inference on distributions, quantiles, and quantile...
TRANSCRIPT
Distributions Quantiles IDEAL Simulations
Inference on distributions, quantiles,and quantile treatment effectsusing the Dirichlet distribution
David M. KaplanUniversity of Missouri (Economics)
Statistics Colloquium22 April 2014
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 1 / 66
Distributions Quantiles IDEAL Simulations
Papers and coauthor
Top 3 (currently) under “Working Papers” on my research webpage:
“IDEAL quantile inference via interpolated duals of exactanalytic L-statistics” (with Matt Goldman, UC San Diego)
“IDEAL inference on conditional quantiles”
“True equality (of pointwise sensitivity) at last: a Dirichletalternative to Kolmogorov-Smirnov inference on distributions”(with Matt)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 2 / 66
Distributions Quantiles IDEAL Simulations
Outline
1 Inference on distributions
2 Inference on quantiles
3 Quantile inference: theory and methods
4 Quantile simulations
5 Conclusion
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 3 / 66
Distributions Quantiles IDEAL Simulations
1 Inference on distributions
2 Inference on quantiles
3 Quantile inference: theory and methods
4 Quantile simulations
5 Conclusion
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 4 / 66
Distributions Quantiles IDEAL Simulations
Wilks (1962)
Yiiid∼ F , continuous Ô⇒ F (Yi) d= Ui iid∼ Uniform(0,1)
Yn∶k: kth order statistic (kth-smallest value in sample);
F (Yn∶k) d= Un∶k ∼ β(k,n + 1 − k){F (Yn∶1), . . . , F (Yn∶n)} ∼ Dir∗(1, . . . ,1; 1)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 5 / 66
Distributions Quantiles IDEAL Simulations
Wilks (1962)
Yiiid∼ F , continuous Ô⇒ F (Yi) d= Ui iid∼ Uniform(0,1)
Yn∶k: kth order statistic (kth-smallest value in sample);
F (Yn∶k) d= Un∶k ∼ β(k,n + 1 − k)
{F (Yn∶1), . . . , F (Yn∶n)} ∼ Dir∗(1, . . . ,1; 1)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 5 / 66
Distributions Quantiles IDEAL Simulations
Wilks (1962)
Yiiid∼ F , continuous Ô⇒ F (Yi) d= Ui iid∼ Uniform(0,1)
Yn∶k: kth order statistic (kth-smallest value in sample);
F (Yn∶k) d= Un∶k ∼ β(k,n + 1 − k){F (Yn∶1), . . . , F (Yn∶n)} ∼ Dir∗(1, . . . ,1; 1)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 5 / 66
Distributions Quantiles IDEAL Simulations
−2 −1 0 1 2
0.0
0.2
0.4
0.6
0.8
1.0
N(0,1), n=21
Y
Pro
babi
lity
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 6 / 66
Distributions Quantiles IDEAL Simulations
−2 −1 0 1 2
0.0
0.2
0.4
0.6
0.8
1.0
N(0,1), n=21, k=4, 90% CI
Y
Pro
babi
lity
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 6 / 66
Distributions Quantiles IDEAL Simulations
−2 −1 0 1 2
0.0
0.2
0.4
0.6
0.8
1.0
N(0,1), n=21, k=11, 90% CI
Y
Pro
babi
lity
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 6 / 66
Distributions Quantiles IDEAL Simulations
−2 −1 0 1 2
0.0
0.2
0.4
0.6
0.8
1.0
N(0,1), n=21, k=17, 90% CI
Y
Pro
babi
lity
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 6 / 66
Distributions Quantiles IDEAL Simulations
Inference on distributions
Beta: 1 − α̃ CI for F (⋅) at each Yn∶k
Dirichlet: which α̃ makes 1 − α uniform confidence band
Hypothesis testing, incl. 2-sample/FOSD
n tests controlling FWER
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 7 / 66
Distributions Quantiles IDEAL Simulations
Inference on distributions
Beta: 1 − α̃ CI for F (⋅) at each Yn∶k
Dirichlet: which α̃ makes 1 − α uniform confidence band
Hypothesis testing, incl. 2-sample/FOSD
n tests controlling FWER
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 7 / 66
Distributions Quantiles IDEAL Simulations
−2 −1 0 1 2
0.0
0.2
0.4
0.6
0.8
1.0
N(0,1), n=21
Y
Pro
babi
lity
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 8 / 66
Distributions Quantiles IDEAL Simulations
−2 −1 0 1 2
0.0
0.2
0.4
0.6
0.8
1.0
90% uniform confidence band
Y
F(Y
)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 8 / 66
Distributions Quantiles IDEAL Simulations
Comparison with Kolmogorov–Smirnov
Kolmogorov–Smirnov test (Kolmogorov, 1933; Smirnov, 1939,1948):
Statistic: Dn = supy√n∣F̂ (y) − F (y)∣
Limit: supt∈(0,1) ∣B(t)∣ (Brownian bridge)
“weighted KS”: weight by inverse asymptotic standarddeviation, 1/
√F (y)[1 − F (y)] (Anderson and Darling, 1952)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 9 / 66
Distributions Quantiles IDEAL Simulations
Comparison with Kolmogorov–Smirnov
Similarities with KS:
