Methods for interval-censored data
and testing for stochastic dominance
Angel G. Angelov
Department of Statistics
Ume˚a School of Business, Economics and Statistics Ume˚a 2018
Doctoral Thesis
Department of Statistics
Ume˚a School of Business, Economics and Statistics Ume˚a University
901 87 Ume˚a, Sweden
Copyright c○ 2018 by Angel G. Angelov (agangelov@gmail.com) Statistical Studies No. 53
ISBN 978-91-7601-911-5 ISSN 1100-8989
Electronic version available at http://umu.diva-portal.org Printed by: UmU Print Service, Ume˚a University
Contents
List of papers v
Abstract vii
Sammanfattning (Summary in Swedish) ix
Preface xi
1. Introduction 1
2. Interval-censored data 2
2.1. Basic notions . . . 2
2.2. Self-selected intervals . . . 3
3. Two-sample testing problems 6 4. Summary of the papers 9 4.1. Paper A . . . 9 4.2. Paper B . . . 9 4.3. Paper C . . . 10 4.4. Paper D . . . 10 5. Future research 11 References 12 Papers A–D iii
List of papers
The following papers are included in the thesis:
Paper A
Angelov AG, Ekstr¨om M (2017). Nonparametric estimation for self-selected interval data collected through a two-stage approach. Metrika, 80(4):377–399. https://doi.org/10.1007/s00184-017-0610-7
Paper B
Angelov AG, Ekstr¨om M (2018). Maximum likelihood estimation for survey data with informative interval censoring. AStA Advances in Sta-tistical Analysis. https://doi.org/10.1007/s10182-018-00329-x
Paper C
Angelov AG (2018). Nonparametric two-sample tests for informatively interval-censored data. Manuscript.
Paper D
Angelov AG, Ekstr¨om M, Kristr¨om B, Nilsson ME (2018). Testing for stochastic dominance: Procedures with four hypotheses. Manuscript.
Abstract
This thesis includes four papers: the first three of them are concerned with methods for interval-censored data, while the forth paper is devoted to testing for stochastic dominance.
In many studies, the variable of interest is observed to lie within an interval instead of being observed exactly, i.e., each observation is an interval and not a single value. This type of data is known as interval-censored. It may arise in questionnaire-based studies when the respon-dent gives an answer in the form of an interval without having pre-specified ranges. Such data are called self-selected interval data. In this context, the assumption of noninformative censoring is not fulfilled, and therefore the existing methods for interval-censored data are not necessarily applicable.
A problem of interest is to estimate the underlying distribution func-tion. There are two main approaches to this problem: (i) parametric es-timation, which assumes a particular functional form of the distribution, and (ii) nonparametric estimation, which does not rely on any distribu-tional assumptions. In Paper A, a nonparametric maximum likelihood estimator for self-selected interval data is proposed and its consistency is shown. Paper B suggests a parametric maximum likelihood estimator. The consistency and asymptotic normality of the estimator are proven. Another interesting problem is to infer whether two samples arise from identical distributions. In Paper C, nonparametric two-sample tests suitable for self-selected interval data are suggested and their prop-erties are investigated through simulations.
Paper D concerns testing for stochastic dominance with uncensored data. The paper explores a testing problem which involves four hypothe-ses, that is, based on observations of two random variables 𝑋 and 𝑌 , one wants to discriminate between four possibilities: identical survival functions, stochastic dominance of𝑋 over 𝑌 , stochastic dominance of 𝑌 over𝑋, or crossing survival functions. Permutation-based tests suitable for two independent samples and for paired samples are proposed. The tests are applied to data from an experiment concerning the individual’s willingness to pay for a given environmental improvement.
Keywords. Interval-censored data; Informative censoring; Self-selected intervals; Questionnaire-based studies; Maximum likelihood; Permuta-tion test; Two-sample test; Stochastic dominance; Four-decision test.
Sammanfattning (Summary in Swedish)
Avhandlingen inneh˚aller fyra artiklar: de tre f¨orsta ¨ar inriktade p˚a metoder f¨or intervallcensurerade data, medan det fj¨arde ¨agnas ˚at hy-potespr¨ovning av stokastisk dominans.
