bluuebunny/crossref_metadata_2025_split huggingface dataset:
Kurtosis
Authors: D'Agostino Ralph B. 
Published: July 2005

AbstractKurtosis expresses the extent to which a probability distribution is more (or less) peaked and long‐tailed as compared to the normal distribution. More‐peaked distributions are called leptokurtic, less‐peaked are platykurtic. The usual index, β2, is the ratio of the fourth moment to the square of the second moment, and equals three for a normal distribution, with the associated index γ2(= β2) equal to zero. The indices may be estimated in terms of sample moments. Other measures have been proposed.
Law of Large Numbers
Authors: Janssen Paul 
Published: July 2005

Abstract
It is well known that the sample mean, based on a sequence of independent random variables with common distribution, is a weakly consistent estimator as well as a strongly consistent estimator for the population mean. The first property is the famous weak law of large numbers (Khinchin), the second property is the strong law of large numbers (Kolmogorov). Both laws are key results of modern probability theory. Similar properties hold for more complicated sequences of random variables, for example, for sequences satisfying the stationarity assumption or for martingale sequences. These extensions are of crucial importance for statistical applications in ergodic theory and survival analysis.
Limit Theorems
Authors: Zabell S. L. 
Published: July 2005

Abstract
These concern the asymptotic behavior of distributions of random variables in large samples. Classical results include the weak and strong laws of large numbers, the central limit theorem, and the law of the iterated logarithm. Other results discussed briefly are the ergodic theorem, the martingale convergence theorem, Skorokhod's theorem, and the use of coupling methods.
L
indley's Paradox
Authors: Shafer Glenn 
Published: July 2005

Abstract

A test of significance can disagree strongly with a Bayesian analysis when the Bayesian analysis assigns a substantial probability to the point hypothesis and distributes the remaining probability diffusely. The disagreement is sometimes called
Lindley's paradox
. If the diffuse distribution under the alternative hypothesis is based on evidence, the Bayesian analysis is convincing, but if the distribution is diffuse only because of ignorance, the test of significance may be more convincing.
Location–Scale Family
Authors: Datta Gauri Sankar 
Published: July 2005

Abstract
A set of random variables has a location‐scale family distribution if their joint distribution function can be expressed as a function of the standardized observations (xi ‐μ)/s, where μ and s are, respectively, the location and scale parameters. Important examples are the uniform, normal, and Cauchy distributions. Least squares estimation and minimum risk equivariant estimation are described.
Lognormal Distribution
Authors: Ratnaparkhi M. 
Published: July 2005

AbstractThis is the distribution of a positive‐valued random variable, the natural logarithm of which follows a normal distribution with mean μ and variance σ2. A generalization to the three‐parameter lognormal distribution arises if a location parameter is introduced, giving a nonzero lower bound for the variable. Properties, estimation procedures, and applications are described.
Loss Function
Authors: Tritchler D. 
Published: July 2005

Abstract

In decision theory, the loss function expresses the consequences of a specific decision when a particular state of nature holds. Negative loss is called
gain or utility
. The aim in Bayesian theory is to minimize the expected loss, integrating over the prior distribution. In frequentist theory, various restrictive conditions may be imposed, or a minimax solution may be sought. Statistical procedures such as hypothesis testing and estimation may be interpreted in terms of loss functions. Practical applications include the design of clinical trials and screening procedures.
Mean
Authors: Smith W. 
Published: July 2005

Abstract
The sample arithmetic mean is the average of a set of numerical measurements. For a random variable, the analog is the expectation. In a weighted mean, the individual observations may be given different weights before averaging, perhaps because they represent groups of different sizes. The geometric mean is obtained by averaging logarithms (of positive‐valued measurements) and then taking antilogarithms. The harmonic mean is obtained by taking reciprocals before averaging and then taking the reciprocal of the average. The relationship between the arithmetic, geometric, and harmonic means is described.
Mean Deviation
Authors: Hampel Frank 
Published: July 2005

Abstract
The mean deviation of a set of measurements is the mean of the absolute deviations from some location measure, which may be the sample mean or median, a population mean or some other constant. Its relation to the standard deviation is described. For normal samples, the standard deviation is a more efficient estimator of the population standard deviation, but the reverse is true for some forms of contaminated distributions.
Minimum Variance Unbiased (
MVU
) Estimator
Authors: Bethel J. 
Published: July 2005

Abstract
Determining the minimum variance of an unbiased estimator constitutes a fundamental topic in mathematical statistics. Two primary results are the Cramér–Rao inequality, which gives a lower bound on the variance of an unbiased estimator, and the Rao–Blackwell theorem, which supplies a method for improving the variance of unbiased estimators that are not functions of sufficient statistics. The strongest results are available for sufficient statistics from complete exponential families. However, any functions of sufficient statistics will attain the Cramér–Rao lower bound for sufficiently large sample sizes.
Moment Generating Function
Authors: Koehler Kenneth 
Published: July 2005

