最大限度地减少Mathematica中自定义分发的NEXpectation

这与6月份的早期问题有关：

Calculating expectation for a custom distribution in Mathematica

我有一个自定义混合分布使用第二个自定义分布定义，跟随 @Sasha 在过去一年的许多答案中讨论的行 .

定义分布的代码如下：

nDist /: CharacteristicFunction[nDist[a_, b_, m_, s_], 
   t_] := (a b E^(I m t - (s^2 t^2)/2))/((I a + t) (-I b + t));
nDist /: PDF[nDist[a_, b_, m_, s_], x_] := (1/(2*(a + b)))*a* 
   b*(E^(a*(m + (a*s^2)/2 - x))* Erfc[(m + a*s^2 - x)/(Sqrt[2]*s)] + 
     E^(b*(-m + (b*s^2)/2 + x))* 
      Erfc[(-m + b*s^2 + x)/(Sqrt[2]*s)]); 
nDist /: CDF[nDist[a_, b_, m_, s_], 
   x_] := ((1/(2*(a + b)))*((a + b)*E^(a*x)* 
        Erfc[(m - x)/(Sqrt[2]*s)] - 
       b*E^(a*m + (a^2*s^2)/2)*Erfc[(m + a*s^2 - x)/(Sqrt[2]*s)] + 
       a*E^((-b)*m + (b^2*s^2)/2 + a*x + b*x)*
        Erfc[(-m + b*s^2 + x)/(Sqrt[2]*s)]))/ E^(a*x);         

nDist /: Quantile[nDist[a_, b_, m_, s_], p_] :=  
 Module[{x}, 
   x /. FindRoot[CDF[nDist[a, b, m, s], x] == #, {x, m}] & /@ p] /; 
  VectorQ[p, 0 < # < 1 &]
nDist /: Quantile[nDist[a_, b_, m_, s_], p_] := 
 Module[{x}, x /. FindRoot[CDF[nDist[a, b, m, s], x] == p, {x, m}]] /;
   0 < p < 1
nDist /: Quantile[nDist[a_, b_, m_, s_], p_] := -Infinity /; p == 0
nDist /: Quantile[nDist[a_, b_, m_, s_], p_] := Infinity /; p == 1
nDist /: Mean[nDist[a_, b_, m_, s_]] := 1/a - 1/b + m;
nDist /: Variance[nDist[a_, b_, m_, s_]] := 1/a^2 + 1/b^2 + s^2;
nDist /: StandardDeviation[ nDist[a_, b_, m_, s_]] := 
  Sqrt[ 1/a^2 + 1/b^2 + s^2];
nDist /: DistributionDomain[nDist[a_, b_, m_, s_]] := 
 Interval[{0, Infinity}]
nDist /: DistributionParameterQ[nDist[a_, b_, m_, s_]] := ! 
  TrueQ[Not[Element[{a, b, s, m}, Reals] && a > 0 && b > 0 && s > 0]]
nDist /: DistributionParameterAssumptions[nDist[a_, b_, m_, s_]] := 
 Element[{a, b, s, m}, Reals] && a > 0 && b > 0 && s > 0
nDist /: Random`DistributionVector[nDist[a_, b_, m_, s_], n_, prec_] :=

    RandomVariate[ExponentialDistribution[a], n, 
    WorkingPrecision -> prec] - 
   RandomVariate[ExponentialDistribution[b], n, 
    WorkingPrecision -> prec] + 
   RandomVariate[NormalDistribution[m, s], n, 
    WorkingPrecision -> prec];

(* Fitting: This uses Mean, central moments 2 and 3 and 4th cumulant \
but it often does not provide a solution *)

nDistParam[data_] := Module[{mn, vv, m3, k4, al, be, m, si},
      mn = Mean[data];
      vv = CentralMoment[data, 2];
      m3 = CentralMoment[data, 3];
      k4 = Cumulant[data, 4];
      al = 
    ConditionalExpression[
     Root[864 - 864 m3 #1^3 - 216 k4 #1^4 + 648 m3^2 #1^6 + 
        36 k4^2 #1^8 - 216 m3^3 #1^9 + (-2 k4^3 + 27 m3^4) #1^12 &, 
      2], k4 > Root[-27 m3^4 + 4 #1^3 &, 1]];
      be = ConditionalExpression[