Nonparametric, distribution-freeExact, finite-sample coverage/sizeFast to compute (given pre-computed α̃ ↦ α mapping/lookup)Identifies specific regions (quantiles) where equality is rejected(e.g., where a treatment has a measurable effect)
Primary advantage:
KS is “insensitive in tails” (Brownian bridge variance biggest inmiddle)Weighted KS tries for “equal weights” (p. 203), butoverweights tails (different asymptotics for “extreme” orderstatistics, Yn∶r w/ fixed r as n→∞)Dirichlet: equal relative pointwise type I error rates byconstruction
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 10 / 66
Distributions Quantiles IDEAL Simulations
Comparison with Kolmogorov–Smirnov
Similarities with KS:
Nonparametric, distribution-freeExact, finite-sample coverage/sizeFast to compute (given pre-computed α̃ ↦ α mapping/lookup)Identifies specific regions (quantiles) where equality is rejected(e.g., where a treatment has a measurable effect)
Primary advantage:
KS is “insensitive in tails” (Brownian bridge variance biggest inmiddle)
Weighted KS tries for “equal weights” (p. 203), butoverweights tails (different asymptotics for “extreme” orderstatistics, Yn∶r w/ fixed r as n→∞)Dirichlet: equal relative pointwise type I error rates byconstruction
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 10 / 66
Distributions Quantiles IDEAL Simulations
Comparison with Kolmogorov–Smirnov
Similarities with KS:
Nonparametric, distribution-freeExact, finite-sample coverage/sizeFast to compute (given pre-computed α̃ ↦ α mapping/lookup)Identifies specific regions (quantiles) where equality is rejected(e.g., where a treatment has a measurable effect)
Primary advantage:
KS is “insensitive in tails” (Brownian bridge variance biggest inmiddle)Weighted KS tries for “equal weights” (p. 203), butoverweights tails (different asymptotics for “extreme” orderstatistics, Yn∶r w/ fixed r as n→∞)
Dirichlet: equal relative pointwise type I error rates byconstruction
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 10 / 66
Distributions Quantiles IDEAL Simulations
Comparison with Kolmogorov–Smirnov
Similarities with KS:
Nonparametric, distribution-freeExact, finite-sample coverage/sizeFast to compute (given pre-computed α̃ ↦ α mapping/lookup)Identifies specific regions (quantiles) where equality is rejected(e.g., where a treatment has a measurable effect)
Primary advantage:
KS is “insensitive in tails” (Brownian bridge variance biggest inmiddle)Weighted KS tries for “equal weights” (p. 203), butoverweights tails (different asymptotics for “extreme” orderstatistics, Yn∶r w/ fixed r as n→∞)Dirichlet: equal relative pointwise type I error rates byconstruction
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 10 / 66
Distributions Quantiles IDEAL Simulations
5 10 15 20
0.00
0.01
0.02
0.03
0.04
Pointwise type I error, n=20
Order statistic
Rej
ectio
n pr
obab
ility
Dirichlet KS weighted KS
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 11 / 66
Distributions Quantiles IDEAL Simulations
0 20 40 60 80 100
0.00
0.01
0.02
0.03
0.04
Pointwise type I error, n=100
Order statistic
Rej
ectio
n pr
obab
ility
Dirichlet KS weighted KS
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 11 / 66
Distributions Quantiles IDEAL Simulations
Comparison with KS
Asymptotic pointwise type I error rates (mostly droppingconstants):
KS: at central order statistics, constant (depends on pointwisevariance); extreme: exp{−√n}Weighted KS: central exp{− ln[ln(n)]}, extremeexp{−
√ln[ln(n)]}
Dirichlet: central and extremeexp{−c1(α) − c2(α)
√ln[ln(n)]}
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 12 / 66
Distributions Quantiles IDEAL Simulations
0.0 0.5 1.0 1.5 2.0 2.5
02
46
810
12
Fitted and simulated α~(α, n)
ln[ln(n)]
−ln
(α~)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 13 / 66
Distributions Quantiles IDEAL Simulations
0.0 0.5 1.0 1.5 2.0 2.5
02
46
810
12
Universally fitted and simulated α~(α, n)
ln[ln(n)]
−ln
(α~)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 13 / 66
Distributions Quantiles IDEAL Simulations
H0 ∶ N(0,1); data ∶ N(0.3,1); n = 100, α = 0.1, 106 replications
Overall power: 82.4% KS, 80.5% Dirichlet, 62.4% weighted KS
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
H1:pnorm(qnorm(x, 0.3, 1), 0, 1)
Quantile
G F−1
(q)
H0
True
0 20 40 60 80 100
0.0
0.1
0.2
0.3
0.4
0.5
Pointwise power, n=100H1:pnorm(qnorm(x, 0.3, 1), 0, 1)
Order statistic
Rej
ectio
n pr
obab
ility
Dirichlet KS weighted KS