Abstract
Moment generating functions are used to derive moments of distributions, establish the distribution of sums and differences of independent random variables, and derive limiting distributions of sequences of random variables. It is an integral transform of the density function and the terms of its Taylor expansion about the origin give moments about zero. Central moments are easily obtained. Properties discussed briefly include the derivation of the MGF for a sum as the product of the separate MGFs, the multivariate case and the derivation of the factorial moment generating function.
Moments
Authors: Koehler Kenneth 
Published: July 2005

Abstract

The
r
th moment of a random variable
X
is the expectation of the
r
th power of
X
. Moments may also be taken about a constant
c
by taking powers of
X
−
c
, and central moments take
c
as the mean of the distribution. Related concepts are absolute moments, factorial moments, and cumulants. The method of moments is a method of estimation by equating estimated moments to theoretical values expressed in terms of parameters.
Paired

t

Test
Authors: Hsu Henry , Lachenbruch Peter A. 
Published: July 2005

Abstract

This is used to compare mean differences between treatments when the observations have been obtained in pairs. The difference between the paired values is assumed to be normally distributed, and the null hypothesis that the expectation is zero is tested by Student's
t
test. The robustness properties are discussed, as is the asymptotic relative efficiency of nonparametric alternatives.
P
areto Distribution
Authors: Childs A. , Balakrishnan N. 
Published: July 2005

Abstract

The classical Pareto distribution of the first kind has density function of the form 1–(
x
/σ)
−ν
. Pareto distributions of the second, third, and fourth kind involve extra parameters. The second kind arises as a gamma mixture of exponential distributions, and has been used in survival analysis. A multivariate generalization is described.
Pearson Distributions
Authors: Armitage Peter 
Published: July 2005

Abstract

This is a family of probability distributions, often called Pearson
curves
, developed by Karl Pearson in the period between 1894 and 1895 to provide flexible descriptions of the nonnormal distributions encountered in his biometric research. It includes many well‐known distributions. The types can be distinguished by the standard indices of skewness and kurtosis.
Pitman Efficiency
Authors: Zacks S. 
Published: July 2005

Abstract
The Pitman efficiency is an index for comparing test procedures or estimators. It is especially important for comparing procedures in large samples. If procedure P1 requires n1 observations to attain a certain power of a test, or a specified mean squared error, and procedure P2 requires n2 observations to achieve the same precision, the Pitman relative efficiency of P1 against P2 is n2/n1. Accordingly, if the mean squared error of an estimator E1 is v1/n + o(1/n), and if that of another estimator, E2, is v2/n + o(1/n), then the asymptotic (large sample) relative efficiency of E1 against E2 is v2/v1. Under smoothness regularity conditions, the variance of an unbiased estimator, in the one‐parameter case, is bounded below by the Cramer–Rao (CR) lower bound. Accordingly, the ratio of the CR‐lower bound to the variance of the estimator is defined as the efficiency index of the unbiased estimator. This is the Pitman relative efficiency of the estimator against the best possible one. Moreover, the CR‐lower bound contains the Fisher information function as a factor, and the inverse of the Fisher information function is the asymptotic variance of the maximum likelihood estimator (bf MLE). Therefore, the efficiency of an unbiased estimator is measured relative to the asymptotic variance of the maximum likelihood estimator.

The Pitman relative efficiency of consistent and asymptotically normal test statistics is defined as the ratio of their corresponding
efficacy ratios
. The asymptotic efficacy relates the rate of change of the mean function of the statistic to its asymptotic standard deviation. For further information, see the article.
Probability Theory
Authors: Steele J.M. 
Published: July 2005

Abstract
Probability theory is that part of mathematics relating to phenomena depending on chance or uncertainty. Probability is usually thought of either in terms of outcomes of repeated trials or as a measure of subjective belief. A.N. Kolmogorov provided an axiomatic basis for probability using measure theory. Fundamental notions include random variables and independence. Two basic theorems are the law of large numbers and the central limit theorem. Reference is made to theories for dependent data, involving Markov processes and martingales.
Quantiles
Authors: Milton Roy C. 
Published: July 2005

Abstract

Quantiles are the
N
− 1 points that divide a distribution or sample data into
N
equal and ordered parts. Some sets of these
N
− 1 points have well‐known names: median (
N
= 2) and percentiles (
N
= 100). Quantiles are commonly used in testing of hypotheses, summarizing sample data, and modeling risk relationships.
Random Variable
Authors: Basu Asit P. 
Published: July 2005