     Root[2 Root[
           864 - 864 m3 #1^3 - 216 k4 #1^4 + 648 m3^2 #1^6 + 
             36 k4^2 #1^8 - 
             216 m3^3 #1^9 + (-2 k4^3 + 27 m3^4) #1^12 &, 
           2]^3 + (-2 + 
           m3 Root[
              864 - 864 m3 #1^3 - 216 k4 #1^4 + 648 m3^2 #1^6 + 
                36 k4^2 #1^8 - 
                216 m3^3 #1^9 + (-2 k4^3 + 27 m3^4) #1^12 &, 
              2]^3) #1^3 &, 1], k4 > Root[-27 m3^4 + 4 #1^3 &, 1]];
      m = mn - 1/al + 1/be;
      si = 
    Sqrt[Abs[-al^-2 - be^-2 + vv ]];(*Ensure positive*)
      {al, 
    be, m, si}];

nDistLL = 
  Compile[{a, b, m, s, {x, _Real, 1}}, 
   Total[Log[
     1/(2 (a + 
           b)) a b (E^(a (m + (a s^2)/2 - x)) Erfc[(m + a s^2 - 
             x)/(Sqrt[2] s)] + 
        E^(b (-m + (b s^2)/2 + x)) Erfc[(-m + b s^2 + 
             x)/(Sqrt[2] s)])]](*, CompilationTarget->"C", 
   RuntimeAttributes->{Listable}, Parallelization->True*)];

nlloglike[data_, a_?NumericQ, b_?NumericQ, m_?NumericQ, s_?NumericQ] := 
  nDistLL[a, b, m, s, data];

nFit[data_] := Module[{a, b, m, s, a0, b0, m0, s0, res},

      (* So far have not found a good way to quickly estimate a and \
b.  Starting assumption is that they both = 2,then m0 ~= 
   Mean and s0 ~= 
   StandardDeviation it seems to work better if a and b are not the \
same at start. *)

   {a0, b0, m0, s0} = nDistParam[data];(*may give Undefined values*)

     If[! (VectorQ[{a0, b0, m0, s0}, NumericQ] && 
       VectorQ[{a0, b0, s0}, # > 0 &]),
            m0 = Mean[data];
            s0 = StandardDeviation[data];
            a0 = 1;
            b0 = 2;];
   res = {a, b, m, s} /. 
     FindMaximum[
       nlloglike[data, Abs[a], Abs[b], m,  
        Abs[s]], {{a, a0}, {b, b0}, {m, m0}, {s, s0}},
               Method -> "PrincipalAxis"][[2]];
      {Abs[res[[1]]], Abs[res[[2]]], res[[3]], Abs[res[[4]]]}];

nFit[data_, {a0_, b0_, m0_, s0_}] := Module[{a, b, m, s, res},
      res = {a, b, m, s} /. 
     FindMaximum[
       nlloglike[data, Abs[a], Abs[b], m, 
        Abs[s]], {{a, a0}, {b, b0}, {m, m0}, {s, s0}},
               Method -> "PrincipalAxis"][[2]];
      {Abs[res[[1]]], Abs[res[[2]]], res[[3]], Abs[res[[4]]]}];

dDist /: PDF[dDist[a_, b_, m_, s_], x_] := 
  PDF[nDist[a, b, m, s], Log[x]]/x;
dDist /: CDF[dDist[a_, b_, m_, s_], x_] := 
  CDF[nDist[a, b, m, s], Log[x]];
dDist /: EstimatedDistribution[data_, dDist[a_, b_, m_, s_]] := 
  dDist[Sequence @@ nFit[Log[data]]];
dDist /: EstimatedDistribution[data_, 
   dDist[a_, b_, m_, 
    s_], {{a_, a0_}, {b_, b0_}, {m_, m0_}, {s_, s0_}}] := 
  dDist[Sequence @@ nFit[Log[data], {a0, b0, m0, s0}]];
dDist /: Quantile[dDist[a_, b_, m_, s_], p_] := 
 Module[{x}, x /. FindRoot[CDF[dDist[a, b, m, s], x] == p, {x, s}]] /;
   0 < p < 1
dDist /: Quantile[dDist[a_, b_, m_, s_], p_] :=  
 Module[{x}, 
   x /. FindRoot[ CDF[dDist[a, b, m, s], x] == #, {x, s}] & /@ p] /; 
  VectorQ[p, 0 < # < 1 &]
dDist /: Quantile[dDist[a_, b_, m_, s_], p_] := -Infinity /; p == 0
dDist /: Quantile[dDist[a_, b_, m_, s_], p_] := Infinity /; p == 1
dDist /: DistributionDomain[dDist[a_, b_, m_, s_]] := 
 Interval[{0, Infinity}]
dDist /: DistributionParameterQ[dDist[a_, b_, m_, s_]] := ! 
  TrueQ[Not[Element[{a, b, s, m}, Reals] && a > 0 && b > 0 && s > 0]]
dDist /: DistributionParameterAssumptions[dDist[a_, b_, m_, s_]] := 
 Element[{a, b, s, m}, Reals] && a > 0 && b > 0 && s > 0
dDist /: Random`DistributionVector[dDist[a_, b_, m_, s_], n_, prec_] :=
   Exp[RandomVariate[ExponentialDistribution[a], n, 
     WorkingPrecision -> prec] - 
       RandomVariate[ExponentialDistribution[b], n, 
     WorkingPrecision -> prec] + 
    RandomVariate[NormalDistribution[m, s], n, 
     WorkingPrecision -> prec]];