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 14 / 66
Distributions Quantiles IDEAL Simulations
H0 ∶ N(0,1); data ∶ N(0,0.7); n = 100, α = 0.1, 106 replications
Overall power: 98.6% weighted KS, 92.0% Dirichlet, 65.6% KS
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
H1:pnorm(qnorm(x, 0, 0.7), 0, 1)
Quantile
G F−1
(q)
H0
True
0 20 40 60 80 100
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Pointwise power, n=100H1:pnorm(qnorm(x, 0, 0.7), 0, 1)
Order statistic
Rej
ectio
n pr
obab
ility
Dirichlet KS weighted KS
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 14 / 66
Distributions Quantiles IDEAL Simulations
H0 ∶ N(0,1); data ∶ N(0,1.2); n = 100, α = 0.1, 106 replications
Overall power: 64.2% Dirichlet, 25.5% KS, 2.8% weighted KS
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
H1:pnorm(qnorm(x, 0, 1.2), 0, 1)
Quantile
G F−1
(q)
H0
True
0 20 40 60 80 100
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Pointwise power, n=100H1:pnorm(qnorm(x, 0, 1.2), 0, 1)
Order statistic
Rej
ectio
n pr
obab
ility
Dirichlet KS weighted KS
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 14 / 66
Distributions Quantiles IDEAL Simulations
Empirical example: testing family of quantile treatment effect nullhypotheses
0.0 0.2 0.4 0.6 0.8 1.0
3040
5060
70
Gift wage: library task, period 1
Quantile
Boo
ks e
nter
ed
●
●
●
●
●
●
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●
●
●
●
●
●
●
●●
●● Treatment
Control
0.0 0.2 0.4 0.6 0.8 1.0
020
4060
80
Gift wage: fundraising task, period 1
Quantile
Dol
lars
rai
sed
●●
● ● ● ●● ● ●
●
● ●
●
● ●
● ● ● ●● ● ●
●
●●
●● Treatment
Control
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 14 / 66
Distributions Quantiles IDEAL Simulations
Distributional inference summary
Inference on distributions, one-sample and two-sample
Precise control of familywise error rate and pointwise errorrate, unlike KS
Fast computation from simulation-calibrated α̃(α,n)Extensions: improve power via step-down/step-up; k-FWER;conditional distributions
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 15 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
1 Inference on distributions
2 Inference on quantiles
3 Quantile inference: theory and methods
4 Quantile simulations
5 Conclusion
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 16 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Exact inference on a quantile?
Let n = 11, consider only k = 9
CI for CDF: take quantiles of F (Y11∶9) ∼ β(9,3)
CI for quantile:
P (F (Y11∶9) > 0.53) = 95% = P (Y11∶9 > F−1(0.53)),
so Y11∶9 is endpoint for lower one-sided CI for 0.53-quantile
Exact, finite-sample coverage
But what if I care about the median instead?
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 17 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Exact inference on a quantile?
Let n = 11, consider only k = 9
CI for CDF: take quantiles of F (Y11∶9) ∼ β(9,3)CI for quantile:
P (F (Y11∶9) > 0.53) = 95% = P (Y11∶9 > F−1(0.53)),
so Y11∶9 is endpoint for lower one-sided CI for 0.53-quantile
Exact, finite-sample coverage
But what if I care about the median instead?
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 17 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Quantile inference via asymptotic normality
Ex: inference on median QY (0.5), using iid {Yi}ni=1
√n(Q̂ −Q0)
d→ N(0, σ2Q)1-sided: (−∞, Q̂ + 1.64σ̂Q/
√n)
Why do we do that?
P (Q̂ + 1.64σ/√n < Q0) = 0.05 = α
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
Y
Density
Determination of Upper Endpointn=11, p=0.5, α=0.05, Y~Unif(0,1)
Q̂ + 1.64σ n
5%
But: σ hard to estimate; asymptotic approx (worse in tails)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 18 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Quantile inference via asymptotic normality
Ex: inference on median QY (0.5), using iid {Yi}ni=1
√n(Q̂ −Q0)
d→ N(0, σ2Q)
1-sided: (−∞, Q̂ + 1.64σ̂Q/√n)
Why do we do that?
P (Q̂ + 1.64σ/√n < Q0) = 0.05 = α
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
Y
Density
Determination of Upper Endpointn=11, p=0.5, α=0.05, Y~Unif(0,1)
Q̂ + 1.64σ n
5%
But: σ hard to estimate; asymptotic approx (worse in tails)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 18 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Quantile inference via asymptotic normality
Ex: inference on median QY (0.5), using iid {Yi}ni=1
√n(Q̂ −Q0)
d→ N(0, σ2Q)1-sided: (−∞, Q̂ + 1.64σ̂Q/
√n)
Why do we do that?
