Functions of Order Statistics
01 Apr 1972-Annals of Mathematical Statistics (Institute of Mathematical Statistics)-Vol. 43, Iss: 2, pp 412-427
TL;DR: Two theorems on the asymptotic normality of linear combinations of functions of order statistics are given in this article, one requires a smooth scoring function but the underlying df need not be continuous even and can also depend on the sample size.
Abstract: Two theorems on the asymptotic normality of linear combinations of functions of order statistics are given. Theorem 1 requires a "smooth" scoring function but the underlying df need not be continuous even and can also depend on the sample size. Theorem 2 allows general scoring functions but places additional restrictions on the df. Applications included.
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TL;DR: In this article, the analysis of inequality is placed in the context of recent developments in economics and statistics, and it is shown that inequality can be expressed as a function of economic and statistical factors.
Abstract: The analysis of inequality is placed in the context of recent developments in economics and statistics.
639 citations
TL;DR: In this article, the authors considered the problem of statistical inference with estimated Lorenz curves and income shares and derived the full variance-covariance structure of the (asymptotic) normal distribution of a vector of Lorenz curve ordinates.
Abstract: The paper considers the problem of statistical inference with estimated Lorenz curves and income shares. The full variance-covariance structure of the (asymptotic) normal distribution of a vector of Lorenz curve ordinates is derived and shown to depend only on conditional first and second moments that can be estimated consistently without prior specification of the population density underlying the sample data. Lorenz curves and income shares can thus be used as tools for statistical inference instead of simply as descriptive statistics.
331 citations
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TL;DR: In this article, the analysis of inequality is placed in the context of recent developments in economics and statistics, and it is shown that inequality can be expressed as a function of economic and statistical factors.
Abstract: The analysis of inequality is placed in the context of recent developments in economics and statistics.
311 citations
TL;DR: In this article, the authors considered linear functions of order statistics of the form (S_n = n −1} √ √ J(i/(n + 1))X_{(i)} and showed that the second moment of the population is finite and bounded and continuous a.i.d.
Abstract: This paper considers linear functions of order statistics of the form $S_n = n^{-1} \sum J(i/(n + 1))X_{(i)}$. The main results are that $S_n$ is asymptotically normal if the second moment of the population is finite and $J$ is bounded and continuous a.e. $F^{-1}$, and that this first result continues to hold even if the unordered observations are not identically distributed. The moment condition can be discarded if $J$ trims the extremes. In addition, asymptotic formulas for the mean and variance of $S_n$ are given for both the identically and non-identically distributed cases. All of the theorems of this paper apply to discrete populations, continuous populations, and grouped data, and the conditions on $J$ are easily checked (and are satisfied by most robust statistics of the form $S_n$). Finally, a number of applications are given, including the trimmed mean and Gini's mean difference, and an example is presented which shows that $S_n$ may not be asymptotically normal if $J$ is discontinuous.
259 citations
TL;DR: In this paper, it was shown that the distance between the empirical and quantile processes can be approximated by a sequence of Brownian bridges as well as by a Kiefer process.
Abstract: Let $q_n(y), 0 < y < 1,$ be a quantile process based on a sequence of i.i.d. rv with distribution function $F$ and density function $f.$ Given some regularity conditions on $F$ the distance of $q_n(y)$ and the uniform quantile process $u_n(y),$ respectively defined in terms of the order statistics $X_{k:n}$ and $U_{k:n} = F(X_{k:n}),$ is computed with rates. As a consequence we have an extension of Kiefer's result on the distance between the empirical and quantile processes, a law of iterated logarithm for $q_n(y)$ and, using similar results for the uniform quantile process $u_n(y),$ it is also shown that $q_n(y)$ can be approximated by a sequence of Brownian bridges as well as by a Kiefer process.
178 citations
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300 citations
TL;DR: In this paper, the problem of estimating a location parameter from a random sample when the form of distribution is unknown or there is contamination of the target distribution is attacked by deriving estimators which are efficient over a class of two or more forms (pencils) of continuous symmetric unimodal distributions.
Abstract: The problem of estimating a location parameter from a random sample when the form of distribution is unknown or there is contamination of the target distribution is attacked by deriving estimators which are efficient over a class of two or more forms (“pencils”) of continuous symmetric unimodal distributions. The pencils considered are the normal, double exponential, Cauchy, parabolic, triangular, and rectangular (a limiting case). The estimators considered are special symmetrical linear combinations of order statistics: trimmed means, Winsorized means, “linearly weighted” means, and a combination of the median and two other order statistics. These are also compared asymptotically with a Hodges-Lehmann estimator. The theory required for deriving asymptotic variances is outlined. Efficiences are tabulated for sample sizes of 4 or 5, 8 or 9, 16 or 17, and ∞. Asymptotic efficiences of at least 0.82 relative to the best estimator for any single pencil are achieved by using the best trimmed mean or li...
227 citations
TL;DR: In this paper, the authors consider the problem of proving asymptotic normality of linear combinations of order statistics, where the objective is to find conditions under which a statistic of the form $S_n = \mathbf{\sum}^n_{i=1} c_{in}X_{in}) has a limiting normal distribution as $n$ becomes infinite.
Abstract: The purpose of this paper is to investigate the asymptotic normality of linear combinations of order statistics; that is, to find conditions under which a statistic of the form $S_n = \mathbf{\sum}^n_{i=1} c_{in}X_{in}$ has a limiting normal distribution as $n$ becomes infinite, where the $c_{in}$'s are constants and $X_{1n}, X_{2n}, \cdots, X_{nn}$ are the observations of a sample of size $n$, ordered by increasing magnitude. Aside from the sample mean (the case where the weights $c_{in}$ are all equal to $1/n$), the first proof of asymptotic normality within this class was by Smirnov in 1935 [19], who considered the case that nonzero weight is attached to at most two percentiles. In 1946, Mosteller [13] extended this to the case of several percentiles, and coined the phrase "systematic statistic" to describe $S_n$. Since the publication in 1955 of a paper by Jung [11] concerned with finding optimal weights for $S_n$ in certain estimation problems, interest in proving its asymptotic normality under more general conditions has grown. For example, Weiss in [21] proved that $S_n$ has a limiting normal distribution when no weight is attached to the observations below the $p$th sample percentile and above the $q$th sample percentile, $p < q$, and the remaining observations are weighted according to a function $J$ by $c_{in} = J(i/(n + 1))$, where $J$ is assumed to have a bounded derivative between $p$ and $q$. Within the past few years, several notable attempts have been made to prove the asymptotic normality of $S_n$ under more general conditions on the weights and underlying distribution. These attempts have employed three essentially different methods. In [1] Bickel used an invariance principle for order statistics to prove asymptotic normality when $\mathbf{\sum}_{i
87 citations
TL;DR: In this article, the authors show how to bound ourselves away from 0 and 1 by an amount #N where NiN -3 0 as N - oo at a rate to be specified later.
Abstract: FN'(t) = inf {x:FN(x) > t}; and we write g o FN-l for the composed function g (FN-1). Note that TN = aO g o FN-1 dVN when the signed measure VN puts mass CNi/N at i/N for i = 1, * * , N and puts 0 mass elsewhere. Let v denote a signed measure on (0, 1). (The signed measures VN will not be used, but v is in some sense their limit.) For technical reasons to be displayed below, we bound ourselves away from 0 and 1 by an amount #N where NiN -3 0 as N - oo at a rate to be specified later. Let (1.2) 1N - fANd (1.2) Y~~~~~~.N = fAN g dv
46 citations