Steps Toward Modeling the Distribution of Automobile Retirements 

Brian K. Sliker 

U.S. Department of Commerce, Bureau of Economic Analysis. Contact: brian.sliker@bea.doc Paper prepared for the Federal Committee 

on Statistical Methodology Research Conference, Arlington, VA, November 17-19, 2003. The views expressed are the author’s and are not 

official positions of the Bureau of Economic Analysis or the Department of Commerce. 

As part of its comprehensive revision of the U.S. National Income and Product Accounts, the Bureau of Economic Analysis 

has recently begun to construct new estimates of automobile depreciation. Three steps are involved: modeling the 

retirements of an initial production cohort of new automobiles through time, assigning appropriate prices to survivors, and 

tracking ownership transfers among businesses, governments, and non-business households. The present essay describes 

only the first step: modeling the distribution of automobile lifespans parametrically, including parameter changes across 

model-years. This is essentially a generalized least-squares project that uses covariances among the order statistics of an 

assumed distribution of retirements to re-weight retirement-count data, tamping down heteroskedastic and serially-correlated 

errors in a small-sample framework. The trick, following John S. White, is to calculate the covariances before fitting the 

distribution’s parameters; this reverses the usual course of generalized least squares, which fits provisional parameters first, 

tests residuals for departures from sphericality, and re-weights for valid large-sample results. The first part of the essay 

discusses the data, limitations of which sharply constrain the statistical model of the second part. The third part presents a

constellation of model-year results, leading to pooled estimates in the fourth part, which the Bureau may implement in the 

comprehensive revision due for release December 10, 2003. Shortcomings and extensions conclude. 

Data 

Table 1 presents all the data considered for use in this paper’s statistical procedures. Drawn from various issues of Ward’s 

Automotive Yearbook with an update from Ward’s Automotive Reports, the numbers give counts, to the nearest thousand, of 

automobiles registered at state motor vehicle bureaus for all fifty states plus the District of Columbia as of July 1 of the years 

shown at the head of each column.1

 Within a column, successive entries give the number of autos registered originating with 

the model-years listed at the table’s left edge, from the most recent model-year back to survivors over 15 years old; a catchall row of very old autos, termed “prior,” is near the bottom. This paper takes “age” to be: 

age = registration year – model year + 0.5, 

so that the 2500 (thousand) autos registered as of July 1, 1988 from model-year 19732

 are all treated as 15.5 years old. The 

convention is probably not far from the best single-date guess of a car’s origin: production of new autos typically builds up 

from the summer before the calendar year for which the “model-year” is named and persists around twelve months; sales 

follow by a quarter, and registrations tarry even longer. But statistical procedures taking thus-derived “age” as an argument 

will be subject to an errors-in-variables problem, since the actual ages of individual autos are distributed about the 

conventional age: so this paper treats (functions of) age as the dependent variable in regressions and (functions of) 

registration counts as the independent variable, even though “measured age” (e.g., 0.5, 1.5, 2.5, ..., 15.5) appears so much less 

random than successive registration decrements.3

 Arguments from the relative precisions of measured age and registrations 

would lead to the same ordering. Table 2 plots the rough age distributions of automobiles registered in 1970, 1975, 1980, ... 

2000. “Brand-new” cars (i.e., those up to half a year old) make up less than 6 percent of registered automobiles since 1975, 

with the modal age around two years4

; yet while autos aged more than 14.5 years comprised less than 3 percent of 

registrations in 1970 and 1975, they claimed 15 percent of registrations by 2000. 

Statistical interest in the rest of the essay focuses on the rows of Table 1, which normalized may be thought of as rough 

survivorship curves. This interpretation has several difficulties. First, from the 1969 model-year forward, counts in each row 

increase for a year or two5

 as “new” cars may take over a year to sell and even longer to register. The upshot is that the 

initial, maximal count for a model-year is unknown: the data are “left truncated.” While the number of autos scrapped very 

 

1

 Recent Yearbooks resemble Table 1 but span fewer years. Early Yearbooks give only a column at a time. 2

 Selected because the 1973 row and 1988 column are fairly close to the middle of Table 1. 3

 Age or its logarithm is the dependent variable in typical failure-time regressions (e.g., Meeker and Escobar, chapter 17), but “typical” 

studies track all or most units until they fail: the recorded breakdown-ages in that “typical” case are plainly stochastic, unlike here. 4

 The effects of the 1975 and 1981-2 recessions are visible in the trough at the 5-year-old mark for the 1980 model-year line (in yellow) and 

the bowl in the 2-3 –year range for the 1985 model-year line (in blue).

5

 ...or three, for the 1991 and 1993 model years. 

Page 91
young is surely small, left truncation is severe for early model-years, where only registration “tails” are observed. Second, 

the “tails” themselves are not so long—the data are “right censored”—as no row tracks cars past age 15.5; spans for recent 

model-years are shorter still. Third, the data are “interval censored”: precise retirement ages are not recorded, only retirement 

sums from one July 1st to the next. Given the large number of retirements occurring over a year, it is safe to infer the 

scrappage ranks of units retired very near July 1. For example, in Table 1 for the 1973 model year, the largest observed 

registration count, 11332 (thousand) is at age “2.5.” By age 3.5, the count is down to 11130, so the 202nd retirement (from 

the left-truncated observed maximum) must have happened at or very near age 3.5. Similarly, the 478th retirement is treated 

as having occurred “at” age 4.5, etc., out to the 8832nd retirement at age 15.5, at which 2500 (thousand) 1973-vintage 

automobiles remain. So the longest model-year retirement-rank series have only 14 observations; model-years 1955 and 

1998 have a single useable observation each. Table 3 gives a sense of the problems of treating raw survivor rates as 

“curves,” plotting the fractions of automobile registrations from model-years 1970, 1975, ... 1995 that last from the nearly 

maximal registration-count age of 1.5. Only four of the curves cross the 50-percent mark; the 1990 model-year curve is 

unavailable after age 10.5: not even 20 percent of its units have expired. Moreover, while the median survival age seems to 

increase steadily by model-year6

, the curves trade places several times down to the 75th percentile: “young” data seem 

unavoidably wild, so the short series of the most recent model years are not to be trusted. And some model years are just 

“bad”: the 1975 vintage in particular is plagued by early retirements. 

Model and Technique 

Parametric reliability/failure-time statistical models often use the Negative Exponential or Gamma distributions, or 

logarithmic transforms of the Normal, Logistic, and Smallest- or Largest- Extreme Value distributions. All are restricted to 

nonnegative retirement ages, all typically (but not exclusively) have long right tails, and all but the Negative Exponential 

allow single modes at positive ages (the LogNormal and logarithmic transform of the Largest Extreme Value compel them). 

