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Book ChapterDOI

A Necessary Condition for Semiparametric Efficiency of Experimental Designs

21 Jul 2021-pp 718-725
TL;DR: In this paper, the authors proposed a necessary condition for a semiparametrically efficient experimental design, and derived a formula to determine the efficient distribution of the input variables, which can be applied to the optimal bid design problem of contingent valuation survey experiments.
Abstract: The efficiency of estimation depends not only on the method of estimation but also on the distribution of data. In statistical experiments, statisticians can at least partially design the data-generating process to obtain high estimation performance. This paper proposes a necessary condition for a semiparametrically efficient experimental design. We derived a formula to determine the efficient distribution of the input variables. The paper also presents an application to the optimal bid design problem of contingent valuation survey experiments.

Summary (1 min read)

1 Introduction

  • Imagine that there exist n lightning bulbs, whose life time hours ω1, . . ., ωn are i.i.d. random variables distributed according to µ.
  • Second, one of the bulbs is sampled without replacement at time x and its status is observed.
  • For the purpose, geometric theory of semiparametric estimation is introduced.
  • In Section 4, application examples of the main theorem are given.
  • In particular, the optimal bid design problem of contingent valuation survey experiments is solved.

2.1 The tangent space of a statistical manifold

  • Geometric theory of semiparametric estimation is introduced to formulate the efficient design problem.
  • Terms and definitions given in the following are according to [12].
  • Under conditions (i) and (ii), ℓ̇0 becomes a tangent vector of M at µ.

2.2 The score operator

  • The tangent bundle TP relates each P with TPP.
  • Note that the score operator is linear and continuous under the Fisher-information metrics.

3 Main Results

  • (1) If the lower bound is minimized at ν = ν∗, any small perturbations added to ν∗ would not significantly change the value of l.b.(φ(µ)|ν∗).
  • A proof of the theorem consists of the following lemmas 3.2-3.4.

4.3 The dichotomous choice contingent valuation experiment

  • This is the dichotomous choice contingent valuation (DC-CV) experiment, which is one of the most widely used experimental methods in environment economics [4].
  • Cooper (1992) reports results of Monte Carlo simulations of the DC-CV experiments, showing sensitivity of estimates to the choice of ν [5].
  • They find the optimal design for estimation of the mean.
  • Eω by directly minimizing the asymptotic variances of the maximum likelihood estimators under assumptions that ω has a finite support and that X is a finite set.
  • The formula (19) is valid only if two technical assumptions are satisfied.

5 Monte Carlo Simulations

  • Small sample properties of the efficient design for the DC-CV experiment are tested through Monte Carlo simulations.
  • Inputs x∗1, · · ·, x∗n are sampled from the efficient distribution ν∗, which is determined by the formula (19), while xo1, · · ·, xon are sampled from the opponent distribution νo, which is arbitrarily selected.
  • The problem is numerically solved by the isotonic regression technique, and the estimator is known to be semiparametrically efficient ([7], [9], [10], [14]).

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A Necessary Condition for Semiparametric
Efficiency of Experimental Designs
Hisatoshi Tanaka
Waseda INstitute of Political EConomy
Waseda University
Tokyo, Japan
WINPEC Working Paper Series No.E2024
March 2021

A Necessary Condition for Semiparametric
Efficiency of Experimental Designs
Hisatoshi Tanaka
School of Political Science and Economics, Waseda University
Shinjuku, Tokyo 169-8050, Japan
hstnk@waseda.jp
Abstract. Efficiency of estimation depends not only on a method of
the estimation, but also on the distribution of data. In statistical ex-
p eriments, statisticians can at least partially design the data generating
pro cess to obtain high performance of the estimation. In this paper, a
necessary condition for the semiparametrically efficient experimental de-
sign is prop osed. A formula to determine the efficient distribution of input
variables is derived. An application to the optimal bid design problem of
contingent valuation survey experiments is presented.
Keywords: Optimal Design · Semiparametric Efficiency · Binary Re-
sp onse Mo del · Contingent Valuation Survey Experiments
1 Introduction
In this paper a class of simple statistical experiments described by a 4-tuple,
E = {(µ, ν, ρ, φ) : µ M, ν N}, (1)
is investigated, where M is a set of probability measures on (W, A), N is a set
of probability measures on (X, B), ρ is a measurable map from W ×X to (Y, C),
and φ is a functional on M. In every experiment (µ, ν, ρ, φ) E, input x is
drawn from ν, output y = ρ(ω, x) with ω µ is observed, and the value of φ(µ)
is estimated from (x, y).
For example, imagine that there exist n lightning bulbs, whose life time
hours ω
1
, . . . , ω
n
are i.i.d. random variables distributed according to µ. In order
to estimate the expected life time hours φ(µ) = Eω, the following experiment is
conducted. First, all n bulbs are turned on at time 0. Second, one of the bulbs is
sampled without replacement at time x and its status is observed. If the sampled
bulb is alive, y is set 1. If otherwise, y is set 0. The procedure is repeated n times
until all of the bulbs are sampled. Finally, data of n indep endent pairs (x
1
, y
1
),
···, (x
n
, y
n
) are obtained, and Eω or any other moments of ω will be consistently
estimated by existing efficient estimation methods, such as the nonparametric
maximum likeliho od estimation.

