Confidence Set Estimation from
Rounded/Digital Normal Data
Steve Vardeman
C-S (Johnson) Lee
(JQT 2001, Comm Stat 2002, (2003))
Iliana Vaca
(M.S. Work in Progress)
1
2
Rounding/Digital Nature of Data
• Hardly a new problem … see e.g.
Sheppard, W. (1898). "On the Calculation of the
Most Probable Values of Frequency Constants for
Data Arranged According to Equidistant Divisions
of a Scale." Proceedings of the London
Mathematical Society 29, pp. 353-380
• Metrologists recognize this as a source of error in
physical measurement, but don’t have good ways
of accounting for it
• Elementary statistical methods are implicitly based
on the assumption that this “isn’t a problem”
3
But … Continuous Models
• Are for “real-number”/“infinitely-manydecimal-place” observations
• Even if they DO adequately describe an
underlying physical phenomenon, they
MAY OR MAY NOT adequately describe
what can be observed
4
Observation “To the Nearest ∆”
• Observations y potentially coded as
integers via y′ = ( y − y0 ) / ∆ so that
1.2, 1.2, 1.2, 1.2, 1.3, 1.3, 1.3, 1.3, 1.3, 1.3
could become (with ∆ = .1 )
2, 2, 2, 2, 3, 3, 3, 3, 3, 3
• Suppose that a continuously distributed X
produces a rounded/digital version Y … the
discrete distribution of Y may or may not
look anything like the continuous
distribution of X
5
Two ∆ = 1 Normal Cases
µ = 4.25 and σ = 1.0
(µY = 4.25 and σ Y = 1.0809)
µ = 4.25 and σ = .25
(µY = 4.1573 and σ Y = .3678)
6
Key is the Size of
• If σ ≥ .5∆ then
µ − µY < .005∆
• If σ ≈ 0 then µ − µY
can be nearly .5∆
σ
∆
• Provided σ > .15∆ ,
σ Y > σ . For such σ
σY −σ
σ
– decreases in σ
– for σ ≥ .5∆ is less than
.141
• For small σ , σ Y can
be many times or a
small fraction of σ
7
Naïve Use of Continuous Data
Inference Formulas …
• y estimates µY not µ and in cases where
sy
“zeros-in” on
µY ≠ µ the interval y ± t
n
µY (and thus for large samples has actual
confidence level near 0)
• s y estimates σ Y not σ and unless σ is
large doesn’t have anything like a (root of
2
a) χ distribution
8
Inference Engine: the “Right”
Likelihood
• (Rounded data) one-sample normal
likelihood
n
L ( µ , σ ) = ΠPµ ,σ ( yi − .5∆ < X < yi + .5∆ )
i =1
  yi + .5∆ − µ 
 yi − .5∆ − µ  
= ΠΦ
−Φ

σ
σ




i =1 
n
9
• Log-likelihood
l ( µ , σ ) = log L ( µ , σ )
• Profile log-likelihoods
l ( µ ) = sup l ( µ , σ )
*
σ >0
l
**
(σ ) = sup l ( µ , σ )
µ
10
Standard Simple Asymptotics
Let
M = sup l ( µ , σ )
( µ ,σ )
(the max (sup) log-likelihood), then
2
2 ( M − l ( µ , σ ) ) 
→
χ
2
n 
→∞
L ( µ ,σ )
2 ( M − l ( µ ) ) 
→χ
n 
→∞
*
Lµ
2
1
2
2 ( M − l (σ ) ) 
→
χ
1
n 
→∞
**
Lσ
11
Cartoon for Asymptotically OK
Confidence Sets
• Region for ( µ , σ ) shaded; interval for µ
(similar interval for σ )
12
• Corresponding cartoon for profile loglikelihoods and estimation of µ or σ
13
Practical Problems With the
Asymptotically-OK Sets
• There is under-coverage
– for µ when σ is large
– for σ when σ is either large or (moderately)
small
– for ( µ , σ ) when σ is large
• Computation of the sets is not always
absolutely obvious (the log-likelihood is not
always so nice-looking)
14
Our Plan (in retrospect, anyway)
• Understand the small sample nature of the
log-likelihood and profile log-likelihoods
• Somehow “fix” the under-coverage
problems by finding suitable small sample
2
2
χ
γ
and
χ
(
)
replacements for 1
2 ( γ ) … together
with …?????
– Central idea for replacement: for large σ ,
the distribution of 2 ( M − l ( µ , σ ) ) is perhaps
essentially that of the corresponding random
variable based on the exact x values
15
Nature of the Likelihood
• This depends on the sample range R, and only
when R≥2∆ is it “tame” (nice and mound-shaped)
– An R=0 case:
n = 10 observations
all 1.2, ∆ = .1 ;
(base 10) loglikelihood
16
– An R=∆ case:
original example
with ∆ = .1 ;
(base 10) version of
l (1.25 + ( t + .25 ) σ , σ )
17
– An R=2∆ case:
∆=.1 with one
data value 1.1, seven
data values 1.2, and
two 1.3; (base 10)
log-likelihood
18
Estimation of µ
• Here the “exact data” version of 2 ( M − l * ( µ ) )
2
 
