Investigating a new estimator of the serial correlation coefficient

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1 Ivestigatig a ew estimator of the serial correlatio coefficiet Mafred Mudelsee Istitute of Mathematics ad Statistics Uiversity of Ket at Caterbury Caterbury Ket CT2 7NF Uited Kigdom 8 May Abstract A ew estimator of the lag-1 serial correlatio coefficiet ρ, ɛ 3 i ɛ i+1 / 1 ɛ 4 i, is compared with the old estimator, 1 ɛ i ɛ i+1 / 1 ɛ 2 i, for a statioary AR(1) process with kow mea. The mea ad the variace of both estimators are calculated usig the secod order Taylor expasio of a ratio. No further approximatio is used. I case of the mea of the old estimator, we derive Marriott ad Pope s (1954) formula, with ( 1) 1 istead of () 1, ad a additioal term ( 1) 2. I case of the variace of the old estimator, Bartlett s (1946) formula results, with ( 1) 1 istead of () 1. The theoretical expressios are corroborated with simulatio experimets. The mai results are as follows. (1) The ew estimator has a larger egative bias ad a larger variace tha the old estimator. (2) The theoretical results for the mea ad the variace of the old estimator describe the pricipal behaviours over the etire rage of ρ, i particular, the declie to zero egative bias as ρ approaches uity.

2 1 Itroductio Cosider the followig estimators of the serial correlatio coefficiet ρ of lag 1 from a time series ɛ i (i = 1,..., ) sampled from a process E i with kow (zero) mea: ˆρ = ˆρ = 1 ɛ i ɛ i+1 1 ɛ 2 i 1 ɛ 3 i ɛ i+1 1 ɛ 4 i, (1). (2) Estimators of type (1) are well-kow (e. g. Bartlett 1946, Marriott ad Pope 1954), whereas (2) is ew to my best kowledge. Let E i be the statioary AR(1) process, E 1 N(0, 1), E i = ρ E i 1 + U i, i = 2,...,, (3) with U i i. i. d. N(0, σ 2 ) ad σ 2 = 1 ρ 2. The followig misjudgemet of mie led to the preset study. ˆρ miimises (ɛ i+1 ρɛ i ) 2 ɛ 2 i, which is a weighted sum of squares. However, sice E i has costat variace uity, o weightig should be ecessary. Nevertheless, we compare estimators (1) ad (2) for process (3). We restrict ourselves to 0 < ρ < 1. We first calculate their variaces, respectively meas, up to the secod order of the Taylor expasio of a ratio, similarly to Bartlett (1946), respectively Marriott ad Pope (1954). Sice we cocetrate o the lag-1 estimators ad process (3), o further approximatio is ecessary. These theoretical expressios ad also those of White (1961) for estimator (1) are the examied with simulatio experimets. 2 Series expasios 2.1 Old estimator Variace We write (1) as ˆρ = ɛ i ɛ i+1 1 ɛ 2 i (4) = c v, say, 1

3 ad have, up to the secod order of deviatios from the meas of c ad v, the variace var(ˆρ) = var(c/v) var(c) E 2 (v) 2 E(c)cov(v, c) E 3 (v) + E 2 (c) var(v) E 4 (v). (5) We ow apply a stadard result for quadravariate stadard Gaussia distributios with serial correlatios ρ j (e. g. Priestley 1981:325), cov(ɛ a ɛ a+s, ɛ b ɛ b+s+t ) = ρ b a ρ b a+t + ρ b a+s+t ρ b a s, from which we derive, for ɛ i followig process (3): cov ( 1 ɛ a ɛ a+s, 1 ) ɛ a=1 b ɛ b+s+t = b=1 = 1 2 (ρ b a ρ b a+t + ρ b a+s+t ρ b a s ). (6) Without further approximatio we ca directly derive the various costituets of the right-had side of (5) by the meas of (6). var(c) = var = 1 2 ( ) 1 ɛ i ɛ i+1 = (put s = 1 ad t = 0 i (6)) ρ 2 b a + ρ b a+1 ρ b a 1. We fid, by summig over the poits of the a-b lattice i a diagoal maer, 2 ρ 2 b a = ( 1) + 2 i ρ 2( i 1) = ( 1) ( ) 2 1 ρ (ρ2 4 ρ 2 ) (7) 2 (1 ρ 2 ) 2 (at the last step we have used the arithmetic-geometric progressio), ad 2 ρ b a+1 ρ b a 1 = ( 1) ρ i ρ 2( i 1) = ( 1) (7) ad (8) give var(c). Further, cov(v, c) = cov ( ) 2 1 ρ + 2 ρ (ρ2 4 ρ 2 ). (8) (1 ρ 2 ) 2 ( 1 2 ɛ 2 i, 1 ɛ i ɛ i+1 )