The Weibull distribution, which log-transforms the Smallest Extreme Value distribution, stands out for its flexibility and ease

of use: 

Cumulative Distribution Function Probability Density Function

β /( θ) 1)( s esF − −= β β /(1 θ)

β θ

β )( s essf −− = (1) 

where F(s) = Pr(S ≤ s): the probability that the retirement age, given by the random variable S, occurs by some realized age s

for positive shape parameter β and positive spread parameter θ. The Weibull matches the Negative Exponential for β=1 and 

Rayleigh for β=2, simulates a Normal for β≈3.5, and is left-skewed for β>~3.6. If automobile lifespans follow a Weibull, then 

for known β and θ the average retirement age should be θΓ(1+1/β), the median θ(ln 2)1/β

, and the mode θ(1–1/β)

1/β

 for β>1. 

Moreover, when s=θ, only some 36.8 percent of the original registrations should still survive. Table 3 shows the 36.8th

percentile as a mottled blue line: the 1970, 1975, and 1980 survivor “curves” cross it at about 13.1, 13.5, and 14.3 years, 

respectively.7

 The Weibull form also accommodates truncated data easily. When the number of retirements below age s0 is 

unknown as in the Ward’s data, the CDF becomes: 

 ...8

 (2) β β 0 /( θ /() θ)

0 1)|( ss esssF − −=≥

Fitting β and θ via the truncated CDF enables the reconstruction of the untruncated form and thence an estimate of the 

missing “age-0” maximal registration count. These practical advantages are persuasive: further work in the paper relies 

exclusively on the truncated Weibull CDF. One might now regress the fraction of age-s0 registrations that survive to age s

against e(s0 / θ)

β −(s/ θ)β

 or, almost linearizing, regress: 

( ) εθβ β

0

β

0

+−−=





























 lnssln

s

s lnln

registrati age at count on

registrati age at count on - ...9

 

6

 In Table 3, the median retirement ages for 1970, 1975, 1980, and 1985 –model-year autos look to be 11.2, 11.9, 12.5, and 14.2 years, 

respectively. By comparison, the most recent version of Schmoyer’s unpublished scrappage study puts the 1970 model-year median at 11.5 

years, the 1980 median at 12.5, and the 1990 median at 16.9 (cited by Davis, Edition 23, Table 3.9, p. 3–13). [N.B.: In Edition 20, 

Schmoyer’s 1970, 1980, and 1990 model-year median survival ages were 11.3, 12.2, and 14.0 years, respectively.] 7

 By comparison, interpolating Schmoyer’s results (c.f. footnote 6) puts 36.8 percent of the 1970, 1980, and 1990 model-year autos 

surviving to ages 13.6, 14.7, and 19.7, respectively. 

8

 C.f. the three-parameter Weibull CDF, which forbids retirements below s0: . 

β 0 /)[( θ]

0 1)|( ss esssF −− −=≥ 9

 When s0=0, the right side becomes β(ln s – lnθ) + ε, which is linear in β and βlnθ. 

Page 92
for each model year, with ε a zero-mean error. But the relative errors-in-variables argument given above urges a reversal: 

( ) θ εβ β 0

β

0

β

+ 































+= −

s

s lnlnln ssln

registrati age at count on

registrati age at count on - , (3)

which has the unfortunate but not fatal side-effect of putting the shape parameter on both sides of the equation. That the 

right-side slope regressor is stochastic—though “less” so than (functions of) age—is no obstacle to consistent estimation 

provided the regressor and error term are contemporaneously uncorrelated; in any event the regressor will soon enough be 

replaced by an appropriate expectation. 

Common complaints against reliability regressions (not only of the Weibull distribution) are the undue influence of very 

early or late failures in short samples—already hinted at in early-age criss-crossing of the “curves” of Table 3—and the 

marked serial correlation of residuals. Maximum likelihood is not much help: in small samples ML estimates of β are biased 

upward. A cottage industry has arisen to propose a bewildering variety of remedies,10 which a recent series of papers in the 

IEEE Transactions on Dielectrics and Electrical Insulation sorts out and tests under various Monte Carlo censoring (but not 

truncation) regimes.11 The best or nearly best technique in all cases considered is a weighted least-squares scheme first 

implemented by White (1964, 1969), which the current essay adapts to the left-truncated case. The key insight is to construe 

the sequence of retirements as order statistics, in which the distribution of the i

th retirement out of n units is different from, 

but correlated with, the distribution of the j

th retirement, and both are narrower than the overall distribution.12 The probability 

density function of the i

th retirement from n original units drawn from an overall distribution with CDF F(s) and PDF f(s) is: 

( ) )()(1)( )!()!1(

! )( 1

/ sfsFsF ini

n sf i in

ni

− − − −− = , (4)

which may be interpreted as the product of f (s) and a Beta density function: x a−1

(1− x)

b−1 B(a,b) , with 0≤F(s)≤1 acting as 

x, i as a, and n–i+1 as b. The magnitudes of i and n at age s reshape the Beta PDF to emphasize the early, middle, or late 

reaches of the overall f (s). For example, consider the probability densities of the first, third, seventh, and tenth retirements 

from a sample of ten drawn from an untruncated Weibull distribution (the heavy line, below) for β=2.5 and θ=15: The distributions of the first and tenth retirements, taken from the thin parts of the overall Weibull, are wider than the distributions of 

the third and seventh, which are drawn closer to the “hump”: this is the source of “wild” early and late observations. 

The joint density of the i

th and jth retirements is: 

( )( ) )()()(1)()()( )!()!1()!1(

! ),( 1 1

// ji

jn

j

ij

ij

i

njni i sFsFsF sfsfsF jniji

n ssf − −− − − − −−−− = , (5)

where F(si) and F(sj) are the overall CDFs at the ages of the i

th and jth retirements, respectively, and f(si) and f(sj) are the 

corresponding PDFs. Note that expression (5) cannot be decomposed into the product of f(si/n) and f(sj/n) from expression (4), 

so the i

th and jth retirements are not independent. 

 

10 C.f. “Dr. Bob” Abernethy’s Handbook (2000) and its battery of software and courses; or the WeibPar software package, which 

implements a β correction factor similar to Ross (1994, 1996) and is distributed free-of-charge by Connecticut Reserve Technologies. 11 Cacciari, Mazzanti, and Montanari (1996), followed by Montanari, Mazzanti, Cacciari, and Fothergill (1997a, 1997b, 1998). 12 Mood, Graybill, and Boes (1974), Chapter VI.5, pp. 251-265. 