Intervallcensurering inneb¨ar att en variabels v¨arde ligger i ett visst intervall, samtidigt som det ¨ar ok¨ant var i intervallet v¨ardet ligger. Det vi observerar ¨ar allts˚a ett intervall och inte ett v¨arde. Dylika data kan f¨orekomma i enk¨atstudier, n¨ar respondenter fritt f˚ar ge svar i form av intervall som f¨oruts¨atts inneh˚alla v¨ardet p˚a variabeln ifr˚aga. Vi kallar denna typ av data f¨or sj¨alvvalda intervall, f¨or vilka antagandet om ick-einformativ censurering inte ¨ar uppfyllt. Tidigare existerande metoder f¨or intervallcensurerade data ¨ar d¨armed inte n¨odv¨andigtvis till¨ampliga.
Av intresse ¨ar bl.a. att skatta den underliggande f¨ ordelningsfunkti-onen som h¨or ihop med de ok¨anda punktv¨ardena. Det finns tv˚a huvud-sakliga tillv¨agag˚angss¨att f¨or denna typ av problem: (i) parametrisk skat-tningsmetodik, som f¨oruts¨atter en viss form p˚a f¨ordelningsfunktionen, och (ii) ickeparametrisk skattningsmetodik, som inte beror p˚a n˚agra f¨ordelningsantaganden. I artikel A f¨oresl˚as en ickeparametrisk maxi-mum likelihood-skattning f¨or sj¨alvvalda intervalldata, och det visas att skattningen ifr˚aga ¨ar konsistent. I artikel B f¨oresl˚as en parametrisk maximum likelihood-skattning, som bevisas vara s˚av¨al konsistent som asymptotiskt normalf¨ordelad.
Ett annat problem av intresse ¨ar att fastst¨alla ifall tv˚a oberoende stickprov kan anses komma fr˚an en och samma f¨ordelning. F¨or dy-lika problem f¨oresl˚as i artikel C ett ickeparametriskt test f¨or stickprov med sj¨alvvalda intervalldata, och testets egenskaper studeras via simu-leringar.
Artikel D behandlar test av stokastisk dominans med ocensurerade data. I artikeln unders¨oks ett hypotespr¨ovningsproblem som innefattar fyra hypoteser. Baserat p˚a observationer av tv˚a slumpm¨assiga variabler 𝑋 och 𝑌 , vill man s¨arskilja mellan fyra olika m¨ojligheter: att 𝑋 och 𝑌 har identiska ¨overlevnadsfunktioner, att 𝑋 dominerar 𝑌 stokastiskt, att𝑌 dominerar 𝑋 stokastiskt, eller att de tv˚a ¨overlevnadsfunktionerna korsar varandra. Permutationstest till¨ampliga f¨or s˚av¨al oberoende stick-prov som f¨or parade stickprov f¨oresl˚as. Hypotestestet till¨ampas p˚a data fr˚an ett experiment om respondenters betalningsvilja ang˚aende en viss specifik milj¨of¨orb¨attring.
Preface
The thesis consists of a brief introduction to the research area and four self-contained papers. It should be mentioned that the online ver-sion of the thesis does not contain the papers; they are available in the printed version. A summary in Swedish is included for those who do not speak English but speak Swedish. Those who do not speak English and want to read the whole thesis need to learn English or use a transla-tor. The introductory part (called kappa in Swedish) describes the area where ”new trees have been planted”. It is written in a concise style so that there is ”more wheat than chaff ”.
PhD studies are like walking up a mountain: the final destination is not so important, it is important what you learned and how much fun you had during the journey. I would like to express my gratitude to a number of people without whom my PhD journey would not have been the same.
First and foremost, I want to thank my supervisor Magnus Ekstr¨om. Thank you for your guidance, support and positive attitude. I am glad that I had the opportunity to work with you and learn from you. I am also grateful to my co-supervisor Maria Karlsson for thoroughly reading earlier versions of several of the papers in this thesis and giving valuable comments and suggestions. I would like to thank my co-supervisor and co-author Bengt Kristr¨om and my co-author Mats E. Nilsson for the fruitful collaboration which resulted in Paper D.