is
x −µ 
 

  sx / n 
n ln 1 +
n −1









2
χ
this suggests the replacement of 1 ( γ )
with

2  1+ γ
 tn −1  2

cn ( γ ) = n ln 1 +
n −1









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• Simulations show this works splendidly
– The intervals are conservative (for small σ ) to
exact (for large σ )
– Coverage probabilities are asymptotically
correct since
Lµ
*
2
2
2 ( M − l ( µ ) ) 
→
χ
and
c
γ
χ
(
)
1
n
→ 1 (γ )
→∞
n 
n →∞
– For large R these limits are essentially
y ±t
sy
n
• For usual confidence levels and moderate
sample sizes, R = 0 intervals are
∆
∆

y− ,y+ 
2
2

20
• Profile loglikelihoods for R=0 and R=∆
(cartoons)
R=0
R=∆
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Estimation of σ
• Trying first to cure the large σ under-coverage
**
problems, the exact data version of 2 M − l (σ )
is
2
(
)
 nσ 2  ( n − 1) sx
n ln 
n
+
−
2 
2

n
1
s
−
σ
(
)
x 

which has the distribution of
 n
U n = n ln 
W

2
 + W − n for W ∼ χ1

2
χ
and suggests replacing 1 ( γ ) with
d n (γ ) = the γ -quantile of the distribution of U n
22
• This cures the large σ problem (makes the
method exact for large σ ), but does not
completely cure the small σ under-coverage
• Think: small σ will often produce R = 0 or R = ∆
23
• We find that the R = 0 and R = ∆ log2
χ
likelihoods are such that using 1 ( γ ) or d n ( γ )
the (naturally one-sided) intervals have
(upper) endpoints
σ 0 < σ ∆ ,1 < σ ∆ ,2 < < σ  n 
∆, 
2
where
σ 0 = the "R = 0" endpoint
σ ∆ , j = the "R = ∆ and smaller count = j" endpoint
• ????Replace these values with (minimally)
larger ones????
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• An obvious necessary condition for correctto-conservative coverage probabilities is
that
(*)
Pµ0,σ + Pµ∆,σ ≤ 1 − γ ∀µ ,σ
for
η
µ ,σ
P
= Pµ ,σ ( R = η and interval fails to cover σ )
Do brute force computations for the
∆ = 1 case of replacements for σ 0 and σ 1, j
that will guarantee (*)
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• Find (for ∆ = 1 )
σ = minimum σ with max Pµ ,σ ( R = 0 ) ≤ 1 − γ
*
0
µ
• Find (for ∆ = 1 )
σ 1,* j = minimum σ with
 Pµ ,σ ( R = 0 )



j
≤
−
γ
max 
1
µ
+ ∑ Pµ ,σ ( R = 1 and smaller count = l )
 l =1

• Replace σ 0 with ∆σ 0* and σ ∆ , j with ∆σ 1,* j
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• Simulations indicate that with the d n (γ )
and R = 0 and R = ∆ “corrections”
– The intervals are rarely liberal, and when they
are, they are only slightly so
– For large σ (where naïve use of the “usual”
formulas makes sense) these intervals are
somewhat shorter on average than the equal-tail
intervals (no real surprise … equal-tail intervals
aren’t optimized for average length)
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Example
• For the example data set (with 4 values 1.2
and 6 values 1.3) 95% intervals are
– (1.226,1.294 ) for µ (not unlike naïve use of a t
interval in this particular case)
– ( 0,.0851) for σ (not unlike naïve use of a
2
χ
one-sided interval in this case) ( s = .0518 )
y
28
(Joint) Estimation of ( µ , σ )
(in progress)
• Could, for example, be used to create
simultaneous confidence limits for all
values of the cdf
• The “exact data” version of
2 ( M − l ( µ ,σ ))
is
n ln
 ( n − 1) s 
n
 x −µ 
+
−n+

2
2
 ( n − 1) sx   σ
σ / n 



2
 σ

2
x
2
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and the exact data distribution of this is that
n
of
Qn = n ln + W − n + V
W
for independent W ∼ χ n2−1 and V ∼ χ12
• Numerical computation of the cdf and thus
quantiles of such a Qn is easy enough
• We expect that with
qn ( γ ) = the γ -quantile of the distribution of Qn
the prescription
1


( µ , σ ) | M − l ( µ , σ ) < qn ( γ ) 
2


will give reliable confidence sets
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