P (Q̂ + 1.64σ/√n < Q0) = 0.05 = α
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
Y
Density
Determination of Upper Endpointn=11, p=0.5, α=0.05, Y~Unif(0,1)
Q̂ + 1.64σ n
5%
But: σ hard to estimate; asymptotic approx (worse in tails)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 18 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Normal Approximation for Uniform Sample 0.50-quantile, n = 11
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
Normal Approximation for Uniform Sample Quantiles, n=11
u
Density
ExactNormal approx
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 19 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Normal Approximation for Uniform Sample 0.58-quantile, n = 11
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
Normal Approximation for Uniform Sample Quantiles, n=11
u
Density
ExactNormal approx
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 19 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Normal Approximation for Uniform Sample 0.67-quantile, n = 11
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
Normal Approximation for Uniform Sample Quantiles, n=11
u
Density
ExactNormal approx
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 19 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Normal Approximation for Uniform Sample 0.75-quantile, n = 11
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
Normal Approximation for Uniform Sample Quantiles, n=11
u
Density
ExactNormal approx
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 19 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Normal Approximation for Uniform Sample 0.83-quantile, n = 11
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
Normal Approximation for Uniform Sample Quantiles, n=11
u
Density
ExactNormal approx
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 19 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Normal Approximation for Uniform Sample 0.92-quantile, n = 11
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
Normal Approximation for Uniform Sample Quantiles, n=11
u
Density
ExactNormal approx
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 19 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Quantile inference via order statistics
Ex: inference on median QY (0.5), using iid {Yi}ni=1Order statistic: Yn∶k is kth smallest out of n in {Yi}ni=1Approach: (−∞, Yn∶k] as lower one-sided CI; which k?
Want: α = P (Yn∶k < F−1Y (0.5)) = P (Un∶k < 0.5)
Use: Uiiid∼ Unif(0,1)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 20 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Quantile inference via order statistics
Ex: inference on median QY (0.5), using iid {Yi}ni=1Order statistic: Yn∶k is kth smallest out of n in {Yi}ni=1Approach: (−∞, Yn∶k] as lower one-sided CI; which k?
Want: α = P (Yn∶k < F−1Y (0.5)) = P (Un∶k < 0.5)
Use: Uiiid∼ Unif(0,1)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 20 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Endpoint selection using beta distribution
α = 0.05 = P (Un∶k < 0.5)
Un∶k ∼ Beta(k,n + 1 − k), k ∈ {1,2, . . . , n}
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
u
Density
Determination of Upper Endpointn=11, p=0.5, α=0.05
α = 5%
k = 8.69
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 21 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Endpoint selection using beta distribution
α = 0.05 = P (Un∶k < 0.5)
Un∶k ∼ Beta(k,n + 1 − k), k ∈ {1,2, . . . , n}
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
u
Density
Determination of Upper Endpointn=11, p=0.5
k = 6
P(Un:6 < 0.5) = 50%
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
u
Density
Determination of Upper Endpointn=11, p=0.5, α=0.05
α = 5%
k = 8.69
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 21 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Endpoint selection using beta distribution
α = 0.05 = P (Un∶k < 0.5)
Un∶k ∼ Beta(k,n + 1 − k), k ∈ {1,2, . . . , n}
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
u
Density
Determination of Upper Endpointn=11, p=0.5
k = 6
P(Un:6 < 0.5) = 50%k = 8
11.3%
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
u
Density
Determination of Upper Endpointn=11, p=0.5, α=0.05
α = 5%
k = 8.69
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 21 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Endpoint selection using beta distribution
α = 0.05 = P (Un∶k < 0.5)
Un∶k ∼ Beta(k,n + 1 − k), k ∈ {1,2, . . . , n}
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
u
Density
Determination of Upper Endpointn=11, p=0.5
k = 6
P(Un:6 < 0.5) = 50%k = 8
11.3%
k = 9
3.3%
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
u
Density
Determination of Upper Endpointn=11, p=0.5, α=0.05
α = 5%
k = 8.69
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 21 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Endpoint selection using beta distribution
α = 0.05 = P (Un∶k < 0.5)
Un∶k ∼ Beta(k,n + 1 − k), k ∈ {1,2, . . . , n} Ô⇒ k ∈ [1, n] ⊂ R
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
u
Density
Determination of Upper Endpointn=11, p=0.5
k = 6
P(Un:6 < 0.5) = 50%k = 8
11.3%
k = 9
3.3%
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
u
Density
Determination of Upper Endpointn=11, p=0.5, α=0.05
α = 5%
k = 8.69
