Distributions Quantiles IDEAL Simulations Inference on distributions, quantiles, and quantile treatment effects using the Dirichlet distribution David M. Kaplan University of Missouri (Economics) Statistics Colloquium 22 April 2014 Dave Kaplan (Mizzou Economics) Dirichlet-based distributional and quantile inference 1 / 66
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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);
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
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
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●● 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
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
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
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
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