The performance of approximations for 𝔼[Lτ(ρ, x)]

ρ =
τ = 5
𝔼[Lτ(ρ, x)]0.0810.2710.4101.077.18
𝔼1[Lτ(ρ, x)]0.0810.2230.2270.2310.272
𝔼2[Lτ(ρ, x)]0.0790.1990.2000.2000.200
𝔼3[Lτ(ρ, x)]0.1000.2500.2500.2500.250
τ = 10
𝔼[Lτ(ρ, x)]0.0650.1200.1230.1280.169
𝔼1[Lτ(ρ, x)]0.0650.1060.1060.1060.106
𝔼2[Lτ(ρ, x)]0.0630.1000.1000.1000.100
𝔼3[Lτ(ρ, x)]0.1000.1110.1110.1110.111
τ = 25
𝔼[Lτ(ρ, x)]0.0380.0430.0430.0430.043
𝔼1[Lτ(ρ, x)]0.0370.0410.0410.0410.041
𝔼2[Lτ(ρ, x)]0.0370.0400.0400.0400.040
𝔼3[Lτ(ρ, x)]0.0420.0420.0420.0420.042
  • All calculations were made at x = 1/2, where 𝔼[Lτ(ρ, x)] can be calculated numerically. Note that ρ can be thought of either in terms of the (scaled) genetic length of the fragment or as a physical length (given assumptions about the recombination rate per base pair).