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 21 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Endpoint selection using beta distribution
α = 0.05 = P (Un∶k < 0.5)
Un∶k ∼ Beta(k,n + 1 − k), k ∈ {1,2, . . . , n} Ô⇒ k ∈ [1, n] ⊂ R
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
u
Density
Determination of Upper Endpointn=11, p=0.5
k = 6
P(Un:6 < 0.5) = 50%k = 8
11.3%
k = 9
3.3%
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
u
Density
Determination of Upper Endpointn=11, p=0.5, α=0.05
α = 5%
k = 8.69
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 21 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Fractional order statistics
α = P (Un∶k < 0.5), U In∶k ∼ Beta(k,n + 1 − k)
Connecting back to reality, k ∈ [1, n] ⊂ R:
(observed) ULn∶k = (1 − ε)Un∶⌊k⌋ + εUn∶⌊k⌋+1, ε ≡ k − ⌊k⌋
(Jones 2002) U In∶kd= (1 −Cε)Un∶⌊k⌋ +CεUn∶⌊k⌋+1, Cε ∼ Beta(ε,1 − ε)
Hutson (1999): suggests this method but w/o theoreticaljustification
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 22 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Fractional order statistics
α = P (Un∶k < 0.5), U In∶k ∼ Beta(k,n + 1 − k)Connecting back to reality, k ∈ [1, n] ⊂ R:
(observed) ULn∶k = (1 − ε)Un∶⌊k⌋ + εUn∶⌊k⌋+1, ε ≡ k − ⌊k⌋
(Jones 2002) U In∶kd= (1 −Cε)Un∶⌊k⌋ +CεUn∶⌊k⌋+1, Cε ∼ Beta(ε,1 − ε)
Hutson (1999): suggests this method but w/o theoreticaljustification
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 22 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Fractional order statistics
α = P (Un∶k < 0.5), U In∶k ∼ Beta(k,n + 1 − k)Connecting back to reality, k ∈ [1, n] ⊂ R:
(observed) ULn∶k = (1 − ε)Un∶⌊k⌋ + εUn∶⌊k⌋+1, ε ≡ k − ⌊k⌋
(Jones 2002) U In∶kd= (1 −Cε)Un∶⌊k⌋ +CεUn∶⌊k⌋+1, Cε ∼ Beta(ε,1 − ε)
Hutson (1999): suggests this method but w/o theoreticaljustification
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 22 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Fractional order statistics
α = P (Un∶k < 0.5), U In∶k ∼ Beta(k,n + 1 − k)Connecting back to reality, k ∈ [1, n] ⊂ R:
(observed) ULn∶k = (1 − ε)Un∶⌊k⌋ + εUn∶⌊k⌋+1, ε ≡ k − ⌊k⌋
(Jones 2002) U In∶kd= (1 −Cε)Un∶⌊k⌋ +CεUn∶⌊k⌋+1, Cε ∼ Beta(ε,1 − ε)
Hutson (1999): suggests this method but w/o theoreticaljustification
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 22 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Recap:
Endpoint is Yn∶k (instead of Q̂ + 1.64σ̂/√n)
Determine exact, fractional k
Approx Y In∶k by Y L
n∶k, Interpolated Dual of Exact AnalyticL-statistic (IDEAL): O(n−1) coverage probability error
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 23 / 66
Distributions Quantiles IDEAL Simulations Normality Order statistics Dirichlet
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 24 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
1 Inference on distributions
2 Inference on quantiles
3 Quantile inference: theory and methods
4 Quantile simulations
5 Conclusion
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 25 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Assumptions
Assumption (PDF)
For each quantile uj , the PDF f(⋅) satisfies (i) f(F−1(uj)) > 0;(ii) f ′′(⋅) is continuous in some neighborhood of F−1(uj).
Assumption (sampling)
Independent samples {Xi}nxi=1 and {Yi}ny
i=1 are drawn iid fromrespective CDFs FX and FY . Respective quantile functions aredenoted QX(⋅) ≡ F −1
X (⋅) and QY (⋅) ≡ F−1Y (⋅).
Assumption (sample sizes)
Sample sizes grow as limnx→∞
√ny/nx ≡ µ > 0.
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 26 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Theorem: fractional order statistic processes
Theorem
Wherever PDF≥ δ,
supk∈(1,n)
∣XIn∶k −XL
n∶k∣ = Op(n−1[logn]).
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 27 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Lemma (Dirichlet PDF approximation)
Basically, precise approximation of Dirichlet PDF and derivative atvalues that for our purposes (i.e., for quantile inference) are drawnwith probability quickly approaching one.
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 28 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Theorem (CDF approximation)
Let LI = lin. com. of idealized fractional order statistics,
LL = lin. com. of linearly interpolated fractional order statistics.
(i) For samples {Xi}ni=1 with Xiiid∼ F , and for a given constant K,
P(LL <K) − P(LI <K)
= C1(K,n, . . .)n−1 +O(n−3/2 log(n)).
(ii) For samples {Xi}ni=1 with Xiiid∼ F ,
supK∈R
[P(LL <K) − P(LI <K)]
= C2(n, . . .)n−1 +O(n−3/2 log(n))
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 29 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Theorem (CDF approximation)
Let LI = lin. com. of idealized fractional order statistics,
LL = lin. com. of linearly interpolated fractional order statistics.
(iii) Given independent samples {Xi}nxi=1 and {Yi}ny
i=1 with
Xiiid∼ FX and Yi
iid∼ FY , with nx ∝ ny ∝ n,
supK∈R
[P (LLX +LLY <K) − P (LIX +LIY <K)] = O(n−1)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 29 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
CPE: single quantile
Lower one-sided coverage probability for Q(u) is
P(XLn∶k(α) > Q(u))
Thm 3= P(XIn∶k(α) > Q(u)) + εh(1 − εh)z1−α exp{−z
21−α/2}
uh(α)(1 − uh(α))√2π
n−1
+O(n−3/2 log(n)),
P(XIn∶k(α) > Q(u)) = P(U In∶k(α) > u) = 1 − α.
Ô⇒ one- or two-sided CI: CPE is O(n−1)Ô⇒ calibrated CI: CPE is O(n−3/2 log(n))
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 30 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
CPE: quantile treatment effect
Coverage probability (lower one-sided CI)
= P{Y Ln∶m(α) −X
Ln∶k(α) > QY (p) −QX(p)}
= P(Y In∶m(α) −X
In∶k(α) > QY (p) −QX(p)} +O(n−1)
= P{F−1Y (U I,1
n∶m(α)) − F−1
Y (p)
− [F−1X (U I,2
n∶k(α)) − F −1
X (p)] > 0} +O(n−1)
= P⎧⎪⎪⎪⎨⎪⎪⎪⎩
U I,1n∶m(α)
− pfY (F−1
Y (p))−U I,2n∶k(α)
− pfX(F −1
X (p))> 0
⎫⎪⎪⎪⎬⎪⎪⎪⎭+ T +O(n−1)
= P{γ(U I,1n∶m(α)
− p) > U I,2n∶k(α)
− p} + T +O(n−1)
= 1 − α +C + T +O(n−1)
C: conditioning on γ̂; T: Taylor remainder; O(n−1): interpolation
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 31 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
CPE: joint, linear combinations
Linear combination: object of interest is ∑Jj=1ψjQ(uj). CPE is
same as for QTE: one-sided O(n−1/2), two-sided O(n−2/3)
Joint: object of interest is (Q(u1), . . . ,Q(uJ)). One- or two-sidedCPE is O(n−1)
Setting Type CPE (two-sided)
1-sample single quantile O(n−3/2 log(n))1-sample joint (multiple quantiles) O(n−1)1-sample linear combination O(n−2/3)2-sample treatment effect O(n−2/3)
CPE ≡ P (θ0 ∈ CI) − (1 − α)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 32 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Nonparametric conditional model
0.0 0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.0
0.5
1.0
Data with IDEAL 95% Confidence Intervals
X
Y
Data
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 33 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Nonparametric conditional model
0.0 0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.0
0.5
1.0
Data with IDEAL 95% Confidence Intervals
X
Y
DataPointwise
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 33 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Nonparametric conditional model
Complements Fan and Liu (2013): they show that the samemethod is valid in wide range of sampling assumptions; I showhigh-order accuracy under somewhat stronger assumptions
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 33 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Setup
Object of interest: QY ∣X(p;X = x0)Observables: (Xi, Yi) ∈ Rd ×R
Discrete X easily added: no bias
Definition (effective sample)
effective sample window: Ch ≡ {x ∶ ∥x − x0∥ ≤ h}effective sample: {Yi ∶Xi ∈ Ch}
effective sample size: Nn ≡#{Yi ∶Xi ∈ Ch}
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 34 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Single quantile: pointwise, joint over multiple x0
0.0 0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.0
0.5
1.0
Data with IDEAL 95% Confidence Intervals
X
Y
DataPointwiseJoint
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 35 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Joint over multiple quantiles
0.0 0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.0
0.5
1.0
Pointwise (by x0) IDEAL 95% CIs forconditional joint upper and lower quartiles
X
Y
DataUpper quartile CILower quartile CI
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 36 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Linear combinations (interquartile range)
0.2 0.4 0.6 0.8
0.0
0.2
0.4
0.6
0.8
1.0
Pointwise IDEAL 95% CIs forconditional interquartile range (IQR)
X
IQR
True IQRIDEAL CI
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 37 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Conditional quantile treatment effects
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 38 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Approach
Inference on QY ∣X(p;x0) using Yi in shrinking Ch
Use 1-sample IDEAL method on effective sample; CPE interms of Nn
Derive nonparametric bias from using Xi ≠ x0Optimal bandwidth; overall CPE
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 39 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Conditional: assumptions
(1) cts (Xi, Yi) ∈ Rd ×R, Yi = QY ∣X(p;Xi) + εi, εi ⊥⊥ εj ∣ {Xk}nk=1,(εi ∣Xi = x) ∼ Fε∣X=x. (More general sampling: Fan and Liu, 2013working paper, Gaussian limits)
Assumption (bias)
(2) fX(x0) > 0; fX(⋅) has smoothness sX = kX + γX > 1 near x0(3) ∀u in neighborhood of p, QY ∣X(u; ⋅) has smoothnesssQ = kQ + γQ > 2(4i) As n→∞, h→ 0
Assumption (IDEAL)
(4ii) As n→∞, nhd/[log(n)]2 →∞ (Ô⇒ Nna.s.→ ∞)
(5) fY ∣X(QY ∣X(p;x0);x0) > 0 (unique cond. quantile)(6) Smoothness: ∀y in neighborhood of QY ∣X(u;x0), ∀x inneighborhood of x0, fY ∣X(y;x) ∈ C2 (wrt y)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 40 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Bias: QY ∣Ch(p) −QY ∣X(p;x0)
Theorem (2)
Under Assumptions 2–4(i) and 6, ξp ≡ QY ∣X(p;x0):
Bias = −h2fX(x0)F (0,2)
Y ∣X(ξp;x0) + 2f ′X(x0)F (0,1)
Y ∣X(ξp;x0)