I thank all my colleagues at the Department of Statistics for creating a good work environment. I want to thank Tanya for being a nice office-mate, for interesting discussions, and for teaching me a few Ukrainian words.
Finally, I am grateful to my teachers in mathematics and statistics from whom I have learned a lot.
Angel G. Angelov Ume˚a, August 2018
1.
Introduction
The human mind treats a new idea the same way the body treats a strange protein; it rejects it.
Peter Medawar
Many phenomena can be modeled by a random variable. Often, the probability distribution of a random variable is unknown but can be estimated on the basis of some data and this is one of the fundamental problems in statistics. There are two main approaches to this problem: (i) parametric estimation, which assumes that the distribution belongs to a particular parametric family, and (ii) nonparametric estimation, which does not rely on any distributional assumptions. Each of these approaches has its advantages and disadvantages. Parametric estima-tors are usually more efficient, given that the distributional assumption is true. On the other hand, choosing the right parametric family of dis-tributions is not a trivial task. In nonparametric estimation, one does not need to assume a particular family of distributions, which makes the approach more flexible. However, the number of parameters to be esti-mated might be quite large and this leads to different issues (although it is called nonparametric estimation, there can still be parameters that need to be estimated). Papers A and B deal with the problem of esti-mating the underlying distribution function for certain data collection schemes. A nonparametric estimator is suggested in Paper A, and a parametric estimator in Paper B.
Another important statistical problem is to infer whether the ran-dom variables corresponding to two different phenomena have identical distributions, in other words, whether two phenomena have the same probabilistic behavior. There are various approaches to this problem, depending on what assumptions about the random variables are made. When no particular family of distributions is assumed, the procedure for handling such questions is called nonparametric test. Papers C and D treat problems of this kind and propose nonparametric two-sample tests for different types of data and different specification of the hypotheses. The next section introduces interval censoring and, in particular, self-selected interval data. Section 3 describes the two-sample testing problem and some variations of it. Section 4 summarizes the papers included in the thesis. Section 5 presents some ideas for further research.
2.
Interval-censored data
An approximate answer to the right question is far better than an exact answer to the wrong question.
John Tukey
2.1. Basic notions
In many circumstances, the variable of interest is not observed ex-actly. Instead, we observe an interval in which the true value of the vari-able lies. For example, we observe that 4< 𝑋 < 7 instead of observing that𝑋 = 5. Data of this kind are referred to as interval-censored data. It often occurs when the variable of interest is the time to some event and in this case the data can be called time-to-event data, survival data, failure time data, lifetime data, or duration data (seeSun 2006;Bogaerts
et al. 2017;Kalbfleisch and Prentice 2002;Lawless 2003). The problem
of analyzing time-to-event data arises in many areas like medicine, epi-demiology, engineering, economics, and demography. Interval censoring can be viewed as a type of data coarsening, which is a more general con-cept including also missing data and grouped data as special cases (see, e.g., Heitjan and Rubin 1991). Grouped data also appear in the form of intervals but should not be confused with interval-censored data. In grouped data the intervals for any two subjects are either identical or have no overlapping, while in interval-censored data the intervals may overlap arbitrarily. The statistical methods for interval-censored data are usually more complicated than those for uncensored (exact) obser-vations since we have only rough information about the true values of the variable of interest.
Interval-censored data may also appear in questionnaire-based stud-ies when the respondent is requested to give an answer in the form of an interval without being given any pre-specified ranges. Response for-mats of this kind are suitable for questions that are hard to answer with an exact amount and for sensitive questions because they allow partial information to be elicited from respondents who are unable or unwilling to provide exact values. One such response format, in which the respon-dent is asked to give an interval containing the true value of the quantity of interest, was considered byBelyaev and Kristr¨om(2010,2012, 2013,
2015). They refer to this type of data as self-selected interval data. 2
Similar format, called respondent-generated intervals, was proposed by
Press and Tanur (2004a,b), where the respondent is asked to provide
both a point value (a best guess for the true value) and an interval. In this case, only the data consisting of the intervals are regarded as interval-censored data.
