Berger

To appear in Biometrika

12 July 2007

A jackknife variance estimator for unistage
stratified samples with unequal probabilities
By YVES G. BERGER
Southampton Statistical Sciences Research Institute,
University of Southampton, Southampton, SO17 1BJ, U.K.
y.g.bergersoton.ac.uk

SUMMARY

Existing jackknife variance estimators used with sample surveys can seriously
overestimate thetrue variance under unistage stratified sampling without replacement
with unequal probabilities. A novel jackknife variance estimator is proposed which is as
numerically simple as existing jackknife variance estimators. Under certain regularity
conditions, the proposed variance estimator is consistent under stratified sampling
without replacement with unequal probabilities. The high entropyregularity condition
necessary for consistency is shown to hold for the Rao-Sampford design. An empirical
study of three unequal probability sampling designs supports our findings.

Some key words: Consistency; Design-based inference; Finite population correction;
Sample survey; Smooth function of means; Stratification.

1. INTRODUCTION

Jackknife methods are widely used for varianceestimation in sample surveys (Wolter,
1985; Shao & Tu, 1995). Properties of various forms of the jackknife variance estimator

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To appear in Biometrika

have been studied both theoretically and empirically (Lee,1973; Jones, 1974; Kish &
Frankel, 1974; Krewski & Rao, 1981; Rao & Wu, 1985; Kovar et al., 1988; Rao et al.,
1992; Shao & Tu, 1995).
Since the mid 1950s, there has been awell-developed theory of sample survey
design inference embracing complex designs with stratification and unequal probabilities
(Smith, 2001). However, customary jackknife variance estimators, henceforth called
jackknife estimators, are not always consistent under these sampling designs (Demnati &
Rao, 2004). We proposed a novel jackknife estimator which is consistent under unistage
stratified samplingwith inclusion probability proportional to size.
The probability sample s is defined as follows. Assume that the sample s is
randomly selected by a unistage stratified probability sampling design p( s ). Let

U1 , K , U H denote H strata, where
and

H

∑ h =1 N h = N .

H

U h =1U h

= U . The size of U h is denoted by N h ,

Suppose that a sample sh of fixed size nh is selectedwithout
H

replacement with unequal probabilities from U h . The complete sample is s = U h =1 sh ,
H
and the size of s is n = ∑ h =1 nh . This sampling design is often used by statistical

agencies, and for surveys of rare animal population (Thompson & Seber, 1996, p. 134).

2. THE CLASS OF POINT ESTIMATORS

Assume that the parameter of interest θ can be expressed as a function of meansof Q
survey variables; that is, θ = g ( µ1 ,K, µQ ), where g (⋅) is a smooth differentiable
function (Shao & Tu, 1995, Ch. 2). The parameter µ q is the finite population mean
defined by µ q = N −1 ∑i∈U y qi . The set U = {1, K , N } is a finite population containing

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To appear in Biometrika

N units. The quantity yqi is the value of the qth survey variable ( q = 1, K , Q ) associatedwith the unit labelled i ∈ U . This definition of θ includes parameters of interest arising in
common survey applications, such as ratios, subpopulation means, correlation and
regression coefficients. It excludes parameters such as L-statistics (Shao, 1994) and
coefficients of logistic regression which cannot be expressed as function of means. For
simplicity, assume that the survey variablesare free from errors due to nonresponse and
measurement.

ˆ
ˆ
ˆ
ˆ
The point estimator θ is the substitution estimator θ = g ( µ1 ,K, µQ ), in which
ˆ
µq = ∑ wi yi

(1)

i∈s

is the Hájek (1971) estimator of µ q , where s is the sample, wi = α i (∑k∈s α k )−1 and

α i = 1 / π i denotes the Horvitz & Thompson (1952) sampling weights. The quantity π i
denotes the first-order...