Asymptotics for AR(1)

Summary

Consider the SIA step intervention model with Tpre-intervention observations.  As n -> ∞, \(TraditionalForm\`\(\(var(ω\&^) ≐ \(γ(0)\)/
          T\)\(,\)\)\) where

\(TraditionalForm\`γ(
        0) = \(σ\_a\%2\) \(∑\+\(k = \(-∞\)\)\%∞ γ\_k\)\)

for stationary disturbances with f = 0.   WLOG we assume (Subscript[σ, a]^2) = 1.

Tiao et al. (1990) showed that 90% power was achieved on a two sided test at the 5% level of (Subscript[ℌ, 0]) : ω = 0 for parameter values, ω = (ω^′), where \(TraditionalForm\`ω\^′ > 
        3.3\ \(\(σ\_ω\&^\)\(.\)\(\ \)\)\) This means that (δ^′) = (γ^′)/(Sqrt[T])is the detection limit, where

(γ^′) = (Sqrt[γ(
                0)/(Subscript[γ, 0])]).

If an intervention produces more that (δ^′), it will be detected 90% of the time using a two-sided test at the 5% level.

Derivations

AR(1) Case

Needs["LinearAlgebra`MatrixManipulation`"];

Needs["Statistics`NormalDistribution`"];

Needs["Statistics`DescriptiveStatistics`"];

a = SEωAR1[𝔫, T, ϕ]

asq = a^2;

Limit[asq, 𝔫 -> ∞,  
      Assumptions -> T > 0  &&  0 < ϕ < 1]

b = Denominator[%]

Expand[b]

Clear[g0]

b

FullSimplify[b]

b /. {ϕ -> 0.5, T -> 25}

Detection Level

DetectionLevel[n_?NumberQ, T_?NumberQ, phi_?NumberVectorQ, 
          theta_?NumberVectorQ, opts___?OptionQ] := 
        Module[{},];

Code Development

phi = {0.5}; theta = {}; n = 50; T = 25;

gvar = First[TacvfARMA[phi, theta, 1]];

gsum = (Total[Flatten[{1, -theta}]]^2)/(Total[
                Flatten[{1, -phi}]]^2);

3.3*Sqrt[gsum/(T*gvar)]

Final Function

DetectionLevel[n_?NumberQ, T_?NumberQ, phi_?NumberVectorQ, 
          theta_?NumberVectorQ] := 
        Module[{gvar, gsum}, 
gvar = 
            First[TacvfARMA[phi, theta, 1]]; 
          gsum = (Total[Flatten[{1, -theta}]]^2)/(Total[
                    Flatten[{1, -phi}]]^2); 
3.3*
            Sqrt[gsum/(T*gvar)]
];

DetectionLevel[50, 25, {0.5}, {}]

Asymptotic Plot

a = DetectionLevel[50, 25, {0.5}, {}]; n = (10^9);

a

n = (10^9); 
    ap = PowerIA[n, 25, a*Sqrt[1/(1 - (0.5^2))], {0.5}, {}]

gEdX = 2 + 0.1; gEdY = -0.1; 
    epi = {Dashing[{ .005, 0.01}], 
        Line[{{-gEdX, ap}, {-a, ap}, {-a, gEdY}}], 
        Line[{{gEdX, ap}, {a, ap}, {a, gEdY}}]};

flab = (StyleForm[#1, 
                FontWeight -> "Bold"] &) /@ {"δ", "\
β"};

\(\(gOUT = 
        Plot[{PowerIA[n, 25, δ*Sqrt[1/\((1 - 0.5\^2)\)], {0.5}, {}], 
            PowerIA[50, 
              25, δ*
                Sqrt[1/\((1 - 0.5\^2)\)], {0.5}, {}]}, {δ, \(-2\), 
            2}, 
Axes -> False, Frame -> True, 
          FrameLabel -> flab, RotateLabel -> False, 
          PlotStyle -> {Dashing[{0.07, 0.02}], 
              Dashing[{}]}, 
Epilog -> {epi, 
              Text[\*""-\!\(δ\^′\)"", {\(-a\), 0.02}, {1, 
                  0}], Text[\*""\!\(δ\^′\)"", {a, 
                  0.02}, {\(-1\), 0}]}
];\)\)

Export Graph

Export["d:/math/2003/PowerIA/AR1Asymptotic.eps", 
        gOUT, "EPS"];

Exact Detection Level

dta =. ;

FindRoot[
      PowerIA[(10^9), 25, dta*Sqrt[1/(1 - (0.5^2))], {0.5}, {}] == 
        0.9, {dta, 1.5}]