Most of the existing methods for interval-censored data rely on the assumption of noninformative censoring, which implies that the observa-tion process that generates the censoring is independent of the variable of interest (seeSun 2006, p. 244). However, for self-selected interval data this is not a reasonable assumption as it is the respondent who chooses the interval. Thus, the standard methods (e.g., Peto 1973; Turnbull 1976) are not appropriate because they can produce biased results, as indicated by the simulation studies in Papers A and B.
Sometimes interval-censored data are analyzed using midpoint im-putation, that is, each finite interval is replaced by its midpoint and then the analysis is performed as if the midpoints were exact observations. As one might expect, this simple approach has its drawbacks. It has been shown that midpoint imputation may lead to biased estimates, es-pecially when the observed intervals are long (seeLaw and Brookmeyer
1992;Bogaerts et al. 2017, pp. 8–12).
2.2. Self-selected intervals
Here we give a brief overview of self-selected interval data and the respective methods for estimation. The exposition is valid for the set-tings ofBelyaev and Kristr¨om (2012,2015) and Papers A–C; however, the specific nuances of each paper are skipped and the notation may be different.
The data collection scheme can be summarized as follows. The re-spondent is first asked to state an interval containing his/her value of the quantity of interest (question Qu1). Then, the stated interval is split into sub-intervals, and the respondent is asked to select one of these sub-intervals (question Qu2). The points of split are chosen from a set of endpoints which is defined using some previously collected data (in a pilot stage) or based on other considerations. Estimating the un-derlying distribution using only the data from Qu1 would require some generally untestable assumptions related to how the respondent chooses the interval. Asking a follow-up question (Qu2) allows avoiding such assumptions.
Let 𝑑0 < 𝑑1 < . . . < 𝑑𝑘−1 < 𝑑𝑘 be the endpoints of all observed
intervals. We assume that the endpoints are rounded and that they are bounded from above by some large number, which implies that 𝑘 remains fixed for large sample sizes. This is essential for the asymp-totic results in Papers A and B. Let us define 𝒱 = {v𝑗}, where v𝑗 =
(𝑑𝑗−1, 𝑑𝑗], 𝑗 = 1, . . . , 𝑘, and let 𝒰 = {uℎ} be the set of all intervals
that can be expressed as a union of intervals from 𝒱, in other words, 𝒰 is the set of all possible intervals with endpoints from {𝑑0, . . . , 𝑑𝑘};
see Figure1 for an illustration. The interval-censored values𝑥1, . . . , 𝑥𝑛
of the quantity of interest are assumed to be values of independent and identically distributed random variables 𝑋1, . . . , 𝑋𝑛 with distribution
function𝐹 (𝑥) = P (𝑋𝑖 ≤ 𝑥). Let 𝑞𝑗 be the probability mass placed on
the intervalv𝑗 = (𝑑𝑗−1, 𝑑𝑗],
𝑞𝑗 =P (𝑋𝑖 ∈ v𝑗) =𝐹 (𝑑𝑗) −𝐹 (𝑑𝑗−1), 𝑗 = 1, . . . , 𝑘,
andq = (𝑞1, . . . , 𝑞𝑘). We define𝐻𝑖, 𝑖 = 1, . . . , 𝑛, to be independent and
identically distributed random variables such that 𝐻𝑖 = ℎ if the 𝑖-th
respondent has stated intervaluℎ at Qu1. Let us denote
𝑤ℎ|𝑗 =P (𝐻𝑖=ℎ | 𝑋𝑖∈ v𝑗).