4 = (put s = 0 ad t = 1 i (6)) = 2 2 ρ b a ρ b a+1. We fid 2 ρ b a ρ b a+1 = ρ (2 i + 1) ρ 2( i) 3 = ( 1) 2 ρ ( ) 1 1 ρ (ρ2 5 ρ 3 ) + (ρ2 3 ρ 1 ). (9) (1 ρ 2 ) 2 (1 ρ 2 ) (At the last step we have used the arithmetic-geometric ad also the geometric progressio.) (9) gives cov(v, c). Further, var(v) = var = 2 2 ( 1 ɛ 2 i ) ρ 2 b a, which is give from (7). We further eed = (put s = t = 0 i (6)) E(v) = 1 (10) ad E(c) = ρ( 1). (11) All three terms cotributig to var(ˆρ) have a part ( 1) 2. However, these parts cacel out: var(ˆρ) (1 ρ2 ) ( 1). (12) Bartlett (1946) already gave, up to the order () 1, which caot be distiguished from our result. var(ˆρ) (1 ρ2 ), (13) 3

5 2.1.2 Mea We have, up to the secod order of deviatios from the meas of c ad v, the mea E(ˆρ) = E(c/v) E(c) cov(v, c) + E(c) var(v) E(v) E 2 (v) E 3 (v). (14) As above, we derive cov(v, c) ad var(v) from (9) ad (7), respectively, ad have E(v) ad E(c) give by (10) ad (11), respectively, yieldig E(ˆρ) ρ 2 ρ ( 1) + 2 (ρ ρ 2 1 ). (15) ( 1) 2 (1 ρ 2 ) Marriott ad Pope (1954) ivestigated the estimator ˆρ = ad gave, up to the order () 1, ɛ 2 i E(ˆρ ) ρ 2 ρ, ɛ i ɛ i+1 which caot be distiguished from the first two terms of our result Ukow mea of the process I have also tried to calculate up to the same order of approximatio the mea ad the variace of the estimator ( 1 ɛi 1 ) ( 1 1 j=1 ɛ j ɛi+1 1 ) 1 1 j=1 ɛ j+1 ( 1 ɛi 1 ) 1 2, 1 j=1 ɛ j for the case whe the mea of the process is ukow. That would require to calculate the followig sums over the poits of a cubic lattice: a,b,c=1 ρ b a ρ c a s for s = 0 ad 1. However, I was uable to obtai exact formulas. 2.2 New estimator Variace We write (2) as ˆρ = ɛ 3 i ɛ i+1 1 ɛ 4 i = d w, say. 4

6 var(ˆρ ), up to the secod order of the Taylor expasio, is give by (5), substitutig d for c ad w for v, respectively. We eed the followig result for octavariate stadard Gaussia distributios with serial correlatios ρ j (Appedix), cov(ɛ 3 aɛ a+s, ɛ 3 bɛ b+s+t ) = 9 ρ b a ρ b a+t + 9 ρ b a+s+t ρ b a s from which we derive, for ɛ i followig process (3): ( 1 cov ɛ 3 aɛ a+s, 1 ) ɛ 3 a=1 bɛ b+s+t = = 1 2 b= ρ s ρ b a ρ b a+s+t + 18 ρ b a ρ b a s ρ s+t + 18 ρ s ρ 2 b a ρ s+t + 18 ρ 2 b a ρ b a+s+t ρ b a s + 6 ρ 3 b a ρ b a+t, (16) ( 9 ρ b a ρ b a+t + 9 ρ b a+s+t ρ b a s + 18 ρ s ρ b a ρ b a+s+t + 18 ρ b a ρ b a s ρ s+t + 18 ρ s ρ 2 b a ρ s+t + 18 ρ 2 b a ρ b a+s+t ρ b a s + 6 ρ 3 b a ρ b a+t ). (17) Without further simplificatio we ca directly obtai the various costituets of var(ˆρ ) by the meas of (17). ( ) 1 var(d) = var ɛ 3 i ɛ i+1 = (put s = 1 ad t = 0 i (17)) = ρ ρ 2 b a ρ ρ 4 b a. ρ b a+1 ρ b a 1 ρ b a ρ b a ρ ρ 2 b a + 18 ρ b a ρ b a 1 ρ 2 b a ρ b a+1 ρ b a 1 The first ad the fifth sum is give by (7), the secod sum by (8) ad the third by (9). We fid, ρ b a ρ b a 1 = 5 ρ b a ρ b a+1,

