Fix end point in smooth kernel distribution density

Question

I am using some extreme value fitting method which results in a parametric distribution for values exceeding some threshold, all values $\geq 0$.

For smaller values I'd like to use a smooth kernel distribution.

Pasting these two together, there is usually a small gap or jump in the distribution. So I looked for the possibility to 'fix' a point in the smoothed distribution, but found no such thing. Is there any way 'force' the smooth kernel distribution density to take a right end value?

It doesn't make a big difference, but there is simply no real reason for such a jump in the density.

Update: Of course it is possible to shift the empirical smoothed density and then scale the empirical density part to obtain a continuous density, but I am not really convinced. To me, it does not seem like the optimal solution. I'll attach two plots to illustrate the problem. Thank you!

Answer 1

Here is a possible solution.

Given a cut point c we can create a kernel density estimate kde truncated on {-Infinity, c} and a tail estimate tail truncated on {c, Infinity} with the additional restriction that the PDF of the tail at c must be equal to PDF[kde, c].

To solve for this point in a way that is robust enough for a variety of data sets and distributions we need to set up special rules for dealing with likelihoods.

The function bound will be used to ensure that indeterminate values of the LogLikelihood for our restricted tail will allow optimization functions to keep searching.

bound[Infinity] := $MinMachineNumber 
bound[-Infinity] := $MinMachineNumber
bound[n_?NumericQ] := n

Now for the tail estimator. The 1000 $MachineEpsilon is to nudge us away from restricted boundaries (and may need adjusted). The initial call to FindDistributionParameters attempts to get good starting values.

tailEstimate[dist_, data_, prob_, c_] := 
 Block[{params, est, tdist},
  params = FindDistributionParameters[data, dist];
  tdist = TruncatedDistribution[{c, \[Infinity]}, dist];
  est = Apply[
    FindMaximum, {{bound[
       LogLikelihood[tdist, Cases[data, x_ /; x > c]]], 
      DistributionParameterAssumptions[tdist] && 
       PDF[tdist, c + 1000 $MachineEpsilon] == prob}, 
     params /. Rule -> List}];
  tdist /. est[[2]]
  ]

Now to put it together with the kernel density estimator.

kde[data_, bw_, c_, tailModel_] :=
 Block[{kern, tail},
  kern = TruncatedDistribution[{-\[Infinity], c}, KernelMixtureDistribution[data, bw]];
  tail = tailEstimate[tailModel, data, PDF[kern, c], c];
  MixtureDistribution[{1, 1}, {kern, tail}]
  ]

Here is an example using heavy tailed data.

data = RandomVariate[ParetoDistribution[1, .5], 1000];
mdist = kde[data, .5, 5, ParetoDistribution[a, b]];
Plot[PDF[mdist, t], {t, 0, 20}, PlotRange -> {0, .2}]