If a respondent has answered with intervaluℎ at Qu1 and intervalu𝑠
at Qu2, then the contribution to the likelihood is ∑︀
𝑗𝛼𝑠𝑗𝑤ℎ|𝑗𝑞𝑗, where
𝛼𝑠𝑗 = 1{v𝑗 ⊆ u𝑠}. For convenience, if a respondent has given uℎ at
Qu1 and no answer at Qu2, we regard this as if the respondent stated uℎ at both Qu1 and Qu2. Let 𝑛ℎ;𝑠 denote the number of respondents
who stated uℎ at Qu1 and u𝑠 at Qu2; the data can be summarized in
a square matrix with elements𝑛ℎ;𝑠. Thus, up to an additive constant,
the log-likelihood function is
log𝐿(q) =∑︁ ℎ,𝑠 𝑛ℎ;𝑠log (︃ ∑︁ 𝑗 𝛼𝑠𝑗𝑤ℎ|𝑗𝑞𝑗 )︃ . (1)
We should mention that in scheme/design B in Papers B and C, two questions are asked after Qu1; however, in this case, the log-likelihood function has the same form as (1).
The conditional probabilities𝑤ℎ|𝑗are nuisance parameters which are
estimated using the data from Qu2. The estimates are then inserted into the log-likelihood (1).
If we want to estimate 𝐹 (𝑥) without making any distributional as-sumptions (nonparametric approach), we maximize the log-likelihood log𝐿(q) with respect to q. In this case, the estimated distribution func-tion will be a step funcfunc-tion with jumps at the points 𝑑1, . . . , 𝑑𝑘 (see
Figure2). If we assume that𝐹 (𝑥) is a known function of some unknown parameter 𝜃 = (𝜃1, . . . , 𝜃𝑑), thenq = q(𝜃) and in order to estimate 𝐹 (𝑥)
we need to estimate 𝜃, that is, to maximize log𝐿(𝜃) = log 𝐿(︀q(𝜃))︀ with respect to 𝜃.
𝑑0 𝑑1 𝑑2 𝑑3 𝑑4 𝑑5
u8
v1 v2 v3 v4 v5
Figure 1. An example with intervals v1, . . . , v5∈ 𝒱 and u8 ∈ 𝒰 .
0 50 100 150 200 0.0 0.2 0.4 0.6 0.8 1.0 x F(x) true c.d.f. parametric est. nonparametric est.
Figure 2. An example using simulated data. True distribution function 𝐹 (𝑥) = 1 − exp(−(𝑥/𝜎)𝜈) with 𝜈 = 1.5 and 𝜎 = 50, parametric estimate
(𝜈 = 1.49,̂︀ 𝜎 = 53.16), and nonparametric estimate (stepwise curve).̂︀
3.
Two-sample testing problems
If the result confirms the hypothesis, then you’ve made a measurement. If the result is contrary to the hypothesis, then you’ve made a discovery.
Enrico Fermi
Comparing two variables/samples is one of the most common prob-lems in applied research. For example, in medical studies, researchers often want to find out whether a new treatment is better than a control treatment. Research questions of this kind give rise to the statistical problem of inferring whether two samples come from identical distri-butions. Let 𝑋 and 𝑌 be random variables with survival functions 𝑆𝑋(𝑡) = P (𝑋 > 𝑡) and 𝑆𝑌(𝑡) = P (𝑌 > 𝑡). Let 𝑥1, . . . , 𝑥𝑛 be
observa-tions from 𝑆𝑋 and 𝑦1, . . . , 𝑦𝑚 be observations from 𝑆𝑌. Based on the
data (𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑚), we would like to infer whether 𝑆𝑋 =𝑆𝑌.
The general two-sample problem is defined as testing the null hypothesis 𝐻0: 𝑆𝑋 =𝑆𝑌
against the alternative hypothesis 𝐻1: 𝑆𝑋 ̸= 𝑆𝑌.
In this general setting, no assumptions about𝑋 and 𝑌 are made. With more assumptions, the problem can get much simpler. For example, if we assume a shift model, that is,𝑆𝑋(𝑡) = 𝑆𝑌(𝑡−∆) for some ∆, the problem
reduces to testing whether ∆ = 0, or if we assume that 𝑋 and 𝑌 are normally distributed with equal variances, the problem reduces to that of comparing the means of𝑋 and 𝑌 . Note that the testing problem can be equivalently formulated in terms of the distribution functions 𝐹𝑋(𝑡) =
1 −𝑆𝑋(𝑡) and 𝐹𝑌(𝑡) = 1 − 𝑆𝑌(𝑡). In Paper C, the two-sample problem
is considered for the case of self-selected interval data and appropriate tests are proposed.
