Asymptotic results for Normal-Cauchy model

Transcription

1 Asymptotic results for Norml-Cuchy model John D. Cook Deprtment of Biosttistics P. O. Box 342, Unit 49 The University of Texs, M. D. Anderson Cncer Center Houston, Texs , USA September, 2 Abstrct This report proves symptotic results for the posterior men when smpling from norml distribution with Cuchy prior on the loction prmeter. Problem sttement Define φθ; σ) = ) exp θ2 σ 2σ 2 nd cθ) = π + θ 2 ). Let y i be smples from normlθ, σ 2 ) distribution where θ hs Cuchy, ) prior. Let y be the smple men of the y i vlues. The posterior men of θ is given by θ φy θ; σ/ n) cθ) dθ φy θ; σ/ n) cθ) dθ.

2 We will estblish symptotic results for the posterior men of θ s y nd s n. 2 Preliminries 2. Fourier trnsforms We will use the following results from Fourier nlysis to evlute the integrls below. First, define the Fourier trnsform of function f L R) s Ff)x) = fθ) expixθ) dθ. The inverse Fourier trnsform is then given by Ff)x) = fθ) exp ixθ) dθ. Define the convolution of two functions f nd g by f g)x) = fθ) gx θ) dθ. Then nd f g = g f Ff g) = Ff)Fg). 2.2 Integrting the exponentil of qudrtic We will lso use the following result. Clim. For > nd b C, exp x 2 + bx) dx = ) π exp 4 2 erfc b 2 ). ) 2

3 Proof. Use the fctoriztion x 2 bx = to come up with the chnge of vribles Now ) x 2 b2 2 w = x exp x 2 + bx) dx = = = = b 2. b/2 ) 2 ) b exp w 2 2 ) exp dw 2 exp b/2 exp w 2 ) dw ) π exp π b/2 exp w 2 ) dw ) π exp 4 2 erfc b ) 2. 3 Integrtion results Clim 2. where φy θ; σ) cθ) dθ = 2 σ hz) = expz 2 ) erfcz). { ) )} + iy iy h + h 2σ 2σ 2) Proof. The integrl in eqution 2) equls φ c)y) nd so its Fourier trnsform is Fφ) Fc). These Fourier trnsforms re well known: Fφ)x) = exp σ 2 x 2 /2) Fc)x) = exp x ). 3

4 Applying the inverse Fourier trnsform we hve the following. φy θ; σ) cθ) dθ = φ c)y) = F F φ c)y))) = exp σ 2 x 2 x ixy) dx = + exp σ 2 x 2 /2 x ixy) dx exp σ 2 x 2 /2 + x ixy) dx. The finl two integrls cn be evluted in terms of the complementry error function erfc using Clim. Clim 3. where s before i θφy θ; σ) cθ) dθ = 2 σ hz) = expz 2 ) erfcz). { ) )} + iy iy h h 2σ 2σ 3) Proof. As before we tke the Fourier trnsform of the integrl nd then trnsform bck. Define the function Here we use the fct tht ηθ) = Fη)x) = θ π + θ 2 ). i x x exp x ). The function η is not in L R) nd so the elementry definition of the Fourier trnsform does not hold. But η L 2 R) nd when the Fourier trnsform is extended to L 2 R), the right side of the eqution bove is its trnsform.) θφy θ; σ) cθ) dθ = φ η 4

5 = F exp σ2 x 2 ) ) i x 2 x exp x ) x = i x exp σ2 x 2 /2 x ixy) dx = i exp σ2 x 2 /2 x ixy) dx i exp σ2 x 2 /2 + x ixy) dx. As before, the finl two integrls cn be evluted using Clim. 4 Asymptotic results Next we pply the symptotic series erfcz) = exp z2 ) ) πz 2z 2 + to the integrls bove. This series is vlid for rgz) < 3π/4. Since we will only be interested in vlues of z with positive rel prt, the series is vlid for our use. For the right side of eqution 2) the first term of the symptotic series is sufficient. Clim 4. As y, φy θ) cθ) dθ π + y 2 ). Proof. Intuitively, s y becomes lrge, the function cθ) becomes very flt. Multiplying by φy θ) nd integrting essentilly smples the function cθ) t y. To prove this ssertion, pply the symptotic pproximtion erfcz) exp z2 ) πz 5

6 to eqution 2). With little rerrngement, the rguments inside the exponentil functions become zero nd the integrl reduces to 2 exp) σ ) + exp) π iy ) 2σ π +iy = π + y 2 ). 2σ Next we pply the symptotic series for erfc to eqution 3). Clim 5. As y, θφy θ) cθ) dθ y π { + y 2 ) 2 + σ 2 y 2 3) + y 2 ) 3 Proof. Here we pply the two-term symptotic pproximtion erfcz) exp z2 ) ) πz 2z 2. As before, the rguments of the exponentil functions become zero nd the integrl reduces to { iy }. ) ) σ 2 ) )} σ 2 iy + iy + iy which further reduces to { y + y 2 ) 2 + σ 2 y 2 } 3) π + y 2 ) 3. Consider single smple y from normlθ, σ 2 ) distribution where θ hs conjugte norml, τ 2 ) prior. It is well-known tht the posterior distribution on θ hs men τ 2 τ 2 + σ 2 y. 6

7 Clim 6. If we use Cuchy, ) prior rther thn norml, τ 2 ) prior on θ bove, the posterior men of θ is ) y O y s y. Proof. Simply tke the rtio of the results of Clim 5 nd Clim 4. Next consider tking multiple smples y i from normlθ, σ 2 ) distribution. Denote the smple men of the y i vlues by y. We exmine the posterior men of θ under norml nd Cuchy priors s the number of smples n increses. With norml, τ 2 ) prior on θ, the posterior men of θ fter observing n smples with men y is )) y = σ2 + σ2 nτ 2 + O n 2 y. nτ 2 Clim 7. If θ hs Cuchy, ) prior, then the posterior men of θ fter smpling n vlues with smple men y is y + y2 3)y σ 2 ) + y 2 ) 2 n + O n 2. Proof. Observing n vlues from normlθ, σ 2 ) distribution is the sme s observing one vlue y from normlθ, σ 2 /n) distribution. The clim cn be estblished nlogous to Clim 5 letting n rther thn y. The rte t which the posterior men converges to y depends on τ in the cse of the norml prior nd on y in the cse of the Cuchy prior. For ny vlue of τ, the convergence is fster under the Cuchy prior for sufficiently lrge vlues of y. 5 Acknowledgment The uthor wished to thnk Aleksey Pichugin for suggesting tht the integrls in this report could be computed vi Fourier trnsforms. 7