摘要:

In the problem of selecting the best ofkpopulations, a natural rule is to select the population corresponding to the largest sample value of an appropriate statistic. As a retrospective analysis, a conservative lower confidence bound on the probability of a correct selection is derived when the probability density function has the monotone likelihood ratio property under the location parameter setting. The result is applied to the normal populations with both known and unknown common variance. Tables to implement the confidence bound are provided.

ISSN:0162-1459    

DOI:10.1080/01621459.1986.10478366

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

There are many nonparametric tests available for the analysis ofkindependent samples. For example, the Kruskal—Wallis test (Wilcoxon rank sum test whenk= 2) is widely used for detecting location differences, and Mood or Ansari—Bradley or Sigel—Tukey rank statistics are often suggested for detecting scale differences. If more general alternatives are of interest, then one can use omnibus tests such as the generalized Kolmogorov—Smirnov and Cramér—von Mises statistics [see Conover (1980, secs. 6.3 and 6.4) or Kiefer (1959)]. Unfortunately, “significant” results from these omnibus tests are hard to interpret. Instead, I would like to propose some new tests based on a 4×ktable of linear rank statistics that are simple to interpret because the four rows of the table are aimed at specific alternatives: location, scale, skewness, and kurtosis. Row summary statistics consist of weighted sums of squares of the individual entries. The first two of these are the Kruskal—Wallis statistic and Mood'sk-sample statistic for testing scale differences, and the last two are newk-sample test statistics relating to skewness and kurtosis alternatives. An overall omnibus statistic GLOBE is then formed by adding all four of these row summary statistics. GLOBE is also a weighted sum of column summary statistics that are computed by taking the sum of squares of each column of the 4×ktable.

ISSN:0162-1459    

DOI:10.1080/01621459.1986.10478367

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

A method is proposed for testing normality, in the case of general Type II censored data—that is, data for which only a subset of the order statistics are available. Three test statistics are proposed, which are generalizations of the statistics proposed in LaRiccia (1986), and have many of the same properties. Specifically, they are designed to be asymptotically optimal with respect to specific alternatives and are easily adjusted to be asymptotically optimal with respect to many other types of alternatives. Under the null hypothesis, irrespective of the type or amount of censoring, the proposed test statistics are asymptotically distributed as chi-squared random variables. Further, results of a simulation study are presented, indicating that these statistics converge quite rapidly in distribution to the appropriate chi-squared random variables and that the asymptotic critical values provide a useful approximation to the small sample critical values even forn= 25. The results of a simulation study comparing the power of the proposed tests with some standard tests of normality are presented. These results indicate that, for the cases considered, these statistics compare favorably with the standard procedures.

ISSN:0162-1459    

DOI:10.1080/01621459.1986.10478368

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

Randomly censored data consist of iid pairs of observations (Xi, δi), i= 1, …,n; if δi= 0,Xidenotes a censored observation, and if δi= 1,Xidenotes an exact “survival” time, which is the variable of interest. For estimating the distributionFof the survival times, the product-limit estimator proposed by Kaplan and Meier (1958) has been studied extensively and it has been shown to enjoy a number of optimality properties (Wellner 1982). See Gill (1980, chaps. 1–4; 1983) for a modern treatment and for references. With censored data, bootstrapping can be carried out using two different resampling plans introduced by Efron (1981) and Reid (1981), respectively. With Efron's plan one takes a random sample with replacement from (X1), …, (Xn, δn), whereas with Reid's plan one takes a random sample from the Kaplan—Meier estimator. The purpose of this article is to study the asymptotic behavior of the bootstrapped Kaplan—Meier estimator with both resampling plans. The approach adopted uses the theory of martingales for point processes. This extends our understanding of the asymptotic behavior of bootstrapped estimated distribution functions to the situation in which random censoring occurs. The uncensored case has been studied by Bickel and Freedman (1981) and Shorack (1982). The new result is used to obtain bootstrap confidence bands for a survival distribution under random censoring using Efron's approach. These bands are the only available ones that are valid even when the survival distribution has a discrete component. Further, it is demonstrated that Reid's proposal for bootstrapping does not produce asymptotically correct confidence bands. Results from simulation studies assessing the finite sample performance of the bootstrap confidence bands are included.

ISSN:0162-1459    

DOI:10.1080/01621459.1986.10478369

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

Nomination sampling is a sampling process in which every observation is the maximum of a random sample from some population. Assuming that all samples are taken from a single underlying distributionF, data may be viewed as consisting of pairs (Xi,Ki), whereKiis the size of theith sample and, givenKi=ki, Xiis distributed according toFki. Willemain (1980), who discussed nomination sampling in the context of health care delivery, proposed an estimator for the median ofFunder the assumption thatKi=N, a fixed integer. In this article, the assumption of a fixed sample sizeNfrom each population is relaxed; withKtaken as random, the problem of nonparametric estimation of the distribution functionFis considered. The nonparametric maximum likelihood estimator ofFis derived, its consistency is demonstrated, and its asymptotic behavior as a stochastic process is identified. Conditions are given under which these asymptotic results hold for nonrandomK. A by-product of this development is the consistency of Willemain's estimator of the median. Several applications are considered. For example, nomination sampling arises naturally in certain reliability experiments; the applicability of the derived estimator in problems involving designed redundancy is noted. A detailed analysis of data on 34 yearly maximum floods of the Nidd River is presented, and the estimator of the underlying flood distributionFis displayed.