7 which is give by (9), ad 2 ρ 2 b a ρ b a+1 ρ b a 1 = ( 1) ρ i ρ 4( i 1) = ( 1) ( ) 2 1 ρ + 4 ρ (ρ4 8 ρ 4 ) (18) (1 ρ 4 ) 2 2 ρ 4 b a = ( 1) + 2 i ρ 4( i 1) = ( 1) ( ) 2 1 ρ (ρ4 8 ρ 4 ). (19) 4 (1 ρ 4 ) 2 (7), (8), (9), (18) ad (19) give var(d). Further, cov(w, d) = cov ( 1 ɛ 4 i, 1 ɛ 3 i ɛ i+1 ) = (put s = 0 ad t = 1 i (17)) = The last sum of this expressio, ρ 3 b a ρ b a+1 = ( 1) ρ + (7), (9) ad (20) give cov(w, d). Further, var(w) = var ρ b a ρ b a ρ ρ 3 b a ρ b a+1. 2 i ρ 4( i) 3 + ρ 2 b a 2 ( 1 = ( 1) ρ 1 ρ ρ ρ 1 ) 5 ρ i ρ 4( i) 5 + (ρ4 7 + ρ 4 9 ) (1 ρ 4+4 ) (1 ρ 4 ) 2. (20) ( 1 = ɛ 4 i ) = (put s = t = 0 i (17)) ρ 2 b a ρ 4 b a,

8 which is give from (7) ad (19), respectively. We further eed ad E(w) = E(d) = 3 ( 1) (21) 3 ρ ( 1). (22) As for the old estimator, the terms ( 1) 2 which cotribute to var(ˆρ ) cacel out: Mea var(ˆρ ) 5 (1 ρ 2 ) 3 ( 1). (23) We use (14) with d istead of c ad w istead of v, respectively. We derive cov(w, d) from (7), (9) ad (20), further, var(w) from (7) ad (19), ad we have E(w) ad E(d) give by (21) ad (22), respectively. This yields, up to the secod order of deviatios from the meas of d ad w, ( E(ˆρ 1 4 ) ρ ( 1) 3 ρ 5 2 ) 1 + ρ Remark + [ 1 4 (ρ ρ2 1 ) ( 1) 2 (1 ρ 2 ) (ρ 3 ρ 4 1 ] ). (24) (1 ρ 2 ) (1 + ρ 2 ) 2 ˆρ is foud to have a larger egative bias ad also a larger variace tha ˆρ. I the simulatio experimet, we ited ot oly to prove that. We are also iterested to examie the sigificace of the term ( 1) 2 i (15). I the compariso we iclude the theoretical results of White (1961) who studied the old estimator (1) for process (3). He calculated the kth momet of (c/v) via the joit momet geeratig fuctio m(c, v), with c ad v defied as i (4) without the factors 1/, E(ˆρ k ) = 0 vk v2... k m(c, v) c k dv 1 dv 2... dv k. c = 0 He expaded the itegrad to terms of order 3 ad ρ 4 ad gave the followig results (the idex W refers to his study): var(ˆρ) W ( ) ( ) ρ ρ4, (25) E(ˆρ) W ( ) ρ ρ ρ5. (26) 7