Page 93
White’s second insight was to connect a two-parameter Weibull distribution to a zero-parameter Smallest Extreme Value 

distribution (which White called a “Reduced Log-Weibull”), and from there to calculate the means, variances, and 

  covariances of ln ln registration count at age s  -  for use in generalized least squares. Applying White’s approach to the      registration count at age s0 

left-truncated Weibull CDF, recall (2) but consider the cumulative distribution function of the random variable Y: 

, (6)

y e yFyY e− 1)()Pr( −==≤

where realized values of Y and S are related monotonically as: y = ln[(s /θ)

β

–(s0/θ)

β

] = ln(s

β

–s β

0 ) – βlnθ, with y covering the 

entire real line. The distribution function in (6) has no parameters, so its moments are immediately calculable: e.g., E(Y) = −γ

and Var(Y) = π2

/6. More to the point, the moments of its order statistics are (numerically) calculable also: 

∫

∞

∞−

−−−−− − −− = dyeee ini

n yYE y eyin y ei y e

ni )()1( )!()!1(

! )( 1

/ (7)

2

/

2 1

/ )()1( )( )!()!1(

! )( ni

y eyin y ei y e

ni EYdyeee ini

n yYVar − − −− = ∫

∞

∞−

−−−−− (8a)

∫ ∫ ∞

∞−

∞

∞−

−−− −−−−− −−+ − − −−−− = ji

y

e y jn eyy y

eij ye y ei y e

njni ji eeeee dydy jniji

n YYCov yy j i ji j j i i

()1( )() )!()!1()!1(

! ),( 1 1

// (8b) 

− EYEY // njni

Apply expectations to the relationship between Y and S, and rearrange: E[ln(si

β

−s0

β

)]/β = lnθ + E(Yi/n)/β. So from (3), find: 

(YE ni s

s lnlnE /

0

=

















































registrati age at count on

registrati age at count on - ). Next compare 































0s

s ln

registrati age at count on

registrati age at count on ln - to (Yi/n) with data, 

say, from the “1973” model-year row of Table 1; the match is excellent: 

Age: 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5

Regis. Count, c: 11130 10854 10559 9965 9151 8458 7629 6798 5881 4883 3929 3161 2500

ln(-ln(c/11332)): -4.0181 -3.1443 -2.6500 -2.0514 -1.5429 -1.2292 -0.9272 -0.6714 -0.4217 -0.1721 0.0576 0.2443 0.4130

i: 202 478 773 1367 2181 2874 3703 4534 5451 6449 7403 8171 8832

EYi/11332: -4.0207 -3.1454 -2.6507 -2.0518 -1.5432 -1.2295 -0.9274 -0.6716 -0.4219 -0.1723 0.0574 0.2442 0.4128

To transform the “errors-in-the-variables” problem to an “errors-in-the-regression” problem, remove expectations but keep 

E(Yi/n) as the slope regressor, shunting (YE ni s

s ln /

0

−































registrati age at count on

registrati age at count on ln - ) into13 the error term: 

( ) θβ ( ) / εβ β

0

β +=− YElnssln ni + (9)

Regressions using form (9) should work out essentially the same as those using (3). 

 

13 ...or “as”: in the single model-year weighting procedures that follow I take ε to be strictly proportional to 

  registration count at age s 

ln-   − (Y ) and so neglect what ε ln E in (3) might already contain, not least the error-in-the-equation      registration at age s  i / n count 0 

brought about by the use of imprecisely-measured age in the dependent variable. An extension of the current project would consider ε the 

sum of   registration count at age s 

ln-ln  − E   (Y and the average of the discrepancies between the actua es of surviving autos and   i / n ) l ag

  registration count at age s0 

the reported conventional age. On the plausible view that the distribution of production/purchase/registration dates is independent of the 

distribution of lifespans, constraining the “registration-count errors” as in the text might allow residual identification of the second moment 

of the “age errors,” even though the data are aggregated. 

Page 94
The payoff to using a tightly-specified error term is the ability to calculate V=E(εε′) in advance, from formulas (8a) and (8b), 

for use as a weighting matrix in generalized least squares -type procedures, which should produce tighter, more stable 

parameter estimates than unweighted estimators when errors are heteroskedastic and serially correlated. Again taking modelyear 1973 as the example, observe immediately below that the variance-covariance matrix of the errors, normalized so its 

trace equals the number of observations, is nothing like the σ2

I matrix common to ordinary least squares. The diagonal 

elements are heteroskedastic—the last observation would get )774.5/1/()159./1( = 6.026 times the weight of the first in 

precision-weighted (i.e., diagonals-only) least squares—while the off-diagonal elements display strong positive correlation, 

even though the particular retirements being counted are thousands of units apart: 

“V” matrix for retirements of 1973 model-year autos: 

5.774 2.406 1.467 0.806 0.485 0.354 0.262 0.203 0.158 0.123 0.098 0.081 0.069

2.406 2.437 1.486 0.816 0.491 0.359 0.265 0.205 0.160 0.125 0.099 0.082 0.069

1.467 1.486 1.507 0.828 0.498 0.364 0.269 0.208 0.162 0.126 0.100 0.083 0.070

0.806 0.816 0.828 0.853 0.513 0.375 0.277 0.214 0.167 0.130 0.103 0.086 0.072

0.485 0.491 0.498 0.513 0.536 0.391 0.289 0.224 0.175 0.136 0.108 0.090 0.076

0.354 0.359 0.364 0.375 0.391 0.408 0.301 0.233 0.182 0.142 0.113 0.093 0.079

0.262 0.265 0.269 0.277 0.289 0.301 0.318 0.246 0.192 0.150 0.119 0.099 0.083

0.203 0.205 0.208 0.214 0.224 0.233 0.246 0.262 0.204 0.159 0.127 0.105 0.089

0.158 0.160 0.162 0.167 0.175 0.182 0.192 0.204 0.221 0.172 0.137 0.114 0.096

0.123 0.125 0.126 0.130 0.136 0.142 0.150 0.159 0.172 0.191 0.152 0.126 0.107

0.098 0.099 0.100 0.103 0.108 0.113 0.119 0.127 0.137 0.152 0.172 0.143 0.121

0.081 0.082 0.083 0.086 0.090 0.093 0.099 0.105 0.114 0.126 0.143 0.163 0.138

0.069 0.069 0.070 0.072 0.076 0.079 0.083 0.089 0.096 0.107 0.121 0.138 0.159

Generalized least-squares procedures augment expression (9) as: 

( ) ( θβ ( ) εβ ) /

β

0

β +=− YElnPsslnP ni + (10) 

where the matrix P is chosen so that (P′P)

–1=V.14 In the case of precision-weighted least squares—e.g. White (1969) but not 

White (1964)—P is a diagonal matrix with elements that are the reciprocals of the square roots of the diagonal elements of V. 

Whatever the choice of P, least-squares procedures all seek β and θ to minimize: 

( ) ( ) θβ ( ) β ( ( ) θβ ( ) β) /

β

0

1 β

/

β

0

β

ni ni −−− −−− YElnsslnVYElnssln ′ − . 

The specification is intrinsically nonlinear in β, preventing unbiased estimation. Nonetheless, “pre-fit” V–1

 remains 

appropriate for small samples. Contrast this to the usual generalized least-squares approach, where V–1

 depends on firstround parametric estimates and so is appropriate for large samples. 