To be noted here is that efficiency of the estimation depends not only on the
estimation method, but also on the distribution ν of x
1
, ···, x
n
. In an extreme
case where x
1
= . . . = x
n
= 0, trivial outcomes y
1
= . . . = y
n
= 1 will be
obtained unless some of the bulbs are with initial failure. In the opposite extreme
case where x
1
= . . . = x
n
= +, y
1
= . . . = y
n
= 0 will occur with probability
one. In both cases, data are so poorly informative that consistent estimation of
Eω is not possible. To find the best distribution ν of x, with which the experiment
produces the most informative data, is therefore an interesting problem.
The paper is organized as follows. In Section 2, the problem of the paper is
formally stated. For the purpose, geometric theory of semiparametric estimation
is introduced. In the theory, every statistical model is considered as a point
on an infinite dimensional manifold, and the efficient design is formulated as a
minimizer of the Fisher-information norm of the gradient of a functional on the
manifold. In Section 3, a necessary condition for the efficient design is proposed.
In Section 4, application examples of the main theorem are given. In particular,
the optimal bid design problem of contingent valuation survey experiments is
solved. In Section 5, results from small Monte Carlo simulations are reported.
It is numerically confirmed that the efficiently designed estimations outperform
opponents even with small samples.
2 The Model
2.1 The tangent space of a statistical manifold
In this section, geometric theory of semiparametric estimation is introduced
to formulate the efficient design problem. Terms and definitions given in the
following are according to [12]. Equivalent definitions are also found in [1], [2],
[3], and [11].
Let µ be a probability measure on (W, A). Let M be a set of probability
measures, which are absolutely continuous with respect to µ. A map t 7→ µ
t
from (ϵ, ϵ) R to M such that µ
0
= µ is differentiable in quadratic mean at
t = 0 if there exists α L
2
(µ) such that
lim
t0
t
t
1
2
α
2
= 0. (1)
Proposition 1. A map t 7→ µ
t
is differentiable in quadratic mean at t = 0 if
(i) a map t 7→
t
(ω) :=
t
/dµ(ω) is continuously differentiable on (ϵ, ϵ) and
if (ii) a map t 7→
(
˙
t
/ℓ
t
)
2
t
becomes continuous on (ϵ, ϵ), where
˙
t
(ω) =
(dℓ
t
/dt)(ω). Under conditions (i) and (ii),
˙
0
becomes a tangent vector of M at
µ.
Proof. See e.g. Proposition 1 in page 13 of [3].
A collection of those differentiable maps t 7→ µ
t
is denoted by M(µ). A
tangent space T
µ
M of M at µ is a set of tangent vectors α as in (1). A tangent
2

bundle T M relates each µ with T
µ
M. A pair (M, T M) is a statistical manifold,
which is an infinite dimensional analog of a standard finite-dimensional manifold.
On (M, T M), the Fisher-information metric µ 7→ ⟨·, ·⟩
1/2
µ
is defined by
α, α
µ
=
W
αα
(2)
for every α and α
in T
µ
M. The Fisher-information norm ·
µ
is also given by
α
µ
= α, α
1/2
µ
. The following proposition characterizes T
µ
M.
Proposition 2 ([10], [13]). Let T
µ
P(W) be the closure of a tangent space
T
µ
P(W) with respect to ·
µ
, then
T
µ
P(W) = L
0
2
(µ) :=
α L
2
(µ)
α = 0
. (3)
Proof. Choose an arbitrary α L
0
2
(µ) and M > 0. Let α
0
M
= α
M
α
M
,
where α
M
= α · {|α| M}. Define a map t 7→ µ
t
by
t
=
t
= exp
0
M
γ
t
, γ
t
= log
exp(
0
M
)
. (4)
Since |α
0
M
| M, (4) is well-defined and t 7→
t
(ω) becomes continuously differ-
entiable with derivative
˙
t
(ω) =
d
dt
t
(ω) =
α
0
M
(ω)
α
0
M
exp(
0
M
)
exp(
0
M
)
exp
0
M
(ω) γ
t
(5)
at every ω W. A map t 7→
(
˙
t
/ℓ
t
)
2
t
is also well-defined and continuous
in t (ϵ, ϵ ), hence t 7→ µ
t
is differentiable in quadratic mean at t = 0 with
derivative
˙
0
= α
0
M
. Let M , then α α
0
M
µ
0. Thus, α T
µ
M is
shown.
On the other hand, for every (µ
t
)
t(ϵ,ϵ)
M(µ) and α L
2
(µ), let ξ
k
=
k(
1/k
) (α/2)
for k N. Then, as k ,
ξ
2
k
0 and
α
=
α
2
2
ξ
2
k
1/2
· 2
1/2
+
1
k
ξ
k
+
α
2
2
o(1) +
1
k
o(1) +
1
2
α
µ
2
0,
which implies T
µ
M L
0
2
(µ).
3