In some studies, the samples are paired. For example, we can make measurements on each subject before and after receiving certain medi-cal treatment. Such designs generate pairs of observations (𝑥𝑖, 𝑦𝑖), 𝑖 =
1, . . . , 𝑛. In this case, the observations on 𝑋 are not independent of those on𝑌 because 𝑥𝑖 and 𝑦𝑖 are measured on the same subject. This
dependence should be taken into account in the statistical analysis and specific methods are needed for paired samples.
If we are interested whether one medical treatment is better than another, we actually need to consider one-sided alternative hypotheses. To formalize them, we use the notion of stochastic dominance. We say that𝑋 stochastically dominates 𝑌 if
𝑆𝑋(𝑡) ≥ 𝑆𝑌(𝑡) for all 𝑡 with strict inequality for some 𝑡.
The stochastic dominance defined above is sometimes called stochas-tic ordering or first order stochasstochas-tic dominance (see, e.g.,Ledwina and
Wy lupek 2012;Donald and Hsu 2016). Four possible hypotheses about
𝑋 and 𝑌 can be distinguished:
𝐻0: 𝑋 and 𝑌 have identical survival functions;
𝐻≻: 𝑋 stochastically dominates 𝑌 ;
𝐻≺: 𝑌 stochastically dominates 𝑋;
𝐻cr: the survival functions of 𝑋 and 𝑌 cross one another.
These hypotheses are illustrated in Figure3.
It is common in the literature to consider a testing problem with a null hypothesis of stochastic dominance (see, e.g., McFadden 1989;
Donald and Hsu 2016). However, it would be more sensible to formulate
the testing problem so that the alternative hypothesis is the hypothesis which we are trying to confirm, that is, stochastic dominance (cf.
David-son and Duclos 2013). On the other hand, the alternative hypothesis
of stochastic dominance can be mixed up with crossing survival func-tions, for example, when the test statistic takes positive values not only under stochastic dominance but also when the two survival functions cross. Therefore, it is reasonable to consider a testing problem involving all four hypotheses, that is, null hypothesis𝐻0 and three ”alternative”
hypotheses: 𝐻≻, 𝐻≺, and 𝐻cr. A testing procedure with four
hypoth-esis was proposed byBishop et al. (1989); see also Bishop and Formby
(1999), Tse and Zhang (2004), Knight and Satchell (2008), Heathcote
et al.(2010). These tests, however, do not provide adequate control over
the error probabilities related to the different types of wrong decisions in a four-hypothesis test. Bennett (2013) modified the decision rule of
Bishop et al.(1989) and suggested a procedure that has better power to
detect crossing survival functions and allows finer control over the differ-7
ent error probabilities based on asymptotic properties of the one-sided Kolmogorov–Smirnov statistics.
The existing dominance tests with four hypotheses assume indepen-dent samples. Using the four-decision rule ofBennett(2013), we suggest in Paper D dominance testing procedures suitable for paired and for in-dependent samples. −4 −2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 (i) t S(t) N(0, 1) N(0, 1) −4 −2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 (ii) t S(t) N(0, 1) N(0.7, 1) −4 −2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 (iii) t S(t) N(0.7, 1) N(0, 1) −4 −2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 (iv) t S(t) N(0, 1) N(0.2, 1.7)
Figure 3. Possible hypotheses for two random variables: (𝑖) identical survival functions, (𝑖𝑖) and (𝑖𝑖𝑖) stochastic dominance, (𝑖𝑣) crossing sur-vival functions.
4.
Summary of the papers
The essential is invisible to the eyes.