6fX(x0)fY ∣X(ξp;x0)+ o(h2)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 41 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Optimal bandwidth and CPE rates (two-sided)
CPE-optimal: h∗ = argminhCPE(h)CPE = CPEGK +CPEBias = O(N−1
n ) +O(Nnh4)
Nn ≍ nhd
Theorem (3)
Under Assumptions 1–6, two-sided CPE-optimal:
h∗ ≍ n−1/(2+d) CPE = O(n−2/(2+d))
Ex: d = 1 Ô⇒ h∗ ≍ n−1/3, CPE = O(n−2/3)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 42 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Optimal bandwidth and CPE rates (two-sided)
CPE-optimal: h∗ = argminhCPE(h)CPE = CPEGK +CPEBias = O(N−1
n ) +O(Nnh4)
Nn ≍ nhd
Theorem (3)
Under Assumptions 1–6, two-sided CPE-optimal:
h∗ ≍ n−1/(2+d) CPE = O(n−2/(2+d))
Ex: d = 1 Ô⇒ h∗ ≍ n−1/3, CPE = O(n−2/3)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 42 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Optimal bandwidth and CPE rates (two-sided)
CPE-optimal: h∗ = argminhCPE(h)CPE = CPEGK +CPEBias = O(N−1
n ) +O(Nnh4)
Nn ≍ nhd
Theorem (3)
Under Assumptions 1–6, two-sided CPE-optimal:
h∗ ≍ n−1/(2+d) CPE = O(n−2/(2+d))
Ex: d = 1 Ô⇒ h∗ ≍ n−1/3, CPE = O(n−2/3)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 42 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
O(n−2/3)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 43 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Optimal CPE comparison (two-sided)
IDEAL: CPE = O(n−2/(2+d))Local polynomial/asy. normality: CPE = O(n−[sQ/(sQ+d)]/2)d = 1 or d = 2: IDEAL better even if assume sQ =∞d = 3: IDEAL better unless assume kQ ≥ 11, 11th-degree localpolynomial (364 terms)
d→∞: IDEAL better unless kQ ≥ 4
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 44 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
IDEAL CPE summary
Setting Type CPE (two-sided)
1-sample single quantile O(n−3/2 log(n))1-sample joint (multiple quantiles) O(n−1)1-sample linear combination O(n−2/3)2-sample treatment effect O(n−2/3)
conditional single quantile O(n−2/(2+d))conditional joint O(n−2/(2+d))conditional linear combination O(n−8/(12+7d))conditional treatment effect O(n−8/(12+7d))
CPE ≡ P (θ0 ∈ CI) − (1 − α)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 45 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
General kernel?
Above: implicit uniform kernel K(x/h) = 1{−h < x < h}Let f̃(x) = f(x)K(x/h)/E[K(x/h)]; can show bias is O(hr)for rth-order kernel for p-quantile from density
∫R fY ∣X(y;x)f̃(x)dxIf K(⋅) ≥ 0, then can perform rejection sampling (from fullsample of n) to get iid draws from this distribution; the f(x)cancels out, so probability of acceptance only depends onK(⋅), which is known—P (accept) =K(xi/h)/ supxK(x)But if r > 2, then some K(x) < 0: can’t have negativeacceptance probability (. . . right?)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 46 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
1 Inference on distributions
2 Inference on quantiles
3 Quantile inference: theory and methods
4 Quantile simulations
5 Conclusion
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 47 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Unconditional QTE: power
Location difference of one unit between medians of X and Ynx = ny = 25, FX = FY , α = 0.05, p = 0.5
N(0,1) Logistic(0,1) Exp(1) LogN(0,1)
IDEAL 0.79 0.39 0.91 0.75K11 0.70 0.31 0.86 0.64Hut07 0.75 0.38 0.87 0.57BS-t (99) 0.70 0.35 0.84 0.68BS-t (999) 0.71 0.36 0.86 0.68Horowitz98 0.62 0.27 0.87 0.66
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 48 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Conditional: other methods
quantreg::rqss (nonparametric; est. asy. covariance matrix)
quantreg::rq (parametric; bootstrap)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 49 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Conditional: setup
Koenker rqss simulations: iid errors (various), σ(X) = 0.2 or0.2(1 +X), median, n = 400, α = 0.05
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
x(1 − x)sin⎛⎝2π(1 + 2−7 5) (x + 2−7 5)⎞⎠
x
QY|X(p;x)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 50 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Conditional: runtime
Runtimes (log-log)
Sample size
Runt
ime
(min
utes
)
200 1,000 5,000 20,000 100,000
0.01
0.1
110
100
IDEALBS(99)rqss
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 51 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Pointwise CP: Gaussian, homoskedastic
0.2 0.4 0.6 0.8
020
4060
80100
Pointwise Coverage Probability
X
CP (%
)
NominalIDEALrqssrq(perc)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 52 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Pointwise CP: Cauchy, homoskedastic
0.2 0.4 0.6 0.8
020
4060
80100
Pointwise Coverage Probability
X
CP (%
)
NominalIDEALrqssrq(perc)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 53 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Pointwise CP: centered χ23, heteroskedastic
0.2 0.4 0.6 0.8
020
4060
80100