2.2 The score operator
Let N be a class of probability measures on (X, B), and let P be a class of
probability measures on (X × Y, σ(B × C)). For every P P, let P(P ) be a
collection of differentiable maps t (ϵ, ϵ) 7→ P
t
P such that P
0
= P . Let
T
P
P be the tangent space of P at P . The tangent bundle T P relates each P
with T
P
P. The Fisher-information norm on (P, T P) is ·
P
such that β
P
=
β(x, y) dP (x, y)
1/2
for every β T
P
P. The closure of T
P
P with respect to
·
P
is L
0
2
(P ) as shown in Proposition 2.
Given a measurable map ρ : W×X 7→ Y, at every ν N, a map ρ
ν
: M 7→ P
defined by
ρ
ν
(µ)(D) =
{(x, ρ(ω, x)) D}µ()ν(dx), D σ(B × C), (6)
is a differentiable map between (M, T M) and (P, T P). To see this, note that
ν
(µ
t
)
ν
(µ)
(x, y) = E
µ,ν
t
(ω)
x, y
(7)
because
ρ
ν
(µ
t
)(D) =
t
(ω){(x, ρ(ω, x)) D}µ()ν(dx)
= E
µ,ν
E
µ,ν
t
(ω)
x, y
{(x, y) D}
.
Particularly when
t
/dµ = exp( γ
t
), where α L
0
2
(µ) is bounded and
γ
t
= log
exp() , t 7→
ρ
t
(x, y) :=
ν
(µ
t
)/dρ
ν
(µ)(x, y) is continuously dif-
ferentiable with derivative
˙
ρ
t
(x, y) :=
d
dt
ν
(µ
t
)
ν
(µ)
(x, y)
= E
µ,ν
α
α exp()
exp()
exp ( γ
t
)
x, y
. (8)
Since t 7→
(
˙
ρ
t
/ℓ
ρ
t
)
2
ν
(µ
t
) is continuous, t 7→ ρ
ν
(µ
t
) is a differentiable path on
P with a tangent vector
˙
ρ
0
(x, y) = E
µ,ν
(α|x, y).
The derivative of ρ
ν
: M 7→ P at µ is the score operator (
ν
)
µ
: T
µ
M 7→
L
2
(ρ
ν
(µ)), which maps every α T
µ
M to
((
ν
)
µ
α)(x, y) = E
µ,ν
(α|x, y), (x, y) X × Y. (9)
Then, a tangent space of a submanifold ρ
ν
(M) := {ρ
ν
(µ) P |µ M}, which
is a set of statistical models to be estimated in experiment (µ, ν, ρ, φ), is the
range of the score operator: that is,
T
ρ
ν
(µ)
ρ
ν
(M) = (
ν
)
µ
(T
µ
M) = R((
ν
)
µ
), (10)
4