ISSN:0162-1459    

DOI:10.1080/01621459.1986.10478370

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

The estimation of the function exp(aξ + bσ2) of normal parameters ξ and σ2on the basis of a random sampleX1, …,Xnis considered. This class of functions includes the mean, the median, and all moments of lognormal distribution. I show that the minimum variance unbiased estimator suggested by Finney (1941) can be substantially improved in terms of mean squared error. A similar result is established for the maximum likelihood estimator. I suggest for practice use the following generalized Bayes estimator when m = [(n+1)/2]: δ(X,Y) = exp(aX + (γ - β)Y) ∑k=0m(2m - k)!/k!(m - k)!(2βY)k/∑k=0m(2m - k)!/k!(m - k)!(2γY)k. Here X = ∑InXj/n, Y2= ∑1n(X1- X)2, and constants β and γ are determined by formulas β2= γ2− 2b + 2a/n, γ = 1.5(b-3a2/(2n)). This estimator is shown to be locally optimal for both small and large values of σ2. The results of numerical study of the quadratic risk show the superiority of this estimator over the mentioned traditional procedures.

ISSN:0162-1459    

DOI:10.1080/01621459.1986.10478371

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

Poverty is usually defined as the extent to which individuals in a society or community fall below a minimal acceptable standard of living. An index of poverty is generally based on the proportion (α) of the poor people and their income distribution through the income gap ratio (β) (i.e., the average income gap of the poor people from the poverty line ω, taken as a ratio to ω itself) and other measures of income inequalities. In this context, the well-known Gini coefficient of income inequality plays a vital role. In view of the fact that the poverty indexes all relate to the income pattern of the poor, interpreted in some way or other, usually a censored (Gω) or truncated (Gα) version of the classical Gini coefficient (G) is incorporated in the formulation of such indexes. Among the various poverty indexes, πA= αβ, πτ = Gcω, and πs=α{β+(1-β)Gα} have been used more extensively than the others. Although each of πSand πTis justified on the grounds of certain plausible axioms, from a statistical point of view, generally, for smaller values of α and β, πτ is somewhat more conservative whereas πSis more anticonservative than they should have been ideally. This phenomenon is mainly due to the two forms of the Gini coefficientGcωandGα, which behave rather differently with the variation of α, β, and the income inequalities among the poor people. This calls for a more intensive study of the behavior of the Gini coefficient under various patterns of income inequality and its role in the formulation of poverty indexes. This examination leads to consideration of a more robust version of πS, namely, π*=αβ1-Gα. In this context, TTT transformations (usually arising in reliability theory and life testing problems) are incorporated to provide a new interpretation of the Gini coefficient; in light of this, the relationship betweenGαandGcωand suitable bounds for either of these indexes are studied in detail. These results, in turn, facilitate the study of the relative and absolute interpretations of these poverty indexes, in the light of which the weakness of the index πτ and the relative strengths of πSand π* have also been discussed.

ISSN:0162-1459    

DOI:10.1080/01621459.1986.10478372

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

Empirical Bayes methods are becoming increasingly popular in statistics. Robbins (1955) introduced the method in the context of nonparametric estimation of a completely unspecified prior distribution. Subsequently, the method has been explored very successfully in a series of articles by Efron and Morris (1973, 1975, 1977) in a parametric framework. In the Efron—Morris setup, a family of parametric distributions is used as possible priors, but only when one or more of the parameters of the family of prior distributions is estimated from the data. Morris (1983) listed a number of areas where empirical Bayes methods are used.

ISSN:0162-1459    

DOI:10.1080/01621459.1986.10478373

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

Outliers in sample data are a perennial problem for applied survey statisticians. Moreover, it is a problem for which traditional sample survey theory offers no real solution, beyond the sensible advice that such sample elements should not be weighted to their fullest extent in estimation.

ISSN:0162-1459    

DOI:10.1080/01621459.1986.10478374

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

Most statistical methodology in use today is designed for sampling schemes or experimental designs in which the sample size or number of data points is regarded as a fixed, preset parameter. This is in part for mathematical convenience but is also suggested by the intuition that sampling is restricted by a total cost of data collection and that the cost of sampling remains the same from unit to unit. The cost of sampling, however, may vary from unit to unit and the costs may be unknown before the sample is taken. If so, the objective of taking a sample of preset cost may force one to sample sequentially until achieving the preset cost and then stop. That is, the objective of taking a sample of fixed size may be turned against itself if the cost of sampling varies from unit to unit. A cost may be monetary or general in character.

ISSN:0162-1459    

DOI:10.1080/01621459.1986.10478375

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor