Involving Order Statistics Research Articles

On classes of consistent tests for the Type I Pareto distribution based on a characterization involving order statistics

We propose new classes of goodness-of-fit tests for the Pareto Type I distribution. These tests are based on a characterization of the Pareto distribution involving order statistics. We derive the limiting null distribution of the tests and also show that the tests are consistent against fixed alternatives. The finite-sample performance of the newly proposed tests are evaluated and compared to some of the existing tests, where it is found that the new tests are competitive in terms of powers. The paper concludes with an application to a real world data set, namely the earnings of the 22 highest paid participants in the inaugural season of LIV golf.

Order statistics and the length of the best-n-of-(2n − 1) competition

ABSTRACTThe best-4-of-7 series is a popular playoff format to decide the champion in most North American professional sports. World Series (best-4-of-7) type competitions give rise to interesting probabilistic and statistical questions. We determine the expected length of this type of series by relating it to a problem involving order statistics. We also calculate the variance of the length and provide a simple formula for series of fair games. The method can be extended to derive higher order moments. This novel approach leads to new results that can be formulated in closed forms in terms of the distribution function of various binomial distributions. The emphasis is on establishing the connection to order statistics and obtaining closed forms. The relation to the negative binomial distribution as well as to the sooner waiting time problem in sequential testing is also discussed. We also consider the case when ties are allowed in the single games.

A Consistent Method of Estimation For The Three-Parameter Gamma Distribution

For the three-parameter gamma distribution, it is known that the method of moments as well as the maximum likelihood method have difficulties such as non-existence in some range of the parameters, convergence problems, and large variability. For this reason, in this article, we propose a method of estimation based on a transformation involving order statistics from the sample. In this method, the estimates always exist uniquely over the entire parameter space, and the estimators also have consistency over the entire parameter space. The bias and mean squared error of the estimators are also examined by means of a Monte Carlo simulation study, and the empirical results show the small-sample superiority in addition to the desirable large sample properties.

On power and sample size computation for multiple testing procedures

Power and sample size determination has been a challenging issue for multiple testing procedures, especially stepwise procedures, mainly because (1) there are several power definitions, (2) power calculation usually requires multivariate integration involving order statistics, and (3) expansion of these power expressions in terms of ordinary statistics, instead of order statistics, is generally a difficult task. Traditionally power and sample size calculations rely on either simulations or some recursive algorithm; neither is straightforward and computationally economic. In this paper we develop explicit formulas for minimal power and r -power of stepwise procedures as well as complete power of single-step procedures for exchangeable and non-exchangeable bivariate and trivariate test statistics. With the explicit power expressions, we were able to directly calculate the desired power, given sample size and correlation. Numerical examples are presented to illustrate the relationship among power, sample size and correlation.

Moment-Based Approximations of Probability Mass Functions with Applications Involving Order Statistics

It is shown in this article that a technique that was previously introduced to approximate the density functions of certain continuous random variables can be successfully applied to discrete distributions. The probability mass function approximants are expressed as the product of an appropriate base density function and a polynomial adjustment. A degree selection criterion that is based on the integrated squared difference between approximants of successive degrees is being proposed. The methodology, which is conceptually simple and easily implementable, is applied to a binomial random variable, the largest order statistic in a binomial sample, a Poisson distribution, and two rank-sum test statistics.

A Fast Algorithm for the Minimum Covariance Determinant Estimator

The minimum covariance determinant (MCD) method of Rousseeuw is a highly robust estimator of multivariate location and scatter. Its objective is to find h observations (out of n) whose covariance matrix has the lowest determinant. Until now, applications of the MCD were hampered by the computation time of existing algorithms, which were limited to a few hundred objects in a few dimensions. We discuss two important applications of larger size, one about a production process at Philips with n = 677 objects and p = 9 variables, and a dataset from astronomy with n = 137,256 objects and p = 27 variables. To deal with such problems we have developed a new algorithm for the MCD, called FAST-MCD. The basic ideas are an inequality involving order statistics and determinants, and techniques which we call “selective iteration” and “nested extensions.” For small datasets, FAST-MCD typically finds the exact MCD, whereas for larger datasets it gives more accurate results than existing algorithms and is faster by orders of magnitude. Moreover, FASTMCD is able to detect an exact fit—that is, a hyperplane containing h or more observations. The new algorithm makes the MCD method available as a routine tool for analyzing multivariate data. We also propose the distance-distance plot (D-D plot), which displays MCD-based robust distances versus Mahalanobis distances, and illustrate it with some examples.

A Fast Algorithm for the Minimum Covariance Determinant Estimator

The minimum covariance determinant (MCD) method of Rousseeuw is a highly robust estimator of multivariate location and scatter. Its objective is to find h observations (out of n) whose covariance matrix has the lowest determinant. Until now, applications of the MCD were hampered by the computation time of existing algorithms, which were limited to a few hundred objects in a few dimensions. We discuss two important applications of larger size, one about a production process at Philips with n = 677 objects and p = 9 variables, and a dataset from astronomy with n = 137,256 objects and p = 27 variables. To deal with such problems we have developed a new algorithm for the MCD, called FAST-MCD. The basic ideas are an inequality involving order statistics and determinants, and techniques which we call “selective iteration” and “nested extensions.” For small datasets, FAST-MCD typically finds the exact MCD, whereas for larger datasets it gives more accurate results than existing algorithms and is faster by orders of magnitude. Moreover, FASTMCD is able to detect an exact fit—that is, a hyperplane containing h or more observations. The new algorithm makes the MCD method available as a routine tool for analyzing multivariate data. We also propose the distance-distance plot (D-D plot), which displays MCD-based robust distances versus Mahalanobis distances, and illustrate it with some examples.