9 3 Simulatio experimets I the first experimet, for each combiatio of ad ρ listed i Table 1, we geerated a set of time series from process (3). For every time series, ˆρ ad ˆρ have bee calculated. The sample meas, µˆρ sim ad µˆρ sim, ad the sample stadard deviatios, σˆρ sim ad σˆρ sim, over the simulatios are listed i Table 1. Therei, our theoretical values from (12), (15), (23) ad (24) are icluded. I case of the meas, these are additioally splitted ito terms ( 1) 2 ad terms of lower order. Also the theoretical values from White s (1961) formula, herei (25) ad (26), are listed. We further ivestigated whether the distributio of ˆρ approaches Gaussiaity as fast as that of ˆρ. Fig. 1 shows histograms of ˆρ, respectively ˆρ, from the first simulatio experimet i compariso with Gaussia distributios N(µˆρ sim, σ 2ˆρ sim), respectively N(µˆρ sim, σ 2ˆρ sim). I the secod simulatio experimet (Fig. 2), formulas (12) versus (25) ad (15) versus (26) are examied over the etire rage of ρ, with particular iterest o ρ large. We plot the egative bias, ρ E(ˆρ), ad var(ˆρ). For each combiatio of ρ ad, the umber of simulated time series after (3) is The error due to the limited umber of simulatios is small eough for ot ifluecig ay iteded compariso. 4 Results ad discussio The first simulatio experimet (Table 1) cofirms, over a broad rage of ρ ad, that ˆρ has a larger bias ad a larger variace, respectively, tha ˆρ. I geeral, the deviatios betwee the theoretical results ad the simulatio results decrease with icreased, as is to be expected. For ay combiatio of ρ ad listed i Table 1, our terms ( 1) 2 i (15), respectively (24), brig these theoretical results closer to the simulatio results, with the exceptio of E(ˆρ) for ρ = 0.9 ad = 800 where the simulatio oise prevets that compariso. For ρ large, respectively small, these secod order terms cotribute heavier. The frequecy distributio of ˆρ (Fig. 1) has a similar shape as that of ˆρ. It is shifted to smaller values agaist that ad also broader, reflectig the larger egative bias, respectively the larger variace. The fuctioal form of the distributio of ˆρ is approximately the Leipik distributio, which is heavier skewed for ρ large ad teds to Gaussiaity with icreased. Both estimators seem to approach Gaussiaity equally fast (Fig. 1). The secod simulatio experimet (Fig. 2) compares White s (1961) theoretical results for ˆρ, (25) ad (26), with ours, (12) ad (15). It shows that his are better performig up to a certai value of ρ. Above, our formulas describe better the simulatio, particularly, the de- 8

10 clie to zero egative bias, respectively zero variace, as ρ approaches uity. The declie to zero egative bias is caused by the term ( 1) 2 i (15). Those declies are reasoable, sice for ρ = 1 all time series poits have equal value ad c = v i (4). For small ρ, his formulas perform better sice they are more accurate with respect to powers of (1/) tha ours. For larger ρ his approximatio becomes less accurate. I particular, for 3 ad ρ > 0, ρ E(ˆρ) W caot d become zero. For 10, (ρ E(ˆρ) dρ W ) caot become egative. For 8, var(ˆρ) W caot become zero. That meas, for those cases White s (1961) formulas caot produce the declie to zero egative bias ad zero variace, respectively. The fact that Bartlett s (1946) formula for the variace, (13), describes the simulatio better tha (12) for ρ 0, is regarded as spurious. These results mea that the expasio formulas for var(ˆρ) ad E(ˆρ), (5) ad (14), respectively, are sufficiet to derive the pricipal behaviours for ρ 1. I case of the mea, however, o further approximatio is allowed which would lead to the term ( 1) 2 i (15) be eglected. 5 Coclusios 1. I case of the statioary AR(1) process (3) with kow mea, the ew estimator, ˆρ, has a larger egative bias ad a larger variace tha the old estimator, ˆρ. Its distributio teds to Gaussiaity about equally fast. 2. Our formula (15) for E(ˆρ) describes the expected declie to zero egative bias for ρ The formula (12) for var(ˆρ) describes the expected declie to zero variace for ρ The secod order Taylor expasio of a ratio is sufficiet to derive these two pricipal behaviours for ρ 1, if o further approximatios are made. This coditio could be fulfilled sice we had cocetrated ourselves o the lag-1 estimator ad process (3). 5. For ukow mea of the process, it is more complicated to derive the equatios with the same accuracy. Ackowledgemets Q. Yao is appreciated for commets o the mauscript. The preset study was carried out while the author was Marie Curie Research Fellow (EU-TMR grat ERBFMBICT971919). 9