Abstract
Random variables are used to describe the uncertain outcomes of a study. Formally, a random variable is a function assigning a real value to each outcome of the sample space, which is the set of all possible outcomes. Related topics described here include the distinction between continuous and discrete random variables, the probability density function and cumulative distribution function, a bivariate distribution, conditional and marginal probability, and independence.
Randomness, Tests of
Authors: Whaley F. S. 
Published: July 2005

AbstractTests of randomness are used to assess whether data are truly random or have some sort of pattern. These include tests performed on data collected over a period of time or over geographical areas. Some tests are used to detect general nonrandomness, while other tests focus on detecting a particular pattern of nonrandomness, such as clustering or trends. Spatial tests of randomness can be used to assess the randomness of the location of data in a geographical region or assess the relationship between a characteristic of a population and its location. Tests of randomness can also be used to assess the randomness of a sequence of pseudo‐random numbers.
Scedasticity
Authors: Morrison Donald F. 
Published: July 2005

Abstract

Homoscedasticity is the condition under which a series of observations have the same degree of variation. Heteroscedasticity is its negation. Homoscedasticity is assumed in the standard models for least squares linear regression, the two‐sample
t
test and the analysis of variance, and the effects of heteroscedasticity in these situations are discussed here.
Simultaneous Confidence Intervals
Authors: Chen Tuhao , Hoppe Fred M. 
Published: July 2005

AbstractThese provide a confidence region for a multivariate parameter, comprising individual intervals for the separate components of the parameter. There is a specified confidence level of the simultaneous correctness of all the statements. The approach is useful for multiple comparisons of treatment means. The methods of Bonferroni, Tukey, Scheffé and Duncan are described, with examples.
Simultaneous Inference
Authors: Geller Nancy L. , O'Brien Peter C. 
Published: July 2005

Abstract
This covers statistical procedures to assess either more than one parameter or one parameter repeatedly during the course of a study. It includes multiple comparisons, multiple endpoints, sequential methods, and longitudinal data analysis. A common approach is to control the per‐family error rate (expected number of false positive conclusions per study) or the experimentwise error rate (probability of at least one false positive). Examples are given, with particular reference to problems of data monitoring of clinical trials.
Statistical Dependence and Independence
Authors: Wermuth Nanny , Cox D. R. 
Published: July 2005

AbstractStatistical dependence is a type of relation between different characteristics measured on the same units. At one extreme is deterministic dependence; at the other is statistical independence, where the distribution of one variable is the same for all levels of the other. With more than two variables, an important distinction is between marginal and conditional dependence. In many contexts, the degree of dependence may be summarized by a suitable measure of association, perhaps as part of a general model. Reference is made to graphical models.
Uniform Distribution
Authors: Deng Lih‐yuan 
Published: July 2005

Abstract
A discrete uniform distribution has constant probability for each integer within a stated range. A continuous uniform distribution has constant probability density within a stated range. The continuous uniform distribution in the range (0, 1) has connections with the probability integral transformation, and with the exponential, logistic, and beta distributions. Some characterizations of functions of two independent random variables are given.
Union–Intersection Principle
Authors: Chinchilli V. M. 
Published: July 2005

AbstractMultivariate analysis is the branch of statistics that involves inferential methods with two or more response (dependent) variables simultaneously. Therefore, multivariate analysis tends to be more complex than univariate analysis (one dependent variable) with respect to its mathematical development and its interpretation. The union–intersection principle is an approach that provides a basic approach for hypothesis testing in certain multivariate situations. We illustrate an application of the union–intersection principle to the situation in which there are pretreatment and posttreatment measurements onpdependent variables for a sample ofnexperimental units.
U

‐Statistics
Authors: Janssen Paul 
Published: July 2005

Abstract

A U‐statistic, calculated from a random sample of size
n
, is an average of a symmetric function of
m
sample values, calculated for all
m
‐tuples in the sample. Examples are the sample variance, the Cramér‐von Mises statistic of goodness of fit, and the Kaplan–Meier and Nelson–Aalen estimators in survival analysis. Asymptotic properties are described.
Wilcoxon–Mann–Whitney Test
Authors: Moses Lincoln E. 
Published: July 2005

AbstractThis is a method of comparing the size of observations in two groups by ranking all the observations and comparing the average ranks of the two groups. The distribution theory is described. The test has high efficiency for the comparison of two normal distributions with the same variance, and is much better than thettest for some nonnormal distributions. Topics discussed include adjustment for tied data and the calculation of confidence limits for a difference in translation between the two distributions.
Combining
P
Values
Authors: Westfall Peter H. 
Published: July 2005

Abstract

When a null hypothesis is tested on different sets of data, the individual
P
values may usefully be combined to provide a more powerful test than any individual one. If the null hypothesis is true, and the statistical model is correct, each
P
value follows a uniform distribution. Various methods of combination have been suggested for situations where the data sets are independent, and some comparisons of power against alternative hypotheses are presented for a particular case. Most of these methods are invalid for dependent data.
Chi‐Square Distribution; Properties
Authors: Klotz Jerome 
Published: July 2005