Some New Results on Covariances Involving Order Statistics from Dependent Random Variables

Formulas for covariance matrix between a random vector and its ordered components are derived for different distributions including multivariate normal,t, andF. The present formulas and related results obtained here lead to some known results in the literature as special cases.

Identities and recurrence relations for order statistics corresponding to ncnidentically distributed variable

In this paper, we derive two simple identities and some recurrencerelations involving order statistics from a sample of size n in case there is a possibility of one or more outliers being present.

Probability inequalities via negative dependence for random variables conditioned on order statistics

Distributions are studied which arise by considering independent and identically distributed random variables conditioned on events involving order statistics. It is shown that these distributions are negatively dependent in a very strong sense. Furthermore, bounds are found on the distribution functions. The conditioning events considered occur naturally in reliability theory as the time to system failure for k-out-of-n systems. An application to systems formed with “second-hand” components is given.

Two identities involving order statistics in the presence of an outlier

In this note, we derive two simple identities involving order statistics from a sample of size n in the presence of an outlier. These generalize the results of Joshi (1973). These identities will be quite useful in checking the computation of the single moments of order statistics from an outlier model.

Some general identities involving order statistics

Some identities involving the density functions of order statistics are derived* These generalize the results of Joshi (1973). These identities are highly useful In checking the computation of the moments of order statistics and also in establishing some interesting combinatorial results.

Characterizations via conditional distributions

For independent random variablesXandY,if the conditional distribution ofXgivenX+Ysatisfies certain conditions, then the joint distribution ofXandYis a member of a certain one-parameter exponential family. Extensions fornindependent random variables are given. A characterization for independent random variables involving order statistics is also given.

Characterizations via conditional distributions

For independent random variablesXandY,if the conditional distribution ofXgivenX+Ysatisfies certain conditions, then the joint distribution ofXandYis a member of a certain one-parameter exponential family. Extensions fornindependent random variables are given. A characterization for independent random variables involving order statistics is also given.

A New Nonlinear Pseudorandom Number Generator

During the next few years a new pseudorandom number generator will become available on many computer systems. A concern for the security of computer data has led to the adoption of a Data Encryption Standard (DES) by the National Bureau of Standards. This standard specifies a nonlinear cryptographic algorithm which can be used inter alia as a source of pseudorandom numbers in software applications, such as those involving order statistics, where the usual linear congruential and generalized feedback shift register generators seem to be inadequate. Results of testing the DES as a pseudorandom number generator indicate that the algorithm is more than satisfactory for this purpose.

Three-Parameter Kappa Distribution Maximum Likelihood Estimates and Likelihood Ratio Tests

Abstract Methods are presented for obtaining maximum likelihood estimates and tests of hypotheses involving the three-parameter kappa distribution. The obtained methods are then applied by fitting this distribution to realized sets of precipitation and streamflow data and testing for seeding effect differences between realized seeded and nonseeded sets of precipitation data. The kappa distribution appears to fit precipitation data as well as either the gamma or log-normal distribution. As a consequence, the sensitivity of test procedures based on the kappa distribution compares favorably with that of previously used test procedures. Since both the density and cumulative distribution functions of the kappa distribution are in closed form, the density and cumulative distribution functions associated with each order statistic are also in closed form. In contrast, the gamma and log-normal cumulative distribution functions are not in closed form. As a consequence, computations involving order statistics are fa...

Two Identities Involving Order Statistics

Two Identities Involving Order Statistics

Two identities involving order statistics

Two identities involving order statistics

Estimation of parameters of the gamma distribution using order statistics

The gamma distribution, of which the chi-squared and exponential distributions are particular cases, is used as a model in various statistical applications. Gupta (1960) has reviewed some of its applications to life tests, extreme values, reliability, and maintenance. Gupta & Groll (1961) have described the use of the gamma distribution in acceptance sampling based on life tests. It has also been used as an approximation to the distribution of quadratic forms in certain cases (see, for example, Box (1954) and references therein). A probability plotting procedure for the gamma distribution has been presented by Wilk, Gnanadesikan & Huyett (1962), and applications to life-test data analysis and to the statistical assessment of homogeneity of variance have been described. A method has been proposed by Wilk & Gnanadesikan (1961) for the graphical analysis of multi-response data which is based on the gamma distribution and involves the estimation of the shape parameter from order statistics. The present paper is concerned with the maximum-likelihood estimation of the scale and shape parameters of a gamma distribution, whose origin parameter is known (say equal to zero), based on order statistics. Tables are provided to facilitate obtaining these estimates and their use is illustrated and discussed. The case of unknown origin parameter is also briefly dealt with. Greenwood & Durand (1960) have considered the problem of maximum-likelihood estimation for the gamma distribution based on a complete sample and have presented tables for that case. Chapman (1956) considered the problem of maximum-likelihood estimation of the parameters of a truncated gamma distribution using a complete sample. The tables of the present paper give those of Greenwood & ]Durand (1960) as a special case and a particularly simple interpolation procedure is given for the circumstance of a complete sample. Two estimation situations involving order statistics may be distinguished: namely, when the size of the complete sample is known, and when the sample size is not known. In the sequel, the former case is considered in the context when the information available involves the 'smallest' observations only. The case where the size of the complete sample is unknown is still being investigated. The problem is formally stated in ? 2, and ?? 3 and 4 contain a discussion of the maximum likelihood estimation of the parameters and some attendant issues. ? 5 presents some special results for the case of estimation using the complete sample. Numerical approximations used are given in Appendix A. Tables to facilitate the solution of the maximumlikelihood equations are given in Appendix B. ? 6 presents some examples of the use of these tables. The case when the origin parameter is also unknown is briefly considered in ?7.