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We derive the exact distributions of order statistics from a finite number of, in general, dependent random variables following a joint ln,p-symmetric distribution. To this end,we first review the special cases of order statistics fromspherical aswell as from p-generalized Gaussian sample distributions from the literature. To study the case of general ln,p-dependence, we use both single-out and cone decompositions of the events in the sample space that correspond to the cumulative distribution function of the kth order statistic if they are measured by the ln,p-symmetric probability measure.We show that in each case distributions of the order statistics from ln,p-symmetric sample distribution can be represented as mixtures of skewed ln−ν,p-symmetric distributions, ν ∈ {1, . . . , n − 1}.

When the assumption of multivariate normality is violated and the sample sizes are relatively small, existing test statistics such as the likelihood ratio statistic and Satorra–Bentler’s rescaled and adjusted statistics often fail to provide reliable assessment of overall model fit. This article proposes four new corrected statistics, aiming for better model evaluation with nonnormally distributed data at small sample sizes. A Monte Carlo study is conducted to compare the performances of the four corrected statistics against those of existing statistics regarding Type I error rate. Results show that the performances of the four new statistics are relatively stable compared with those of existing statistics. In particular, Type I error rates of a new statistic are close to the nominal level across all sample sizes under a condition of asymptotic robustness. Other new statistics also exhibit improved Type I error control, especially with nonnormally distributed data at small sample sizes.

Despite the advantages of the use of the normal distribution in Statistical Quality Control, the normality assumption is too restrictive for modelling real data sets, which usually exhibit asymmetry or tails heavier than the normal tails. But even in potential normal situations, there is often a small to moderate percentage of contamination in the data. In this paper, we analyze the efficiency and robustness of the total median statistic in comparison with the sample mean and the sample median to estimate the mean value of symmetric contaminated normal distributions, close to the normal, but with heavier-than-normal tails. We also compare the performance of the total median and the sample mean charts to monitor the mean value of such processes. The simulation results lead us to suggest the use of the total median statistic due to its efficiency and degree of robustness.

Higher mathematics and Probability and statistics are the core course in university,There is a close relation between the two. By comparison and analysis of the scores of the two courses, this paper attempts to show that higher math scores on the impact of Probability and statistics, Higher mathematics X is in linear relationship with Probability and statistics Y,The coefficients for the two is r=0.541. the correlativity between higher mathematics and Probability and statistics has been investigated by regression analysis,the regression equation between the two is obtained,therefore the quantitative analysis can be carried out.Conclusion: There is a good linear correlation between higher mathematics scores and Probability and statistics scores.Description the quality of higher mathematics achievements on the impact of Probability and statistics can not be ignored, for the construction of Higher mathematics courses provide a strong basis.

Test statistics based on likelihood function, order statistics, population quantiles are suggested to discriminate between log-logistic distribution and Rayleigh distribution. Because of non tractability of their exact sampling distributions, the percentiles of the proposed test statistics are tabulated with the help of simulated sampling distributions of the test statistics. The power of the test statistics are also tabulated and a comparative study w.r.t. the powers for a given sample and the level of signicance are worked out.

The problem of predicting future generalized-order statistics, by assuming the future sample size is a random variable, is discussed. A general expression for the coverage probability of the prediction intervals is derived. Since k-records and progressively type-II censored-order statistics are contained in the model of generalized-order statistics, the corresponding results for them can be deduced as special cases. When the future sample size has degenerate, binomial, Poisson and geometric distributions, numerical computations are given. The procedure for finding an optimal prediction interval is presented for each case. Finally, we apply our results to a real data set in life testing given in Lee and Wang [Statistical methods for survival data analysis. Hoboken, NJ: John Wiley and Sons; 2003, p. 58, Table 3.4] for illustrative the proposed procedure in this paper.

This article considered several test statistics for testing the population signal-to-noise ratio based on parametric, nonparametric, and modified methods. To compare the performance of the proposed...

In this paper, we present some characterizations of distributions based on the regression of generalized order statistics. In the case of adjacent generalized order statistics, the conditional expectation of one generalized order statistic given the other one completely characterizes distributions depending on the type of regression function. In the case of non-adjacent generalized order statistics, the characterization of distributions using conditional expectations becomes more complicated. The results presented in the paper unify and extend some of the existing results involving order statistics and record values.

In this work, the exact finite-sample distribution of the median absolute deviation about the median (MAD) of continuous random variables is derived. The main results are used for providing exact confidence intervals for the scale parameter of a location–scale model, which are robust to the presence of outliers.