References
More filters
Book
31 Jul 1992
TL;DR: In this paper, the authors proposed a nonparametric maximum likelihood estimator for interval censoring, which is based on the Van der Vaart Differentiability Theorem (VDT).
Abstract: I. Information Bounds.- 1 Models, scores, and tangent spaces.- 1.1 Introduction.- 1.2 Models P.- 1.3 Scores: Differentiability of the Model.- 1.4 Tangent Sets P0 and Tangent Spaces P.- 1.5 Score Operators.- 1.6 Exercises.- 2 Convolution and asymptotic minimax theorems.- 2.1 Introduction.- 2.2 Finite-dimensional Parameter Spaces.- 2.3 Infinite-dimensional Parameter Spaces.- 2.4 Exercises.- 3 Van der Vaart's Differentiability Theorem.- 3.1 Differentiability of Implicitly Defined Functions.- 3.2 Some Applications of the Differentiability Theorem.- 3.3 Exercises.- II. Nonparametric Maximum Likelihood Estimation.- 1 The interval censoring problem.- 1.1 Characterization of the non-parametric maximum likelihood estimators.- 1.2Exercises.- 2 The deconvolution problem.- 2.1 Decreasing densities and non-negative random variables.- 2.2 Convolution with symmetric densities.- 2.3 Exercises.- 3 Algorithms.- 3.1 The EM algorithm.- 3.2 The iterative convex minorant algorithm.- 3.3 Exercises.- 4 Consistency.- 4.1 Interval censoring, Case 1.- 4.2 Convolution with a symmetric density.- 4.3 Interval censoring, Case 2.- 4.4 Exercises.- 5 Distribution theory.- 5.1 Interval censoring, Case 1.- 5.2 Interval censoring, Case 2.- 5.3 Deconvolution with a decreasing density.- 5.4 Estimation of the mean.- 5.5 Exercises.- References.

638 citations

Book ChapterDOI
TL;DR: Contingent valuation as mentioned in this paper is defined as the "valuation" estimated from preference information given the respondent is said to be "contingent" on the details of the "constructed market" for the environmental good put forth in the survey.
Abstract: Value estimates for environmental goods can be obtained by either estimating preference parameters as “revealed” through behavior related to some aspect of the amenity or using “stated” information concerning preferences for the good. In the environmental economics literature the stated preference approach has come to be known as “contingent valuation” as the “valuation” estimated obtained from preference information given the respondent is said to be “contingent” on the details of the “constructed market” for the environmental good put forth in the survey. Work on contingent valuation now typically comprises the largest single group of papers at major environmental economics conferences and in several of the leading journals in the field. As such, it is impossible to “review” the literature per se or even cover all of the major papers in the area in some detail. Instead, in this chapter we seek to provide a coherent overview of the main issues and how they fit together. The organization of the chapter is as follows. First, we provide an overview of the history of contingent valuation starting with its antecedents and foundational papers and then trace its subsequent development using several broad themes. Second, we put forth the theoretical foundations of contingent valuation with particular emphasis on ties to standard measures of economic welfare. Third, we look at the issue of existence/passive use considerations. Fourth, we consider the relationship of contingent valuation to information on preferences that can be obtained by observing revealed behavior and how the two sources of information might be combined. Fifth, we look at different ways in which preference information can be elicited in a CV survey, paying particular attention to the incentive structure posed by different elicitation formats. Sixth, we turn to econometric issues associated with these different elicitation formats. Seventh, we briefly consider survey design issues. Eighth, we look at issues related to survey administration and extrapolating the results obtained to the population of interest. Ninth, we describe the major controversies related to the use of contingent valuation and summarize the evidence. Finally, we provide some thoughts on where we think contingent valuation is headed in the future.

342 citations

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255 citations

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TL;DR: In this article, the maximal regular (i.e., with strictly positive densities) of a given reference probability measure is defined as the set of all probability densities equivalent to a given target probability measure.
Abstract: Let $\mathscr{M}_\mu$ be the set of all probability densities equivalent to a given reference probability measure $\mu$. This set is thought of as the maximal regular (i.e., with strictly positive densities) $\mu$-dominated statistical model. For each $f \in \mathscr{M}_\mu$ we define (1) a Banach space $L_f$ with unit ball $\mathscr{V}_f$ and (2) a mapping $s_f$ from a subset $\mathscr{U}_f$ of $\mathscr{M}_\mu$ onto $\mathscr{V}_f$, in such a way that the system $(s_f, \mathscr{U}_f, f \in \mathscr{M}_\mu)$ is an affine atlas on $\mathscr{M}_\mu$. Moreover each parametric exponential model dominated by $\mu$ is a finite-dimensional affine submanifold and each parametric statistical model dominated by $\mu$ with a suitable regularity is a submanifold. The global geometric framework given by the manifold structure adds some insight to the so-called geometric theory of statistical models. In particular, the present paper gives some of the developments connected with the Fisher information metrics (Rao) and the Hilbert bundle introduced by Amari.

249 citations

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TL;DR: In this article, the use of contingent valuation to estimate non-market benefits from separately valued nested goods is considered and survey respondents were asked referendum WTP questions regarding either a scheme to protect the "whole" area of a wetland or a...
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In this paper, a necessary condition for the semiparametrically efficient experimental design is proposed. An application to the optimal bid design problem of contingent valuation survey experiments is presented.