Results for Separate Model-Years 

The discussion so far sets up a “horse race” between four estimators—unweighted nonlinear least squares with the stochastic 

slope regressor 































0 s

s lnln

registrati age at count on

registrati age at count on - , unweighted NLLS on nonstochastic E(Yi/n) instead, diagonal precisionweighted NLLS using E(Yi/n), and generalized P-weighted NLLS on E(Yi/n)—over 40 separate model years: the “winner” will 

advance to the pooled model that may be useful for prediction. The results are too numerous to describe verbally. Consider 

instead Table 4, which plots the fitted values from each estimator of the shape (β) and ln spread (θ) parameters across model 

years, as if the parameters were time series. The table comprises eight charts, organized into two columns—fitted β on the 

left and fitted lnθ on the right—and four rows, one for each estimation technique: from top to bottom, nonlinear least-squares 

using the stochastic slope regressor, NLLS using nonstochastic E(Yi/n), precision-weighted NLLS with E(Yi/n), and general 

 

14 P is not unique but V is; this paper uses the upper-triangular Cholesky factorization of V–1 as P. 

Page 95
NLLS on E(Yi/n). Several features are clear. First, estimates drawn from 10 or more observations, shown as the thick portion 

of each chart, are more plausible than estimates drawn from less data: point values are wild from the 1990 model year onward 

for all eight charts, while point values before the 1964 model year are “smooth” but drop off sharply. Second, within the 

1964-89 model-year range, fitted lnθ moves much more smoothly than fitted β; this is a fairly standard Weibull result, but it 

means that efforts to locate and tame heterogeneity across model-years by intercept dummies or, for predictive purposes, 

homoskedastic error components, might not find very much. Third, while unweighted NLLS—whether with double-logged 

registration ratios or E(Yi/n)—give rocky estimates of the shape parameter even within 1964-89, the β “series” is much 

smoother under precision weighting, and smoothest of all under general weighting, so White’s approach to least-squares 

estimation seems borne out. None of the methods erases the dip in β for the 1975 model year, and a downward trend in β

becomes apparent since the 1980 model year. 

A few specific general-weighted NLLS results for single model years are presented to prepare a comparison with the medianlifespan and “36.8-percent” θ benchmarks from footnotes 6 and 7: 

Model Year Parameters Point Estimates Variance-Covariance of Parameters 

 1970 2.72053 0.0460191 4.88163×10–6 β

ˆ

 2.59414 4.88163×10–6 0.0009977 ^lnθ

2 σˆ 0.018791 

 1975 β 2.4752 0.0606913 -0.000642015 ˆ

 2.59727 -0.000642015 0.00171051 ^lnθ

 0.00983048 2 σˆ

 1980 β 3.26848 0.0232575 -0.000441853 ˆ

^ lnθ 2.65795 -0.000441853 0.000227824 

2 σˆ 0.00297765 

 1985 β 2.68005 0.0278493 -0.00179112 ˆ

 2.7853 -0.00179112 0.000675882 ^lnθ

 0.00543318 2 σˆ

The approximate expected value of the median lifespan is: 















 − + 













 ≈ + + 



















2

θ)( ,( θβ ) β)2( β)( β

)2( 2

2β

)2( 1)2()2( ^

3 2

θ /1 β ^ θ /1 β lnVar lnCov lnln Var lnlnlnln lnelneE ln ln

with approximate standard error: 

,( θβ θ)() β

)2( β)( 2

β

)2( )2( 2

2

2

θ /1 β lnVarlnCov lnln Var lnln lneln − + 













 . 

The approximate expected value of θ is: ( θ)(1 ^

2

θθ 1 eEe lnVar lnln +≈ ), with approximate standard error θ)( θ lnVar ln

e .

15 

Fitted point values, approximate expectations, and approximate standard errors of the median lifespan and θ follow: 

 

15 To approximate the expected value of an expression that is nonlinear in its random variables—here, in the fitted values of β and lnθ—

take expectations of a second-order Taylor expansion about the true values. The expectation operator removes first order terms, leaving the 

point value plus a sum of weighted variances and covariances. To approximate the variance of a nonlinear expression, take only a firstorder Taylor expansion, then apply the standard variance-of-a-sum rule. In neither case are the true values of β and lnθ observed, so use 

fitted values instead. (Mood, Graybill, and Boes, 1974, p. 181.) 

Page 96
Model Year -------------Median Lifespan------------- ----------------------θ---------------------- 

Point Value Approx. Exp’n. Approx. St. Err. Point Value Approx. Exp’n. Approx. St. Err. 

 1970 11.698 11.6947 .38992 13.3851 13.3917 .422786 

 1975 11.5791 11.5728 .498156 13.4271 13.4386 .555321 

 1980 12.7536 12.7519 .191253 14.267 14.2686 .215343 

 1985 14.1334 14.1299 .336126 16.2046 16.2101 .421283 

By comparison, the “benchmark values” from footnotes 6 and 7 are collected here as: 

Model Year -----Median Lifespan Benchmarks---- ------------θ Benchmarks------------ 

Table 3 “eyeball” Schmoyer Table 3 “eyeball” Schmoyer 

1970 11.2 11.5 13.1 13.6 

1975 11.9 13.5 

1980 12.5 12.5 14.3 14.7 

 1985 14.2 

1990 16.9 19.7 

Fitted point values are all quite close to the approximate expectations, and usually not significantly different from the benchmarks: fitted θ

^ differs significantly from Schmoyer’s benchmark for the 1980 model year by some 5 months. There is no 

1985 “eyeball” benchmark for θ, as the 1985 model-year survival curve in Table 3 has not yet crossed the 36.8th percentile: 

confirmation of the point estimate of 16.2 awaits new data. Separate model-year regressions for model-year 1990 were not 

reliable, so tests against the benchmarks must await the pooled results: for 1990 these will amount to predicted values. 

As a final application of the separate model-year procedures, one could apply the fitted regression coefficients and the covariance term to calculate the original “age-zero” cohort size. First, rewrite: 

( ) 













 −−= β ^ ˆ

0

β

ˆ

/ θβˆ ni )( lnsslnEYE

but replace the first s inside the logarithm by s0 before setting –s0

β

 to zero. This retools White’s expectation exercise to 

compare the largest observed registration count, at age s0, to the unobserved “age-0 count”; it also simplifies the logarithm to 

β )( . Then apply the rule for the expectation of a product to find: ˆ 0sln

β, θ) ˆ θβ ( ˆ β )( ˆ ^ ^

largest /( )count observed 0 YE lnCovlnsln ii + −−=

where I have already substituted the regression-fitted values for the proper but unobserved true β and lnθ. For, say, the 1985 

model-year16, this works out to EYi/(i + 10532) = 2.68005[ln(1.5)–2.7853]+.00179112 = –6.37626. The double-exponential 

approximation to the age-zero count follows as 10532 e

e–6.37626 = 10549.9351 ≈ 10550 to the precision of the data. A strict 

equality of expectations solves: 

∫

∞

∞−

−−− − − −

+ =− dyeee i

i

y

y ey y ei y e 1 10532 )()1(!10532)!1(

)!10532( 37626.6

for i= 18.4341, so that 10532+i again rounds to 10550. 