This chapter discusses probability and statistics and includes a discussion on discrete random variables; probability distributions; sample spaces; sampling; the abuse of statistics; variance and standard deviation; hypothesis testing. Statistics is an empirical science that is concerned with the collection, organization, analysis, interpretation and presentation of data. They allow the behaviour of a population to be studied and inferences to be made about the population.

This chapter reviews the different statistical methods used to describe a sample and make inference for a larger population. Despite its apparent simplicity, one should not underestimate the importance of the task, especially in the context of public policies. Providing basic descriptive statistics to point out the issues that must be addressed is a preliminary and necessary step in program evaluation (Sect. 3.1). One-way and two-way tables summarize the data in a very efficient manner (Sect. 3.2). Bar graphs, pie charts, histograms, line graphs and radar charts can also be generated at the evaluator’s convenience (Sect. 3.3). To go further, numerical analysis rests on measures of central tendency (mode, median, and mean), and of dispersion (interquartile range, variance, standard deviation, coefficient of variation) (Sect. 3.4). The asymmetry of a distribution and its “tailedness” can be approximated by the skewness and kurtosis coefficients (Sect. 3.5). Last, in most cases, the description of a database is done in the context of a sample survey. Any generalization to the population of interest thus involves the calculation of confidence intervals (Sect. 3.6). Several examples illustrate the methods.

Our new method uses order statistics to judge the fit of a distribution to data. A test-statistic based on quantiles of order-statistics compares favorably with the Kolmogorov-Smirnov (K-S) and Anderson-Darling (A-D) test statistics. The performance of this new goodness-of-fit test statistic is examined with simulation experiments. For certain hypothesis tests, the test statistic is more powerful than the K-S and A-D test statistics. The new test statistic is calculated using a computer algebra system because of the need to compute exact distributions of order statistics.

Distributional properties of two non-adjacent dual generalized order statistics have been used to characterize distributions. Further, one sided contraction and dilation for the dual generalized order statistics are discussed and then the results are deduced for generalized order statistics, order statistics, lower record statistics, upper record statistics and adjacent dual generalized order statistics.

There are several well-known mappings which transform the first r common order statistics in a sample of size n from a standard uniform distribution to a full vector of dimension r of order statistics in a sample of size r from a uniform distribution. Continuing the results reported in a previous paper by the authors, it is shown that transformations of these types do not lead to order statistics from an i.i.d. sample of random variables, in general, when being applied to order statistics from non-uniform distributions. By accepting the loss of one dimension, a structure-preserving transformation exists for power function distributions.

Abtsrcat In this work, we obtain the best linear unbiased estimator based on order statistics for the scale parameter of skew-normal distribution for some known values of the shape parameter. We also propose a technique for estimating the scale parameter of skew-normal distribution by U-statistics based on best linear functions of order statistics as kernels. The efficiency comparison of the proposed U-statistics with the maximum likelihood esti- mator is also carried out.

A detailed study of the asymptotic efficiency of the spatial median under elliptically symmetric distributions is presented. In particular, it is shown that its asymptotic relative efficiency with respect to an affinely equivariant version of the spatial median does not exceed one, and consequently the spatial median is asymptotically inadmissible at elliptical distributions. The degree of the inefficiency of the spatial median depends on the true underlying scatter matrix parameter, with its inefficiency being quite severe, but not always severe, when the scatter matrix is far from being proportional to the identity. Finally, a simulation study is presented for the finite sample relative efficiency of the spatial median. This simulation study reveals some curious oscillating patterns in the efficiencies as the sample size increases.

Statisticians use words deliberately and specifically, but not necessarily in the way they are used colloquially. For example, in general parlance ‘statistics’ can mean numerical information, usually data. In contrast, one large statistics textbook defines the term ‘statistic’ to denote ‘a characteristic of a sample, such as the average score’, and the word ‘parameter’ to denote ‘a characteristic of a population’. However, for statisticians, statistics means more than just numerical information or a characteristic of a sample. Statistics is also a discipline. It is relevant to all areas of scientific enquiry, and spans study design, data collection, developing methods of analysis and analysing data, interpreting results, and making predictions. Other words that statisticians use carefully, and very particularly, form the cornerstones of statistical reasoning, words such as probability, significance, likelihood. Why does this matter? Well, it may well explain why non-statisticians struggle with the ideas and concepts used by statisticians. The reader is confused by the terminology, not the least because the statistician uses familiar words in different ways, a jargon similar to other professional vocabularies.