这使我能够拟合分布参数并生成 PDF's 和 CDF's . 这个图的一个例子：

Plot[PDF[dDist[3.77, 1.34, -2.65, 0.40], x], {x, 0, .3}, 
 PlotRange -> All]
Plot[CDF[dDist[3.77, 1.34, -2.65, 0.40], x], {x, 0, .3}, 
 PlotRange -> All]

现在我已经定义了 function 来计算平均剩余寿命（参见this question进行解释） .

MeanResidualLife[start_, dist_] := 
 NExpectation[X \[Conditioned] X > start, X \[Distributed] dist] - 
  start
MeanResidualLife[start_, limit_, dist_] := 
 NExpectation[X \[Conditioned] start <= X <= limit, 
   X \[Distributed] dist] - start

第一个没有像第二个那样设置限制需要花费很长时间来计算，但它们都有效 .

现在我需要找到相同分布的最小 MeanResidualLife 函数（或它的一些变体）或最小化它 .

我已尝试过多种变体：

FindMinimum[MeanResidualLife[x, dDist[3.77, 1.34, -2.65, 0.40]], x]
FindMinimum[MeanResidualLife[x, 1, dDist[3.77, 1.34, -2.65, 0.40]], x]

NMinimize[{MeanResidualLife[x, dDist[3.77, 1.34, -2.65, 0.40]], 
  0 <= x <= 1}, x]
NMinimize[{MeanResidualLife[x, 1, dDist[3.77, 1.34, -2.65, 0.40]], 0 <= x <= 1}, x]

这些似乎永远运行或遇到：

Power :: infy：无限表达式1/0遇到 . >>

应用于更简单但形状相似的分布的 MeanResidualLife 函数显示它只有一个最小值：

Plot[PDF[LogNormalDistribution[1.75, 0.65], x], {x, 0, 30}, 
 PlotRange -> All]
Plot[MeanResidualLife[x, LogNormalDistribution[1.75, 0.65]], {x, 0, 
  30},
 PlotRange -> {{0, 30}, {4.5, 8}}]

还有：

FindMinimum[MeanResidualLife[x, LogNormalDistribution[1.75, 0.65]], x]
FindMinimum[MeanResidualLife[x, 30, LogNormalDistribution[1.75, 0.65]], x]

当与 LogNormalDistribution 一起使用时，给我答案（如果首先是一堆消息） .

有关如何使其适用于上述自定义分发的任何想法？

我是否需要添加约束或选项？

我是否需要在自定义分布的定义中定义其他内容？

也许 FindMinimum 或 NMinimize 只需要运行更长时间（我运行它们将近一个小时无济于事） . 如果是这样，我只是需要一些方法来加快找到功能的最小值？有关如何的任何建议？

Mathematica 有另一种方法吗？

Added 9 Feb 5:50PM EST:

任何人都可以从Wolfram技术大会2011研讨会'Create Your Own Distribution' here下载关于在Mathematica中创建发行版的 Oleksandr Pavlyk's 演示文稿 . 下载包括笔记本， 'ExampleOfParametricDistribution.nb' 似乎列出了创建一个可以使用的分发所需的所有部分，就像Mathematica附带的发行版一样 .

它可能会提供一些答案 .