Pooled Regressions: No Simple Error Components 

Results for individual model years are encouraging where series are long enough and errors are adequately disciplined, but 

statistical agencies cannot always wait for data that are “good enough.” Yet if automobile durability develops smoothly 

across model years, it may be feasible to pool the historic data of available cohorts, run regressions using adjustable 

 

16 The largest entry in the “1985” row of Table 1 is 10532, and it occurs at approximate age 1.5. 

Page 97
“summary parameters,” and then predict the retirement patterns of future cohorts. To the extent that variations across the 

coefficients of individual model-year regressions are substantially random, simple trend models might predict years of auto 

retirements concisely and accurately before all the data are in. Such is the conceit of the following charts of generalized 

NLLS fits of β and lnθ for separate model years 1964-89, overlaid with two-part linear trends: 

The trend lines are fitted values of OLS regressions of the GNLLS coefficients on model-year trends; the split between the 

1978 and 1979 model years gave the smallest sum of squared residuals for the β fit. Statistical preconditions for tests of OLS 

fits of the GNLLS results are not satisfied; in particular, wildly different variances across model-year regressions imply that

the various β’s and lnθ’s as-dependent-variables are not drawn from the same distribution.17 Still, the charts suggest the 

following story, representing what little economic content there is to this essay: the U.S. auto industry underwent a “regime 

change” beginning in the late 1970s, possibly as a delayed response to the 1973-74 oil shocks (repeated in 1979) and 

subsequent increases in import penetration, culminating in Chrysler’s brush with bankruptcy in 1979. Improvements in 

automobile “quality”—crudely, increases in θ—sped up, from around 1 to nearly 3 percent a year, inducing premature 

obsolescence of old-regime models. Cars bought at the cusp of the change, nearest substitutes to the new regime, suffered 

most: witness the small dip in the level of lnθ and the jump in β for the 1979-81 model years, as the otherwise right-skewed 

distribution became nearly symmetric. Since the mid-1980s, lifespans and skew have both increased markedly and smoothly. 

This account has the easy plausibility of a business-school case study, even if it lays too much on the time paths of two 

parameters that are strictly about neither the engineering characteristics of automobiles nor the preferences of auto owners. 

Also like a case study, there is not a standard error in sight. Consider then a simple “stacked” model: 

( )

( )

( )

( )

( ) ( )

( ) ( )

( ( )

( )

)

( )



























































+θ +

+θ +

+θ +

+θ +

=

































































−

−

−

−

[89] [89] / [89]]89[ ]89[

[66] [66] / [66]]66[ ]66[

[65] [65] / [65]]65[ ]65[

[64] [64] / [64]]64[ ]64[

[89]

β

0]89[

β

]89[

]89[

[66]

β

0]66[

β

]66[

]66[

[65]

β

0]65[

β

]65[

]65[

[64]

β

0]64[

β

]64[

]64[

ε

ε

ε

ε

β

β

β

β

β

β

β

β

[89] [89]

[66] [66]

[65] [65]

[64] [64]

ni

ni

ni

ni

YElnP

YElnP

YElnP

YElnP

ssln

P

ssln

P

ssln

P

ssln

P

M M

 (11) 

where the boxed subscripts refer to the pertinent model year—i.e., the first “row” of (11) consists of all ten rows of the 1964

model-year GNLLS regression specified by expression (10), the second “row” contains the ten rows of the corresponding 

1965 model-year regression, the third “row” has the eleven rows of the 1966 model-year regression, etc.: 331 observations 

 

17 Estimated σ2

 for separate GNLLS regressions are: 0.00243868, 0.0019129, 0.00396199, 0.00678624, 0.0291893 and 0.0329714 for 

model years 1964-69; 0.018791, 0.0115717, 0.0228268, 0.0011955 and 0.00841734 for model-years 1970-74; 0.00983048, 0.0163441, 

0.019821, 0.0144801 and 0.0164932 for model-years 1975-79; 0.00297765, 0.0130386, 0.00246744, 0.00627537 and 0.00622816 for 

model-years 1980-84; and 0.00543318, 0.00726288, 0.0142753, 0.0381807 and 0.225687 for model-years 1985-89. Bold-type values are 

already in the text, above, in the results reported for model-years 1970, 1975, 1980, and 1985. Fitted σ2

’s plainly increase over the final 

five model years, which are drawn from successively younger (hence wilder) retirement counts, but the 1989 value is disproportionate. 

Page 98
altogether. Under the specification of separate β’s and lnθ’s and the assumption that Eε[m1]ε[ m2]=0 across different model 

years m1 and m2, the “stacked” regression point values would match the bumpy schedules graphed just above, and the 

expected P-weighted error variance-covariance matrix would be non-scalar diagonal, with elements from footnote 17 filling 

positions corresponding to each model year’s observations in the stack. Such a regression is not efficient, as the separate 

model years could be reweighted until the overall P-weighted error variance-covariance matrix is scalar diagonal. But the 

stacked regression is probably not parsimonious either, so to save 44 parameters, replace separate model-year β’s and lnθ’s 

by simple two-part trends, linear in model years. That is: 

 replace {β[64], β[65], β[66], ...β[89]} by 





⋅+ ≥

⋅+ <

ββ year -modelfor *

ββ year -modelfor *

late

trend

late

0

early

trend

early

0

m mm

m mm

K

K

 and {lnθ[64], lnθ[65], lnθ[66], ... lnθ[89]} by . 







⋅

⋅

+ ≥

+ <

θθ year -modelfor *

θθ year -modelfor *

late

trend

late

0

early

trend

early

0

mlnln mm

mlnln mm

K

K

Table 5 presents the pooled parsimonious results, which were iterated until the joint convergence of individual model-year 

σ2s to the overall σ2 in order to precision-weight the model-year blocks. The impression is the same as the “case study” in 

nearly all respects, although the best joint split is m*

=80. The shape parameter through the 1979 model year is about 2.9, 

indicating mild right skew, and has no significant trend. The relatively high standard error (0.76) was already visible in the 

noisy “β series” graph. From 1980 on, the retirement tail lengthens considerably: β falls significantly by –0.85 per model 

year. The spread parameter increases in both regimes: by about 1.1 percent per model year through the 1979 vintage and 2.7 

percent since; both rates are highly significant, as is their gap (t-ratio = 4.51). The dependent variable ( ) β β

0][

β

][ mim ln − ss is 

not observed without a fitted value for β and so is too smooth; nonetheless the unweighted residual sum of squares is less 

than 5 percent of the sum of squared deviations of unweighted ( ) β β

0][

β

][ mim − ssln from its calculated average, while the 

generalized P-weighted residual sum of squares is barely 0.1 percent of the sum of squared deviations of P-weighted 

( ) β β

0][

β

][ mim − ssln from its average. To test whether the eight “summary parameters” deal too roughly with the separate 

model years, form the F-test: 

)52331/(27655.5

44/)27655.554625.5(

−

− = 0.32410 

where 5.54625 is the sum of the squared weighted residuals from Table 5 and 5.27655 is the sum of the products of the 

estimated variances in footnote 17 with their respective degrees of freedom. Rejection at even the 10 percent confidence 

level requires an F-ratio of 1.29219, so it seems the data are not complaining.18

Comparisons against the benchmarks of footnotes 6 and 7 require computing the point values and approximate expectations 

and standard errors of β, θ, and median retirement ages for model years 1970, 1975, ...1990. The summary shape parameter 

β

^

0 + β

^

trend m is linear in its random variables, so its point- and expected values coincide and its variance is simply 

ar(β0)+2mCov(β0,βtrend)+m2

Var(βtrend).19 V The approximate expectation of θ is: 

( ) (1 θ ( ) θ , θ () θ ) ^

2 0 0 trend 2 trend

θ θθ 1 2 0 trend eEe lnlnVarmlnVar lnVar ln mlnln m ≈ + + + +

with approximate standard error ( θ ( 2) θ , θ () θ ) trend

2

0 0 trend

θθ0 trend e lnVarmlnlnVarmlnVar mlnln + + + . The approximate 

expected value of the median lifespan is: 

 

18 The substance of the result does not change when the constrained sum of squares is 5.483 from the noniterated summary 

model (not reported but quite similar to Table 5) instead of 5.54625. 