Summary The classical spatial median is not affine-equivariant, which often turns out to be an unfavourable property. In this paper, the asymptotic properties of an affine-equivariant modification of the spatial median are investigated. It is shown that under some weak regularity conditions, the modified spatial median computed by means of the sample norming matrix is asymptotically equivalent to the one computed by means of the population norming matrix, which yields its asymptotic normality. A consistent estimate of the asymptotic covariance matrix of the modified spatial median is also presented. These results are implemented in a scheme, where the sample norm is determined by means of the sample Dumbgen scatter matrix. The results are utilized in the construction of affine-invariant test statistics for testing the multi-sample hypothesis of equality of location parameters. The performance of the proposed tests is demonstrated through a simulation study.

Based on the notion of the associated copula for order statistics and some L1-distances, we derive a symmetric nonparametric measure of asymptotic dependence between the order statistics under power normalization. This nonparametric criterion of dependence is extended to the generalized, as well as the dual generalized, order statistics.

In this paper, we establish some Pitman closeness results concerning the sample median from a symmetric continuous distribution. We show that when an odd sample size is increased by one, the sample median becomes Pitman-closer to the population median, while when an even sample size is increased by one, the sample median need not be Pitman-closer. We establish the former through probabilistic derivations while the latter is through a counterexample. We also discuss the situation when the sample is increased by two observations.

The hazard rate function of the family of J-shaped distribution is bathtub shaped which makes it an appropriate model for failure data. In this paper we study order statistics from this family of distributions. Expressions for moments and product moments of order statistics are given. Some recurrence relations for moments of order statistics are derived. Bounds based on Jensen’s inequality are constructed.

In multivariate location problems, the sample mean is most widely used, having various advantages. It is, however, very sensitive to outlying observations and inefficient for data from heavy tailed distributions. In this situation, the spatial median is more robust than the sample mean and could be a reasonable alternative. We reviewed several spatial median based testing methods for multivariate location and compared their significance level and power through Monte Carlo simulations. The results show that bootstrap method is efficient for the estimation of the covariance matrix of the sample spatial median. We also proposed bootstrap simultaneous confidence intervals based on the spatial median for multiple comparisons in the multi-sample case.

In this paper, Pitman closeness of order statistics to the median of a distribution is discussed. In particular, it is shown that the sample median is the Pitman-closest order statistic to the population median in some general situations, and expressions for probabilities of closeness are also derived.

It is well known that for distributions on the real line, the variability of sample median as an estimator of the population median remains almost the same when the sample of size 2k increases by 1 (2k + 1). Quasi medians—the averages of two symmetrically placed order statistics—were introduced by Hodges and Lehmann (1967) as alternative estimators of the location parameter, reducing the variability of the estimator in the case of samples with odd sizes. This article explores the analogous idea for the data on the circle. We propose the CQM—a circular mean of the two most central order statistics on a circle—as the estimator of a population median. The proposed estimator improves the well known Mardia median (Mardia, 1972) when the sample size is odd, and coincides with it when the sample size is even.

Distributions of exceedance statistics based on minimal spacing of generalized order statistics are obtained in a random threshold model. As the special cases of the generalized order statistics the ordinary order statistics, Progressively Type II right censored order statistics and record values are considered. The results obtained in the paper imply previous results on exceedance statistics for the variety of models of ordered random variables.

Maria is asked to compare the viscosities from two production lines and uses descriptive statistics and graphs to make a determination.

Distributions of exceedance statistics based on generalized order statistics are obtained for a random threshold model. The ordinary order statistics, progressively Type-II right censored order statistics and record values are considered as special cases. The results obtained in the article imply many results on exceedance statistics for the variety of models of ordered random variables.

SYNOPTIC ABSTRACTThe upper percentiles for the distributions of the test statistics in profile analysis are discussed. These test statistics are Hotelling's T2-type statistic in normal populations and their upper percentiles are expressed by that of the F-distributions. In elliptical populations, however, it is difficult to express them exactly. Then asymptotic expansions for the distributions of the test statistics are derived by the perturbation method and we obtain approximate upper percentiles for the distributions of the test statistics. The effect of nonnormality on approximations is discussed and the approximate accuracy is evaluated via a Monte Carlo simulation study.

Summary We detail a general method for measuring agreement between two statistics. An application is two ratios of directly standardized rates which differ only by the choice of the standard. If the statistics have a high value for the coefficient of agreement then the expected squared difference between the statistics is small relative to the variance of the average of the two statistics, and inferences vary little by changing statistics. The estimation of a coefficient of agreement between two statistics is not straightforward because there is only one pair of observed values, each statistic calculated from the data. We introduce estimators of the coefficient of agreement for two statistics and discuss their use, especially as applied to functions of standardized rates.