11 Refereces Bartlett, M. S., 1946, O the theoretical specificatio ad samplig properties of autocorrelated time-series: Joural of the Royal Statistical Society Supplemet, v. 8, p (Corrigeda: 1948, Joural of the Royal Statistical Society Series B, v. 10, o. 1). Marriott, F. H. C., ad Pope, J. A., 1954, Bias i the estimatio of autocorrelatios: Biometrika, v. 41, p Priestley, M. B., 1981, Spectral Aalysis ad Time Series: Lodo, Academic Press, 890 p. Scott, D. W., 1979, O optimal ad data-based histograms: Biometrika, v. 66, p White, J. S., 1961, Asymptotic expasios for the mea ad variace of the serial correlatio coefficiet: Biometrika, v. 48, p Appedix We assume that ɛ i is draw from a stadard Gaussia distributio with serial correlatios ρ j. We derive (16) by meas of the momet geeratig fuctio. Write cov(ɛ 3 aɛ a+s, ɛ 3 bɛ b+s+t ) = E(ɛ 3 aɛ a+s ɛ 3 bɛ b+s+t ) E(ɛ 3 aɛ a+s ) E(ɛ 3 bɛ b+s+t ) = E(ɛ 3 aɛ a+s ɛ 3 bɛ b+s+t ) 9 ρ s ρ s+t. Now, E(ɛ 3 aɛ a+s ɛ 3 bɛ b+s+t ) is the coefficiet of (t 3 1 t 2 t 3 3 t 4 )/(3! 1! 3! 1!) i the momet geeratig fuctio with m(t 1, t 2, t 3, t 4 ) = exp σ 2 ij t i t j, j=1 σ 11 = σ 22 = σ 33 = σ 44 = 1, σ 12 = σ 21 = ρ s, σ 13 = σ 31 = ρ b a, σ 14 = σ 41 = ρ b a+s+t, σ 23 = σ 32 = ρ b a s, 10

12 σ 24 = σ 42 = ρ b a+t, σ 34 = σ 43 = ρ s+t. Terms (t 3 1 t 2 t 3 3 t 4 ) i m(t 1, t 2, t 3, t 4 ) ca oly be withi the fourth order of the series expasio of the expoetial fuctio, i. e., withi further restrictig, withi σ 4! 2 ij t i t j, j= (t2 1 + t σ 12 t 1 t σ 13 t 1 t 3 further restrictig, withi + 2 σ 14 t 1 t σ 23 t 2 t σ 24 t 2 t σ 34 t 3 t 4 ) 4, (4 σ2 13t 2 1t t 2 1t σ 12 t 3 1t σ 13 t 3 1t σ 14 t 3 1t σ 23 t 2 1t 2 t σ 24 t 2 1t 2 t σ 34 t 2 1t 3 t σ 12 t 1 t 2 t σ 13 t 1 t σ 14 t 1 t 2 3t σ 23 t 2 t σ 24 t 2 t 2 3t σ 34 t 3 3t σ 12 σ 13 t 2 1t 2 t σ 12 σ 14 t 2 1t 2 t σ 12 σ 34 t 1 t 2 t 3 t σ 13 σ 14 t 2 1t 3 t σ 13 σ 23 t 1 t 2 t σ 13 σ 24 t 1 t 2 t 3 t σ 13 σ 34 t 1 t 2 3t σ 14 σ 23 t 1 t 2 t 3 t σ 23 σ 34 t 2 t 2 3t 4 ) 2. Fially, these terms are (2 4 σ2 13t 2 1t σ 12 σ 34 t 1 t 2 t 3 t σ 2 13t 2 1t σ 13 σ 24 t 1 t 2 t 3 t σ 2 13t 2 1t σ 14 σ 23 t 1 t 2 t 3 t t 2 1t σ 12 σ 34 t 1 t 2 t 3 t t 2 1t σ 13 σ 24 t 1 t 2 t 3 t t 2 1t σ 14 σ 23 t 1 t 2 t 3 t σ 12 t 3 1t 2 4 σ 34 t 3 3t σ 13 t 3 1t 3 4 σ 24 t 2 t 2 3t σ 13 t 3 1t 3 8 σ 23 σ 34 t 2 t 2 3t σ 14 t 3 1t 4 4 σ 23 t 2 t σ 23 t 2 1t 2 t 3 4 σ 14 t 1 t 2 3t σ 23 t 2 1t 2 t 3 8 σ 13 σ 34 t 1 t 2 3t σ 24 t 2 1t 2 t 4 4 σ 13 t 1 t σ 34 t 2 1t 3 t 4 4 σ 12 t 1 t 2 t σ 34 t 2 1t 3 t 4 8 σ 13 σ 23 t 1 t 2 t σ 12 t 1 t 2 t σ 13 σ 14 t 2 1t 3 t σ 13 t 1 t σ 12 σ 14 t 2 1t 2 t σ 14 t 1 t 2 3t 4 8 σ 12 σ 13 t 2 1t 2 t σ 12 σ 13 t 2 1t 2 t 3 8 σ 13 σ 34 t 1 t 2 3t σ 13 σ 14 t 2 1t 3 t 4 8 σ 13 σ 23 t 1 t 2 t 2 3). 11