Antoine de Saint-Exupery
4.1. Paper A
This paper builds upon previous research of Belyaev and Kristr¨om
(2012,2015). We consider a two-stage scheme for collecting self-selected interval data where the number of sub-intervals in the second question of the main stage is limited to two or three. Such a scheme would be easier to implement in practice (for example, in a telephone interview) compared to a scheme where the respondents may need to choose from a large list of sub-intervals (this might happen with the scheme ofBelyaev
and Kristr¨om 2012,2015). We propose a nonparametric maximum
likeli-hood estimator of the underlying distribution function. The consistency of the estimator is established under easily verifiable conditions. An it-erative procedure for finding the estimate is proposed. Furthermore, we conduct a simulation study which indicates that the suggested estima-tor has good performance in terms of root mean square error compared to the empirical distribution function of the uncensored observations. We also demonstrate through simulations that ignoring the informative censoring can lead to substantial bias.
4.2. Paper B
In this paper, we consider two schemes for collecting self-selected interval data that extend the sampling schemes studied inBelyaev and
Kristr¨om (2012, 2015) and in Paper A. We suggest a maximum
likeli-hood estimator of the underlying distribution function, assuming that it belongs to a parametric family. In previous papers only nonparametric estimators are considered. The consistency and asymptotic normality of the proposed estimator are established. A simulation study demon-strates that the estimator performs well in comparison with the max-imum likelihood estimator for uncensored observations and also that ignoring the informative censoring can lead to bias. Additionally, we prove the asymptotic validity of a bootstrap procedure which can be used for constructing confidence intervals. The procedure is easier to apply compared to the confidence intervals based on asymptotic
mality where, for example, the derivatives of the log-likelihood need to be calculated. Our simulations indicate similar performance of the two approaches in terms of coverage and length of the confidence intervals.
4.3. Paper C
This paper deals with the problem of testing whether two samples arise from identical distributions. We suggest nonparametric two-sample tests suitable for self-selected interval data. Test statistics that can be seen as modifications of the Cram´er–von Mises statistic are employed. The testing procedures are based on a permutation test approach, which allows computing𝑝-values without relying on large-sample results for the test statistic. A simulation study indicates that the tests have reasonable power properties and control of type I error.
4.4. Paper D
In this paper, we explore a testing problem involving four possible decisions, that is, based on observations of two random variables𝑋 and 𝑌 , we want to discriminate between four hypotheses: identical survival functions, stochastic dominance of 𝑋 over 𝑌 , stochastic dominance of 𝑌 over 𝑋, or crossing survival functions. Employing the four-decision rule of Bennett (2013), we suggest dominance testing procedures suit-able for paired and for independent samples. We are not aware of any existing dominance test with four hypotheses for paired samples. The suggested procedures are based on a permutation test approach, which allows computing𝑝-values without relying on large-sample results for the test statistic. One-sided Cram´er–von Mises and Kolmogorov–Smirnov statistics are used. The tests rely on less assumptions than the existing asymptotic tests which assume, for example, continuous survival func-tions (cf. Bennett 2013). A simulation study suggests that the tests have good power to detect stochastic dominance and control of false detection errors. Finally, the proposed tests are applied to data about the individual’s willingness to pay for an environmental improvement (traffic noise reduction).
5.
Future research
The important thing is not to stop questioning; curiosity has its own reason for existing.
Albert Einstein
The methods proposed in this thesis have been implemented in the language and environment for statistical computing R (seeR Core Team 2017) for the purposes of performing the simulations and analyses re-ported in Papers A–D. Further improving the program code and creating R packages would make the methods accessible for a wider range of users. Extending the methods explored in Papers A and B to the case when the underlying distribution function depends on a set of background variables (covariates) is a possible direction for future investigation.
In Paper B, a parametric estimator for self-selected interval data is suggested. It may be of interest in future research to develop a test for assessing the goodness of fit of a parametric model.
In Papers C and D, tests for comparing two samples are proposed. A natural continuation of this work would be to develop analogous tests for more than two samples.
References
Belyaev, Y. and Kristr¨om, B. (2010). Approach to analysis of self-selected interval data. Working Paper 2010:2, CERE, Ume˚a University and the Swedish University of Agricultural Sciences. http://dx.doi.org/10.2139/ssrn.1582853.