We characterize all symmetric location models for which a linear combination of the median and the sample mean is an asymptotically efficient estimator of the location parameter. The resulting model can be understood as a symmetrized or double truncated normal distribution. A simple algorithm to estimate the parameters is given and an application is presented.

There is now quite an extensive literature based on analysis of multivariate abundances, which have often been collected according to a MANOVA design. Many test statistics have been proposed specifically for this case, yet remarkably the power of these methods has not previously been compared. In this paper, the power of distance-based statistics (e.g., Mantel, analysis of similarities) is compared to variable-based statistics (e.g., redundancy analysis, the sum of ANOVA F statistics), when using permutation tests to assess significance of all statistics. Different choice of transformation, standardization, and distance measure were considered. For 19 data sets taken from the literature, P values for the different statistics were compared. Power simulations were then conducted, where data were generated to mimic the properties of each of the 19 data sets. For transformed data, using different distance measures (Euclidean, Manhattan, Bray-Curtis) and different distance-based statistics had little effect on power. Overall, statistics based on multivariate analysis of variance (MANOVA) were at least as powerful as others, although particular data sets gave different results. The distance-based statistics most commonly used in the literature do not standardize abundances, so these were more powerful when effects are present in taxa that are more variable (on the transformed scale), and less powerful otherwise. There are several reasons to prefer a statistic based on MANOVA to others (e.g., interpretability, generalization to more complex designs), and so we generally recommend that the MANOVA-based statistics used here be preferred to distance-based statistics.

SUMMARY In industry, most of the process observations are assumed to come from a normal population, but usually we merely want to control the process mean value. It is thus sensible to find control statistics, which are ‘robust’ to monitor the process mean, giving the expected rate of false alarms whenever that mean is close to the target value, although not under a normal regime. Simulation studies for a few symmetric and asymmetric distributions allow us to suggest the total median as a robust median estimator. We shall here analyse such a robustness, as well as the robustness of the total median chart comparatively to the sample mean chart, whenever we want to control the mean value of a symmetric underlying parent. Some indication is also provided on the comparative out-of-control behaviour of the two charts. Copyright # 2004 John Wiley & Sons, Ltd.

Data in social and behavioral sciences are often hierarchically organized though seldom normal. They typically contain heterogeneous marginal skewnesses and kurtoses. With such data, the normal theory based likelihood ratio statistic is not reliable when evaluating a multilevel structural equation model. Statistics that are not sensitive to sampling distributions are desirable. Six statistics for evaluating a structural equation model are extended from the conventional context to the multilevel context. These statistics are asymptotically distribution free, that is, their distributions do not depend on the sampling distribution when sample size at the highest level is large enough. The performance of these statistics in practical data analysis is evaluated with a Monte Carlo study simulating conditions encountered with real data. Results indicate that each of the statistics is very insensitive to the underlying sampling distributions even with finite sample sizes. However, the six statistics perform quite differently at smaller sample sizes; some over-reject the correct model and some under-reject the correct model. Comparing the six statistics with two existing ones in the multilevel context, two of the six new statistics are recommended for model evaluation in practice.

Healthcare quality improvement professionals need to understand and use inferential statistics to interpret sample data from their organizations. In quality improvement and healthcare research studies all the data from a population often are not available, so investigators take samples and make inferences about the population by using inferential statistics. This three-part series will give readers an understanding of the concepts of inferential statistics as well as the specific tools for calculating confidence intervals for samples of data. This article, Part 2, describes probability, populations, and samples. The uses of descriptive and inferential statistics are outlined. The article also discusses the properties and probability of normal distributions, including the standard normal distribution.

Healthcare quality improvement professionals need to understand and use inferential statistics to interpret sample data from their organizations. In quality improvement and healthcare research studies all the data from a population often are not available, so investigators take samples and make inferences about the population by using inferential statistics. This three-part series will give readers an understanding of the concepts of inferential statistics as well as the specific tools for calculating confidence intervals for samples of data. This article, Part 1, presents basic information about data including a classification system that describes the four major types of variables: continuous quantitative variable, discrete quantitative variable, ordinal categorical variable (including the binomial variable), and nominal categorical variable. A histogram is a graph that displays the frequency distribution for a continuous variable. The article also demonstrates how to calculate the mean, median, standard deviation, and variance for a continuous variable.