19 Assuming away covariances across model years rules out covariances between “early” and “late” parameters, which 

superscripts are therefore dropped. 

Page 99
[ ] 







+ + 















+

+

+ ≈ + 

















 + +

tr

2

0 tr0

tr0 3

tr0

β θθ ˆ θ /1 β (2 ββ ) β ββ

)2( 2

2(β β )

)2( 2 12 ^ β0 β tr

1

tr0 Varm,CovmVar m

lnln

m

lnln lnelneE ln mlnln m

[ ] (β , θ ( ) () β , θ () β , θ () β , θ ) (β β )

)2( trtr

2

00 tr0 0tr 2

tr0

lnCovmlnCovlnCovmlnCov m

lnln

+ + +

+ − 























+ + + tr

2

tr0 0 θ

2 ( θ , θ ) 2

θ lnVar m lnlnCovm lnVar

with approximate standard error: 

[ ] [ tr

2

tr 0 tr0

2

0 tr0

2

2 tr0

θθ β (2 ββ ) β θ (2 θ , θ ) θ

(β β )

)2( 2 β0 βtr

1

tr0 Varm,CovmVar lnVarmlnlnCovmlnVar m

lnln lne mlnln m + + +











+ + 















+

+ + ]

[ ] ( ) 2

1

trtr 2 00 tr0 0tr 2 tr0

(β , θ () β , θ () β , θ () β , θ ) (β β )

)2( 2 







+ + +

+ − lnCovmlnCovlnCovmlnCov m

lnln

Fitted point values, approximate expectations, and approximate standard errors of β, θ, and the median lifespan implied by 

the results of Table 5 follow: 

Model ----------β---------- ----------------θ---------------- -------Median Lifespan------- 

Year Point=Exp’n St. Err. Point Exp’n. St. Err. Point Exp’n. St. Err. 

1970 2.83868 0.0437131 12.9993 12.9995 0.0803356 11.4247 11.4246 0.0747539 

1975 2.83474 0.060424 13.7441 13.7446 0.117403 12.0772 12.0769 0.106591 

1980 3.1326 0.078876 14.15 14.1507 0.141943 12.5876 12.5873 0.123912 

1985 2.70865 0.0629669 16.2352 16.2363 0.183127 14.1805 14.1802 0.144198 

1990 2.28469 0.148864 18.6277 18.6342 0.491757 15.8668 15.8604 0.371906 

Matches with the benchmarks are good. Pooled estimates of θ and the median lifespan for the 1990 model year are about a 

year short of Schmoyer’s most recently revised results; otherwise differences are statistically or substantively negligible. 

The preceding tests are valid so long as the error structure is correctly specified. In models that pool data from two 

dimensions, however, it is common to allow three independent variance components: an idiosyncratic piece plus a component 

each from the model-year and calendar-year dimensions. Both dimensional errors might be present here. First, deviations by 

the true parameters from their fitted trends would find their way into the error, inducing a small model-year component. 

Second, common shocks to operating costs (e.g., fuel price changes) and a pervasive business cycle might generate a 

calendar-year component. Both pieces could be messy: The “β series” of slope parameters graphed above is much bumpier 

than the “lnθ series” of intercepts, so the model-year errors induced by the two-trend model might be heteroskedastic. The 

“nearest substitute” argument suggests that calendar-time shocks may affect recent model years disproportionately, fading or 

even switching signs in older vintages. Further, applying White’s method to heteroskedastic and serially correlated 

idiosyncratic errors, while successful in itself, muddies the residuals needed to track model-year and calendar-year 

components. Finally, the data are not “rectangular,” and parametric nonlinearity frustrates easy averaging, so the trick of 

computing weighted averages of “between” and “within” estimators does not work so readily. 

To remedy some of these objections, this essay compares the variance-covariance matrix of the residuals of the stacked, 

generalized P-weighted regression of Table 5 against the (approximate) expectation of the same matrix, under the maintained 

hypotheses that White’s method fully corrects the idiosyncratic variance but confounds otherwise-homoskedastic model-year 

and calendar-year components. More realistic structures—e.g., an additional idiosyncratic component due to treating all 

members of a model year as having exactly the same age, a model-year component that is proportional to E(Yi/n), and a time 

component that dies away—are put off to future work. Setting up the comparison is a straightforward, if computationally 

awkward, application of least-squares results. Consider the stacked P-weighted residual variance-covariance matrix: 

( ) ( ) ( ) ( ) ′ ′





 





 −−− 

 



 ˆˆ ′′ ≡ ˆ θβ −−− ni β

ˆ ˆ θβ / ni βˆ PYElnsslnYElnsslnPEPPE β

ˆ ^

0

β

ˆ

/

β

ˆ ^

0

β

ˆ

εε

( −≈ ) ′′′′ ( − XXXXIPPXXXXIE ′′ ) − − ~ ) ~ ~ ( ~ ~ ) ~ ~ ( ~ 1 1 εε

Page 100
where X

~

 is the 331×8 matrix of stacked “regressors”—partial derivatives of P[ln(s

β

–so

β

)/β–lnθ–E(Yi/n)/β] with respect to β

for the “slope” and with respect to lnθ for the “intercept”20—and ε is the three-component stacked error vector, εidio + εm + εt.

21

The construct is a variation on the well-known symmetric and idempotent “M” matrix, with rank = trace 

= 331–8. The equality is only approximate due to the nonlinearity of the slope regressor. Since the three error components 

are taken to be independent, each may be analyzed separately. For the idiosyncratic component, White’s fix is assumed to 

work without complications: 

− XXXXI ′′ − ~ ) ~~( ~ 1

E MPεidioεidio ′ P′M = σidio

2

 M P(P′P)

–1P′M = σidio

2

 M. 