Abstract
This article complements the introductory article on the chi‐square distribution, with details of its derivation as the distribution of the sum of squares of standard normal deviates, the density being a gamma function, and the relationship between its distribution function and that of the Poisson distribution. When the component normal distributions have nonzero mean, the sum of squares has a noncentral chi‐square distribution, depending on a noncentrality parameter, the sum of squares of the normal means. This plays a role in determining the power of a chi‐square test.
Composite Estimators
Authors: Schaible Wes 
Published: July 2005

AbstractA composite estimator is a weighted combination of two (or more) component estimators that, with appropriate weights, will have a mean square error smaller than that of either component estimator. Early applications of composite estimators occurred in conjunction with continuing surveys with rotational designs. These designs allow the production of two distinct component estimators for the current time period—one using the overlap data and the estimate for the previous time period, and the second using only the data from the current time period. A second type of application occurs when the sample size in a particular domain is too small to produce a standard, direct estimate of sufficient reliability. In such cases, a second indirect estimate can be made and a weighted combination of the direct and indirect estimates used to make a composite estimate.
Distance Sampling
Authors: Buckland Stephen T. , Anderson David R. , Burnham Kenneth P. , Laake Jeffrey L. 
Published: July 2005

AbstractDistance sampling covers a range of methods in which distances of detected objects (usually animals) from a line or point are recorded, from which object density or abundance is estimated. The most widely used methods are line transect sampling and point transect sampling. Objects on the line or point are usually assumed to be detected with certainty, and a detection function, representing probability of detection as a function of distance from the line or point, is modeled. This allows estimation of the proportion of objects missed within surveyed strips or circles, extending a specified distance from the line or point. Appropriate survey design allows the resulting object density estimates to be extrapolated to the full survey region, to yield estimates of object abundance.
Graeco–Latin Square Designs
Authors: Martin Richard J. , Nadarajah Saraleesan 
Published: July 2005

AbstractA Graeco‐Latin square is a pair of orthogonal Latin squares (each letter combination occurs exactly once). Such squares exist for all positive integers except for 1, 2, and 6. They, and the extension to sets of mutually orthogonal Latin squares, are useful for constructing experimental designs in several situations in which there are four or more blocking or treatment structures. However, they are limited because essentially the numbers of treatments, blocks, or factor levels must all be the same.
Lattice Designs
Authors: Martin Richard J. 
Published: July 2005

Abstract

Lattice designs are two types of experimental designs in which the number of treatments is the square of
k
. A square lattice design is a resolvable (0, 1) incomplete block design, with block size
k
. For
r
replicates, with
r
more than three, existence requires that there is a set of
r
mutually orthogonal Latin squares. If r = k + 1, the design is balanced. Extensions to cubic lattice designs and some rectangular lattice designs are possible. A lattice square design is a nested row–column design with
k
rows and
k
columns, with each block being a complete replicate.
Randomized Complete Block Designs
Authors: Murray L. 
Published: July 2005

AbstractThis article discusses the general need for blocking when experimental units are not homogeneous. Complete, generalized, and incomplete block designs are discussed, together with issues of randomization, missing cells, and the usual assumption of “no interaction”. Two specific cases of the Randomized Complete Block Design are considered: the fixed case (treatments and blocks are both fixed) and the mixed case (treatments are fixed, blocks are random). For each case, the scalar linear models are defined in full‐rank and non‐full‐rank versions, together with necessary assumptions. Point estimates and standard errors are given for main‐effect means and cell means, and for variance components. The analysis of variance table is given, with expected mean squares, for testing hypotheses.
Resampling Procedures for Sample Surveys
Authors: Frankel Martin R. 
Published: July 2005

AbstractThis entry discusses the two basic resampling techniques that have found widespread application in survey sampling: Jackknife Repeated Replication (JRR) and Balanced Repeated Replication (BRR). To date, resampling methods based on the bootstrap have found only limited application in samples, which involve complex (stratified and clustered) sample design.
Sample Size Adequacy in Surveys
Authors: Penne Michael A. , Levy Paul S. 
Published: July 2005

Abstract
This entry presents concepts involved in determination of sample size adequacy in sample surveys. There is a broad discussion of issues such as stratification, sampling without replacement, finite population corrections, sampling of clusters, sampling in more than one stage, and unequal weighting of observations that have a profound effect on the distribution of estimates and on the sample sizes necessary to construct reliable estimates. We include formulas for determining sample size adequacy for some of the most widely used sample designs and some practical guidelines for statistical consulting on this issue. A set of references to texts and specific articles is included.