Comparison of Sample Sizes

Reference

Tiao,G.C.,Reinsel,G.C.,Xu,D.,Pedrick,J.H.,Zhu,X.,Miller,A.J.,DeLuisi,J.J.,Mateer,C.L.and Wuebbles,D.J.(1990),``Effects of autocorrelation and temporal sampling schemes on estimation of trend and spatial correlation, Journal of Geophysical Research 95 D12,20,507--20,517.

Background

Number of Years, \(TraditionalForm\`\(\(n\^*\)\(,\)\)\) for 90% probability of detecting a prescribed trend, δ, using a two-sided5% test given 30 years of prior data and assuming AR(1) errors with estimated paramete \(TraditionalForm\`\(\(\(ϕ\&^\)\_1\)\(.\)\)\) The last line of the table shows the comparable values given in Tiao et al. (1990, Table 2) for sites: Tateno, Hohen., Wakkan, Bulawayo and Abidajan.  The data for these sites is respectively,

sigmaD = {3.3/100,  4.6/100,  1.6,  1.3,  
          1.5}; 
phiD = {0.32, 0.05, 0.14, 0.43,  0.65};
deltaD = {0.003, 0.003, 0.2, 0.2, 0.2};

Derivation of Table

Let (Subscript[ℛ, t]^((T))) denote the ramp intervention defined in Tiao (1990) so we have (Subscript[ℛ, t]^((T))) = 
        (Subscript[R, t]^((T + 1)))/12.   Then assuming T = 0 the following expression was derived for \(TraditionalForm\`σ\_ω\&^\),

σ g(T, ϕ), where σ = (Subscript[σ, a]^2) /(1 - (ϕ^2)) and  \(TraditionalForm\`g(
          T, ϕ) = \((\((1 + ϕ)\) \((1 - ϕ)\))\)\^\(1/2\)\ 12/\((\(T(T\^2 - 1)\)/12)\)\^\(1/2\)\)

((((1 + ϕ)/(1 - ϕ)))^(1/2)) * 
        12/(((T ((T^2) - 1)/12))^(1/2)) // Simplify

Hence we may write:

𝔤[T_, ϕ_] := 
      24 (((3 (1 + ϕ)/((1 - ϕ) T ((T^2) - 1))))^(1/2))

The number of years needed to detect a 3% annual trend with 90% certainty is the smallest n such that 3  %>3.3 \(σ\_\(ω\&^\)\). which may be written as solving g(n yr,  ϕ) = 
                      0.003/(3.3 σ). This may be obtained by solving 𝔤(n yr, ϕ)=0.003/(3.3 σ).

RealQ[x_] := NumberQ[x] && Im[x] == 0;

c0 = 
      Table[Select[(n /. 
                NSolve[𝔤[12 n,  
                      phiD[[
                        i]]] == 
                    deltaD[[i]]/(3.3*
                          sigmaD[[
                            i]]), n]), RealQ], {i, 
            Length[phiD]}] // Flatten