The P-matrix is useless on the model-year component: one must go through the calculations term by term and in the final 

step set squared elements of εm to a common σm

2

 that can be pulled out, but crossed elements to zero. Let “Mm” be the 

surviving 331×331 symmetric matrix of constants that multiplies σm

2

: 

E MPεmεm′P′M ≡ σm

2 Mm 

It works out the trace of Mm is 682.665 in these data, and that 51336 of the 109561 elements of Mm are zeroes. Likewise for 

the time component, let “Mt” be the 331×331 symmetric matrix of constants that multiplies σt

2

: 

E MPεtεt

′P′M ≡ σt

2 Mt 

Here the trace of Mt is 25178.7, and no element is zero. A simple matrix-weighted sum of variance components follows as 

the expected value of the outer product of the stacked P-weighted residuals: 

t 2tm 2m 2 εε ˆˆ ′′ idio ++= σσσ MMMPEP

When σm

2

=σt

2

=0, the best quadratic unbiased estimator of σidio

2

 equates traces: . The same estimator 

has a sampling variance of 2σidio

4

 /(331–8) if the errors are Normal. In the three-component case fit: 

σˆ ˆˆ )8331/( 2

idio = ′′ PP εε −

′′ +++= uMMMPP (12) t

2

tm

2

m

2 εε ˆˆ idio σσσ

by linear regression, subject to a trace constraint: . In the spirit of ANOVA 

comparisons of “between” and “within” estimators, and to evade sample-size counting issues, restrict the regression to the 

diagonal elements only. A regression using, say, all the upper-triangular elements would presumably be consistent, but the 

reported standard errors would require special attention. The trace-constrained diagonal-elements-only regression gives: 

t

2

tm

2

m

2 ′′ PP εε ˆˆ = σidio )8331( +− σ trM + σ trM

σidio

2

 σm

2 σt2

Point Value: 0.0181474 –0.000134274 –8.88414×10–6 

Standard Error: (0.00105286) (0.000182946) (0.0000125192)

 Sum of Squared Residuals: 0.202319 

 Sum of Squared Deviations of ( εεˆˆ ′PPdiag ′) about mean: 0.297272 

 Number of Observations: 331 

 

20 I’ve abused notation: “β” includes β o 

early, βtrendearly, βo late, and βtrendlate while “the intercept” is lnθ o early+lnβtrendearly m or lnθo late+lnβtrendlate

m. 21 To be very specific, the εm vector takes on values: {εm[64] (10 times), εm[65] (10 times), εm[66] (12 times), εm[67] (13 times), εm[68] (14 times), 

εm[69] (14 times), εm[70] (14 times), εm[71] (14 times), εm[72] (14 times), εm[73] (13 times), εm[74] (14 times), εm[75] (13 times), εm[76] (13 times), 

εm[77] (13 times), εm[78] (13 times), εm[79] (13 times), εm[80] (12 times), εm[81] (14 times), εm[82] (13 times), εm[83] (13 times), εm[84] (13 times), 

εm[85] (13 times), εm[86] (13 times), εm[87] (12 times), εm[88] (11 times), εm[89] (10 times)}, and the εt vector takes on values: {εt[70]...εt[79], 

εt[70]...εt[79], εt[70]...εt[81], εt[70]...εt[82], εt[70]...εt[83], εt[71]...εt[84], εt[72]...εt[85], εt[73]...εt[86], εt[74]...εt[87], εt[76]...εt[88], εt[76]...εt[89], εt[77]...εt[89], 

εt[78]...εt[90], εt[79]...εt[91], εt[80]...εt[92], εt[81]...εt[93], εt[83]...εt[94], εt[83]...εt[96], εt[85]...εt[97], εt[85]...εt[97], εt[86]...εt[98], εt[87]...εt[99], εt[88]...εt[’00], 

εt[89]...εt[’00], εt[90]...εt[’00], εt[91]...εt[’00]}. 

Page 101
The fitted values of σm

2

 and σt

2

 are both very small, the wrong sign, and insignificant. Relaxing the trace constraint gives 

essentially the same result: 

σidio

2

 σm

2 σt2

Point Value: 0.0182023 –0.000133953 –8.88177×10–6 

Standard Error: (0.00173421) (0.000183401) (0.0000125384)

 Sum of Squared Residuals: 0.202318 

 Sum of Squared Deviations of ( εεˆˆ ′PPdiag ′) about mean: 0.297272 

 Number of Observations: 331 

Setting σm

2

=σt

2

=0 and regressing the diagonal elements of εεˆˆ ′PP ′ on the diagonal elements of M gives a point value for σidio

2

 

of 0.0172758, with a standard deviation of 0.001395. Setting σm

2

=σt

2

=0 and reimposing the trace constraint—i.e., pretending 

to run a regression to find σidio

2

 = 0.017191092 (from Table 5)—gives a standard deviation of 0.00139501. In both the σidio

2

 -

only cases, the implied variance, 1.946×10–6, is slightly larger than the best quadratic unbiased sampling variance implied by 

Normality: 1.830×10–6. This is at least provisional evidence that it is sensible to use regressions to infer error components. 

Caveats, Extensions, and Conclusions 

The basic conclusions so far are that White’s method of correcting idiosyncratic nonspherical errors works, while efforts to 

find homoskedastic model-year and calendar-year component errors come up empty. It is tempting to look farther. Table 6 

presents approximate retirement histograms22 together with the Weibull probability density functions implied by the 

generalized NLLS regressions conducted separately on model years 1964-89. Although the regressions were fit to 

cumulative distribution functions rather than probability density functions, and although ordinary least squares usually gives 

a better “eyeball fit” than generalized least squares, still the matches between the histograms and the curves are not bad. 

Several features stand out. First, discrepancies between the histograms and the smooth PDFs are strongly serially correlated. 

Second, before the 1980 model year, the histograms almost always “peak” higher than the PDFs; the tendency is often 

accompanied by a slighter peak at a lower age, most noticeably in model years 1975-79. In fact, the first peak in the 1975 

data seems to have fooled the regression into finding a particularly low point estimate of β. Two modes might imply a mixed 

distribution, with the first mode representing retirements of “lemons” or of cars that were driven into the ground (perhaps 

rentals). I have not thought about how to squeeze a mixed distribution into White’s corrective procedures: the observations 

suggesting two modes are quite few. After the 1979 model year, the early modes disappear, and the quality of fits improves. 

It could be that the “regime change” of 1980 is really about the loss of lemons. Third, histogram points for calendar year 

1992, which show up in every model year since 1979 as large diamonds, are outliers that grow increasingly disruptive as they 

become “newer.” The R.L. Polk Co., source for the Ward’s registration data, revised its tabulations after 1991 to remove 

autos registered in more than one state: the revision shows up as a “blip” in 1992 only. Accounting for the blip might make 

regressions of more recent model years feasible. Such regressions would be a good idea: the shape parameter falls 

uncomfortably quickly in Table 5, such that expected β becomes statistically indistinguishable from 1 by model year 2005—

sixteen years after the latest regressed model year—implying more automobiles will be retired when brand-new than at any 

other age. Surely the declines in β must have leveled off by then. Another expedient would be to reestimate Table 5 using an 

asymptotic summary model for new-regime β.

23

 

22 To construct the histogram, subtract consecutive entries in any particular row of Table 1 and normalize by the double-exponential 

approximation to the age-zero count. Normalizing by the largest entry in the same row gives very nearly the same result, since so few 

retirements occur in the earliest years. 

