thick blue line), ultimately becoming one single pile. Along the way, the maxima change discontinuously from the starting value of $n$ (assuming the kernel has a single maximum, which is almost always the case) to $1.$ The critical width for $k$ maxima is the first (smallest) width that reduces the KDE to a curve with no more than $k$ maxima.






                      share|cite|improve this answer











                      $endgroup$


















                        1












                        $begingroup$

                        I hate animations in Web pages, but this question begs for an animated answer:



                        Figure



                        These are KDEs for a set of three values (near -2.5, 0.5, and 2.5). Their bandwidths continually vary, growing from small to large. Watch as three peaks become two and ultimately one.




                        A KDE puts a pile of "probability" at each data point. As the bandwidth widens, the pile "slumps." When you start with tiny bandwidths, each data value contributes its own discrete pile. As the bandwidths grow, the piles slump and merge and accumulate on top of each other (the thick blue line), ultimately becoming one single pile. Along the way, the maxima change discontinuously from the starting value of $n$ (assuming the kernel has a single maximum, which is almost always the case) to $1.$ The critical width for $k$ maxima is the first (smallest) width that reduces the KDE to a curve with no more than $k$ maxima.






                        share|cite|improve this answer











                        $endgroup$
















                          1












                          1








                          1





                          $begingroup$

                          I hate animations in Web pages, but this question begs for an animated answer:



                          Figure



                          These are KDEs for a set of three values (near -2.5, 0.5, and 2.5). Their bandwidths continually vary, growing from small to large. Watch as three peaks become two and ultimately one.




                          A KDE puts a pile of "probability" at each data point. As the bandwidth widens, the pile "slumps." When you start with tiny bandwidths, each data value contributes its own discrete pile. As the bandwidths grow, the piles slump and merge and accumulate on top of each other (the thick blue line), ultimately becoming one single pile. Along the way, the maxima change discontinuously from the starting value of $n$ (assuming the kernel has a single maximum, which is almost always the case) to $1.$ The critical width for $k$ maxima is the first (smallest) width that reduces the KDE to a curve with no more than $k$ maxima.






                          share|cite|improve this answer











                          $endgroup$



                          I hate animations in Web pages, but this question begs for an animated answer:



                          Figure



                          These are KDEs for a set of three values (near -2.5, 0.5, and 2.5). Their bandwidths continually vary, growing from small to large. Watch as three peaks become two and ultimately one.




                          A KDE puts a pile of "probability" at each data point. As the bandwidth widens, the pile "slumps." When you start with tiny bandwidths, each data value contributes its own discrete pile. As the bandwidths grow, the piles slump and merge and accumulate on top of each other (the thick blue line), ultimately becoming one single pile. Along the way, the maxima change discontinuously from the starting value of $n$ (assuming the kernel has a single maximum, which is almost always the case) to $1.$ The critical width for $k$ maxima is the first (smallest) width that reduces the KDE to a curve with no more than $k$ maxima.







                          share|cite|improve this answer














                          share|cite|improve this answer



                          share|cite|improve this answer








                          edited 1 hour ago

























                          answered 1 hour ago









                          whuber♦whuber

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                          202k33439807






























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