Pointwise Coverage Probability
X
CP (%
)
NominalIDEALrqssrq(perc)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 54 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
0.0 0.2 0.4 0.6 0.8 1.0
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Joint Hypotheses
X
Y
00
0
000
0
0
0000
0
0
0
000000
00000000000000000000000000
0+-
no shift
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 55 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
0.0 0.2 0.4 0.6 0.8 1.0
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Joint Hypotheses
X
Y
00
0
000
0
0
0000
0
0
0
000000
00000000000000000000000000
++
+
+++
+
+
++++
+
+
+
++++++
++++++++++++++++++++++++++
0+-
no shiftshift = +0.1
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 55 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
0.0 0.2 0.4 0.6 0.8 1.0
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Joint Hypotheses
X
Y
00
0
000
0
0
0000
0
0
0
000000
00000000000000000000000000
++
+
+++
+
+
++++
+
+
+
++++++
++++++++++++++++++++++++++- -
-
-- -
-
-
-- - -
-
-
-
--- - - -
---------- -- - - - - - - - - - - - - - -
0+-
no shiftshift = +0.1shift = −0.1
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 55 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Joint power curves: Gaussian, homoskedastic
-0.4 -0.2 0.0 0.2 0.4
020
4060
80100
Joint Power Curves
Deviation of Null from Truth
Rej
ectio
n Pr
obab
ility
(%)
IDEALrqssrq(perc)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 56 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Joint power curves: Cauchy, homoskedastic
-0.4 -0.2 0.0 0.2 0.4
020
4060
80100
Joint Power Curves
Deviation of Null from Truth
Rej
ectio
n Pr
obab
ility
(%)
IDEALrqssrq(perc)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 57 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Joint power curves: centered χ23, heteroskedastic
-0.4 -0.2 0.0 0.2 0.4
020
4060
80100
Joint Power Curves
Deviation of Null from Truth
Rej
ectio
n Pr
obab
ility
(%)
IDEALrqssrq(perc)
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 58 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Pointwise power against ±0.1: Gaussian, homoskedastic
0.2 0.4 0.6 0.8
020
4060
80100
Pointwise Power
Deviation: -0.1 or 0.1X
Rej
ectio
n Pr
obab
ility
(%)
IDEAL rqss
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 59 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Pointwise power against ±0.1: Cauchy, homoskedastic
0.2 0.4 0.6 0.8
020
4060
80100
Pointwise Power
Deviation: -0.1 or 0.1X
Rej
ectio
n Pr
obab
ility
(%)
IDEAL rqss
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 60 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Pointwise power against ±0.1: centered χ23, heteroskedastic
0.2 0.4 0.6 0.8
020
4060
80100
Pointwise Power
Deviation: -0.1 or 0.1X
Rej
ectio
n Pr
obab
ility
(%)
IDEAL rqss
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 61 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Conditional QTE: median
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
x0s
y0s
CPLength
0.925 0.965 0.945 0.920 0.955 0.9500.367 0.354 0.324 0.283 0.271 0.323
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 62 / 66
Distributions Quantiles IDEAL Simulations Unconditional Conditional
Conditional quantile treatment effects
x0 valuep 0.04 0.224 0.408 0.592 0.776 0.96 Joint
Coverage Probability0.75 0.938 0.924 0.938 0.940 0.948 0.960 0.9520.50 0.926 0.962 0.964 0.952 0.944 0.952 0.9460.25 0.948 0.960 0.948 0.930 0.922 0.924 0.932
Median Interval Length0.75 0.366 0.364 0.325 0.281 0.268 0.3220.50 0.348 0.340 0.323 0.282 0.250 0.2970.25 0.373 0.373 0.337 0.316 0.258 0.341
Table: CP and median interval length for IDEAL CIs for conditionalQTEs; 1 − α = 0.95, n = 400 for both treatment and control samples, 500replications.
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 63 / 66
Distributions Quantiles IDEAL Simulations
1 Inference on distributions
2 Inference on quantiles
3 Quantile inference: theory and methods
4 Quantile simulations
5 Conclusion
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 64 / 66
Distributions Quantiles IDEAL Simulations
Recap
Wilks (1962) Dirichlet: inference on distributions
Fractional order statistic theory (Stigler 1977): IDEALinference on quantiles
Methods: single quantile (Hutson 1999), joint, linearcombinations, quantile treatment effects; unconditional,conditional (complementing Fan and Liu 2013)
Results: improved CPE in most common cases, robust
R code, examples, papers on my website
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 65 / 66
Distributions Quantiles IDEAL Simulations
Inference on distributions, quantiles,and quantile treatment effectsusing the Dirichlet distribution
David M. KaplanUniversity of Missouri (Economics)
Statistics Colloquium22 April 2014
Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 66 / 66