23 Something like β∞ + e–βtr m or β∞ + 1/(1+βtr m), with β∞>1 and βtr>0, instead of β0 + βtrm with βtr<0. 

Page 102
References 

Abernethy, Robert B. (2000). The New Weibull Handbook, Fourth Edition: Reliability & Statistical Analysis for Predicting 

 Life, Safety, Survivability, Risk, Cost and Warranty Claims. (November). 

Box, M.J. (1971). “Bias in Nonlinear Estimation.” Journal of the Royal Statistical Society, Series B, vol. 33: 171-201. 

Cacciari, M., G. Mazzanti, and G.C. Montanari (1996). “Comparison of Maximum Likelihood Unbiasing Methods for the 

 Estimation of the Weibull Parameters.” IEEE Transactions on Dielectrics and Electrical Insulation, vol. 3, no. 1 

 (February): 18-27. 

Duffy, Steven F. (1997). WeibPar. Connecticut Reserve Technologies, LLP. 

http://www.crtechnologies.com/EngDiv/mechanics/weibull/index.html

Davis, Stacy C., ed. (2003). Transportation Energy Data Book: Edition 23. (October). U.S. Department of Energy. 

 [Download from: http://www-cta.ornl.gov/data/ , although prior issues seem no longer available on line.] 

Greenspan, Alan and Darrel Cohen (1996). “Motor Vehicle Stocks, Scrappage, and Sales.” Finance and Economics 

 Discussion Series, working paper 40 (October). Federal Reserve Board. 

Meeker, William Q., and Luis A. Escobar (1998). Statistical Methods for Reliability Data. New York: John Wiley & Sons. 

Montanari, G.C., G. Mazzanti, M. Cacciari, and J.C. Fothergill (1997a). “In Search of Convenient Techniques for Reducing 

 Bias in the Estimation of Weibull Parameters for Uncensored Tests.” IEEE Transactions on Dielectrics and Electrical 

 Insulation, vol. 4, no. 3 (June): 306-313. 

__________________________________________________ (1997b). “Optimum Estimators for the Weibull Distribution 

 of Censored Data: Singly-censored Tests.” IEEE Transactions on Dielectrics and Electrical Insulation, vol. 4, no. 4 

 (August): 462-469. 

__________________________________________________ (1998). “Optimum Estimators for the Weibull Distribution of 

 Censored Data: Progressively-censored Tests.” IEEE Transactions on Dielectrics and Electrical Insulation, vol. 5, no. 

 2 (April): 157-164. 

Mood, Alexander M., Franklin A. Graybill, and Duane C. Boes (1974). Introduction to the Theory of Statistics, Third 

 Edition. New York: McGraw-Hill. 

Ratkowsky, David A. (1983). Nonlinear Regression Modeling: A Unified Practical Approach. New York and Basel: 

 Marcel Dekker. 

Ross, R (1994). “Formulas to Describe the Bias and Standard Deviation of the ML-estimated Weibull Shape Parameter.” 

IEEE Transactions on Dielectrics and Electrical Insulation, vol. 3, no. 1 (February): 28-42. 

______ (1996). “Bias and Standard Deviation due to Weibull Parameter Estimation for Small Data Sets.” IEEE 

 Transactions on Dielectrics and Electrical Insulation, vol. 3, no. 1 (February): 28-42. 

Theil, Henri (1971). Principles of Econometrics. New York: John Wiley & Sons. 

Ward’s Automotive Yearbook, various issues. 

White, John S. (1964). “Least Squares Unbiased Censored Linear Estimation for the Log Weibull (Extreme Value) 

 Distribution,” Journal of the Industrial Mathematics Society, vol. 14, part 1: 21-60. 

White, John S. (1969). “The Moments of Log-Weibull Order Statistics.” Technometrics, vol. 11, no. 2 (May): 373-386. 

Page 103
Page 104
Table 2: Age-Distribution of Registered Autos

(as of July 1: 1970, 1975, ... 2000)

02468

10121416

0.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5

8.5

9.5

10.5

11.5

12.5

13.5

14.5

>14.5

Ages

Percent

Page 105
Table 3: Survival Rates by Model Year

(for model-years 1970, 75, ... 95)

0

102030405060708090

100

1.5

2.5

3.5

4.5

5.5

6.5

7.5

8.5

9.5

10.5

11.5

12.5

13.5

14.5

15.5

Age

Percent Still Registered 

Page 106
 

Table 4: Model-Year -Specific Point Estimates 

by Various Methods 

Shape Parameter (Log of) Spread Parameter 

Page 107
Table 5: Pooled Regression Results 

Iterated Generalized Nonlinear Least Squares 

No Error Components 

 

Best-Fit Parameters, with Standard Errors 

 2.89386 1.78485 early

0 β early0 lnθ

 (0.761313) (0.107348) 

 –0.000788386 0.0111435 early

trend β earlytrend lnθ

 (0.0107087) (0.0015101) 

 9.91589 0.450214 late β0

late

0 lnθ

 (1.68042) (0.271172) 

 –0.084791 0.0274938 late βtrend

late

trend lnθ

 (0.0202248) (0.00329571) 

σ2

=0.017191092 

Variance-Covariance Matrix of the Fitted Coefficients 

early

0 β early

trend β early

0 lnθ early

trend lnθ late β0

late βtrendlate

0 lnθ latetrend lnθ

0.5796 -0.008140 -0.0004958 0.00001037 0 0 0 0

early

trend β -0.008140 0.0001147 0.00001035 -1.933×10−7 0 0 0 0

early

0 lnθ -0.0004958 0.00001035 0.01152 -0.0001619 0 0 0 0

early

trend lnθ 0.00001037 -1.933×10−7 -0.0001619 2.280×10−6 0 0 0 0

late β0

0 0 0 0 2.824 -0.03397 -0.1769 0.002160

late βtrend

0 0 0 0 -0.03397 0.0004090 0.002160 -0.00002639

late

0 lnθ 0 0 0 0 -0.1769 0.002160 0.07353 -0.0008934

late

trend lnθ 0 0 0 0 0.002160 -0.00002639 -0.0008934 0.00001086

early

0 β

Sums of Squares 

unweighted residuals: 4.63694 weighted residuals: 5.54625 

unweighted β β ˆ ˆ

0

β

ˆ





 



 ln − ss , about mean: 93.741 weighted β β ˆ ˆ

0

β

ˆ





 



 ln − ss , about mean: 5457.97 

Traces of (P′P)–1 Model-Year Blocks

model year 64 65 66 67 68 69 70 71 72 73 74 75 76

Trace 1.19 1.02 2.43 4.59 21.37 23.86 14.73 8.62 17.44 0.95 6.91 7.71 10.77

model year 77 78 79 80 81 82 83 84 85 86 87 88 89

Trace 13.64 10.22 11.99 2.03 9.40 1.78 4.14 4.23 3.58 4.88 8.59 20.87 114.04

Overall Trace = 331 

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