Empirical mean excess plot of (μ, e(μ))

Spit-balling a little here, based on my best understanding of your definition:

n = 500;
variates = RandomVariate[CauchyDistribution[], n];

ClearAll[meanExcess]
meanExcess[data_, mu_] := 
 Total[#]/Length[#]& @ Cases[data, x_?(# > mu &) :> x - mu]

Plot[meanExcess[variates, mu], {mu, 0, 10}]

Three additional ways to implement empirical mean excess using

1. TruncatedDistribution + EmpiricalDistribution

2. UnitStep + Pick

3. Clip + DeleteCases

ClearAll[eme1, eme2, eme3]

eme1[data_, μ_] := Mean @ TruncatedDistribution[{μ, ∞}, EmpiricalDistribution @ data]

eme2[data_, μ_] := Mean @ Pick[data, UnitStep[data - μ], 1]

eme3[data_, μ_] := Mean @ DeleteCases[Null] @ Clip[data, {μ, ∞}, {Null, ∞}]

Examples:

SeedRandom[1]
rv = RandomVariate[CauchyDistribution[], 100];

Verify that all three methods give the same result when the threshold parameter is within the sample range:

Max @ Chop[{Norm[eme1[rv, #] - eme2[rv, #]], 
     Norm[eme1[rv, #] - eme3[rv, #]], 
     Norm[eme2[rv, #] - eme3[rv, #]]} & /@ RandomReal[MinMax[rv], 500]]

 0

Plot[{eme1[rv, x], eme2[rv, x], eme3[rv, x]}, {x, 0, 10}, 
 PlotStyle -> {Directive[AbsoluteThickness[7], Red], 
   Directive[AbsoluteThickness[3], Blue], Directive[Thin, Green]}, 
 ImageSize -> Large, PlotLegends -> {"eme1", "eme2", "eme3"}]