13 Thus, we fid E(ɛ 3 aɛ a+s ɛ 3 bɛ b+s+t ) = 3! 1! 3! 1! (96 σ σ σ 13 σ σ 14 σ σ 12 σ 13 σ σ 13 σ 23 σ σ 12 σ 2 13σ σ 2 13σ 2 14σ σ 3 13σ 24 ). This gives the fial result cov(ɛ 3 aɛ a+s, ɛ 3 bɛ b+s+t ) = 9 ρ b a ρ b a+t + 9 ρ b a+s+t ρ b a s + 18 ρ s ρ b a ρ b a+s+t + 18 ρ b a ρ b a s ρ s+t + 18 ρ s ρ 2 b a ρ s+t + 18 ρ 2 b a ρ b a+s+t ρ b a s + 6 ρ 3 b a ρ b a+t. It should be oted that this result has bee checked usig the joit cumulat of order eight, i my assessmet without less effort. 12

14 Table 1: First simulatio experimet. The umber of simulated time series after (3) is i each case S.d.: stadard deviatio. The order of the ucertaity of the mea, respectively the stadard deviatio, due to the limited umber of simulatios, is S.d. / Sim: simulatio (sample mea ad sample stadard deviatio, µˆρ sim ad σˆρ sim, respectively µˆρ sim ad σˆρ sim ). T(24, 23): theoretical value, this study, (24), respectively (23). 2d: term ( 1) 2 i (15), respectively (24). 1st: terms ot ( 1) 2 i (15), respectively (24). ˆρ Sim ˆρ Sim ˆρ T(24) 1st ˆρ T(24) 2d ˆρ T(24, 23) ˆρ T(15) 1st ˆρ T(15) 2d ˆρ T(15, 12) ˆρ T(26, 25) ρ =.200 Mea = 10 S.d S.d. / ρ =.200 Mea = 20 S.d S.d. / ρ =.200 Mea = 50 S.d S.d. / ρ =.500 Mea = 10 S.d S.d. / ρ =.500 Mea = 50 S.d S.d. / ρ =.500 Mea = 150 S.d S.d. /

15 Table 1: (Cotiued) ˆρ Sim ˆρ Sim ˆρ T(24) 1st ˆρ T(24) 2d ˆρ T(24, 23) ˆρ T(15) 1st ˆρ T(15) 2d ˆρ T(15, 12) ˆρ T(26, 25) ρ =.900 Mea = 20 S.d S.d. / ρ =.900 Mea = 100 S.d S.d. / ρ =.900 Mea = 800 S.d S.d. / ρ =.980 Mea = 20 S.d S.d. / ρ =.980 Mea = 200 S.d S.d. / ρ =.980 Mea = 2000 S.d S.d. / ρ =.999 Mea = 500 S.d S.d. / ρ =.999 Mea = 2000 S.d S.d. / ρ =.999 Mea = S.d S.d. /

16 ρ = 0.20 = ρ = 0.20 = ρ = 0.20 = Figure 1: First simulatio experimet (cf. Table 1). Histograms of ˆρ (thick lie), respectively ˆρ (thi lie), compared with Gaussia distributios N(µˆρ sim, σ 2ˆρ sim) (heavy lie), respectively N(µˆρ sim, σ 2ˆρ sim) (light lie). The umber of histogram classes follows Scott (1979). 15

17 ρ = 0.50 = ρ = 0.50 = ρ = 0.50 = Figure 1: (Cotiued) 16

18 ρ = 0.90 = ρ = 0.90 = ρ = 0.90 = 800 Figure 1: (Cotiued)

19 ρ = 0.98 = ρ = 0.98 = ρ = 0.98 = Figure 1: (Cotiued) 18

20 ρ = = ρ = = ρ = = Figure 1: (Cotiued)

21 ρ - E(ρ^) 0.10 = SQRT[var(ρ^)] = ρ Figure 2: Secod simulatio experimet. Above: egative bias, below: stadard deviatio. Simulatio results (dots). Theoretical result, our formula, (12) ad (15), respectively, (heavy lie). Theoretical result, White s (1961) formula, (25) ad (26), respectively, (light lie). 20

22 ρ - E(ρ^) = SQRT[var(ρ^)] 0.00 = ρ Figure 2: (Cotiued) 21

23 ρ - E(ρ^) = SQRT[var(ρ^)] 0.00 = ρ Figure 2: (Cotiued) 22

Topic 9: Sampling Distributions of Estimators

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