Belyaev, Y. and Kristr¨om, B. (2012). Two-step approach to self-selected interval data in elicitation surveys. Working Paper 2012:10, CERE, Ume˚a University and the Swedish University of Agricultural Sciences. http://dx.doi.org/10.2139/ssrn.2071077.
Belyaev, Y. and Kristr¨om, B. (2013). Analysis of contingent valuation data with self-selected rounded WTP-intervals collected by two-step sampling plans. In Kollo, T., editor, Multivariate Statistics: Theory and Applications, pages 48–60. World Scientific.
Belyaev, Y. and Kristr¨om, B. (2015). Analysis of survey data containing rounded censoring intervals. Informatics and Applications, 9(3):2–16. Bennett, C. J. (2013). Inference for dominance relations. International
Economic Review, 54(4):1309–1328.
Bishop, J. A. and Formby, J. P. (1999). Tests of significance for Lorenz partial orders. In Silber, J., editor, Handbook of Income Inequality Measurement, pages 315–339, Dordrecht. Springer.
Bishop, J. A., Formby, J. P., and Thistle, P. D. (1989). Statistical infer-ence, income distributions, and social welfare. Research on Economic Inequality, 1:49–82.
Bogaerts, K., Komarek, A., and Lesaffre, E. (2017). Survival Analysis with Interval-Censored Data: A Practical Approach with Examples in R, SAS, and BUGS. CRC Press, Boca Raton, Florida.
Davidson, R. and Duclos, J.-Y. (2013). Testing for restricted stochastic dominance. Econometric Reviews, 32(1):84–125.
Donald, S. G. and Hsu, Y.-C. (2016). Improving the power of tests of stochastic dominance. Econometric Reviews, 35(4):553–585.
Heathcote, A., Brown, S., Wagenmakers, E. J., and Eidels, A. (2010). Distribution-free tests of stochastic dominance for small samples. Journal of Mathematical Psychology, 54(5):454–463.
Heitjan, D. F. and Rubin, D. B. (1991). Ignorability and coarse data. The Annals of Statistics, 19(4):2244–2253.
Kalbfleisch, J. D. and Prentice, R. L. (2002). The Statistical Analysis of Failure Time Data. John Wiley & Sons, Hoboken, New Jersey, 2 edition.
Knight, J. and Satchell, S. (2008). Testing for infinite order stochastic dominance with applications to finance, risk and income inequality. Journal of Economics and Finance, 32(1):35–46.
Law, C. G. and Brookmeyer, R. (1992). Effects of mid-point imputa-tion on the analysis of doubly censored data. Statistics in Medicine, 11(12):1569–1578.
Lawless, J. F. (2003). Statistical Models and Methods for Lifetime Data. John Wiley & Sons, Hoboken, New Jersey, 2 edition.
Ledwina, T. and Wy lupek, G. (2012). Two-sample test against one-sided alternatives. Scandinavian Journal of Statistics, 39(2):358–381. McFadden, D. (1989). Testing for stochastic dominance. In Fomby,
T. B. and Seo, T. K., editors, Studies in the Economics of Uncertainty, pages 113–134, New York. Springer.
Peto, R. (1973). Experimental survival curves for interval-censored data. Journal of the Royal Statistical Society. Series C (Applied Statistics), 22(1):86–91.
Press, S. J. and Tanur, J. M. (2004a). An overview of the respondent-generated intervals (RGI) approach to sample surveys. Journal of Modern Applied Statistical Methods, 3(2):288–304.
Press, S. J. and Tanur, J. M. (2004b). Relating respondent-generated intervals questionnaire design to survey accuracy and response rate. Journal of Official Statistics, 20(2):265–287.
R Core Team (2017). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org.
Sun, J. (2006). The Statistical Analysis of Interval-Censored Failure Time Data. Springer, New York.
Tse, Y. K. and Zhang, X. (2004). A Monte Carlo investigation of some tests for stochastic dominance. Journal of Statistical Computation and Simulation, 74(5):361–378.
Turnbull, B. W. (1976). The empirical distribution function with arbi-trarily grouped, censored and truncated data. Journal of the Royal Statistical Society. Series B (Methodological), 38(3):290–295.
