General scores statistics on ranks in the analysis of unbalanced designs

The authors consider the situation of incomplete rankings in which n judges independently rank k_i ∈ {2, …, t} objects. They wish to test the null hypothesis that each judge picks the ranking at random from the space of k_i! permutations of the integers 1, …, k_i. The statistic considered is a generalization of the Friedman test in which the ranks assigned by each judge are replaced by real‐valued functions a(j, k_i), 1 ≤ j ≤ k_i ≤ t of the ranks. The authors define a measure of pairwise similarity between complete rankings based on such functions, and use averages of such similarities to construct measures of the level of concordance of the judges' rankings. In the complete ranking case, the resulting statistics coincide with those defined by Hájek & ?idák (1967, p. 118), and Sen (1968). These measures of similarity are extended to the situation of incomplete rankings. A statistic is derived in this more general situation and its properties are investigated.     

Integral Representation of a Distribution Associated with a Pure Birth Process

ABSTRACT

This article deals with a distribution associated with a pure birth process starting with no individuals, with birth rates λ _n  = λ for n = 0, 2,…, m ? 1 and λ _n  = μ for n ≥ m. The probability mass function is expressed in terms of an integral that is very convenient for computing probabilities, moments, generating functions, and others. Using this representation, the kth factorial moments of the distribution is obtained. Some other forms of this distribution are also given.     

Small sample set estimation of a baseline ranking

Ranking objects by a panel of judges is commonly used in situations where objective attributes cannot easily be measured or interpreted. Under the assumption that the judge independently arrive at their rankings by making pair wise comparisons among the objects in an attempt to reproduce a common baseline ranking w₀, we develop and explore confidence regions and Bayesian highest posterior density credible regions for w₀ with emphasis on very small sample sizes.     

On the Distribution of Matrix Quadratic Forms

A characterization of the distribution of the multivariate quadratic form given by X A X′, where X is a p × n normally distributed matrix and A is an n × n symmetric real matrix, is presented. We show that the distribution of the quadratic form is the same as the distribution of a weighted sum of non central Wishart distributed matrices. This is applied to derive the distribution of the sample covariance between the rows of X when the expectation is the same for every column and is estimated with the regular mean.     

Characterizing the BMAP/MAP/1 Departure Process via the ETAQA Truncation

Abstract

We propose a family of finite approximations for the departure process of a BMAP/MAP/1 queue. The departure process approximations are derived via an exact aggregate solution technique (called ETAQA) applied to M/G/1-type Markov processes. The proposed approximations are indexed by a parameter n(n > 1), which determines the size of the output model as n + 1 block levels of the M/G/1-type process. This output approximation preserves exactly the marginal distribution of the true departure process and the lag correlations of the interdeparture times up to lag n ? 2. Experimental results support the applicability of the proposed approximation in traffic-based decomposition of queueing networks.     

New Families of Copulas Based on Periodic Functions

Abstract

Although there exists a large variety of copula functions, only a few are practically manageable, and often the choice in dependence modeling falls on the Gaussian copula. Furthermore most copulas are exchangeable, thus implying symmetric dependence. We introduce a way to construct copulas based on periodic functions. We study the two-dimensional case based on one dependence parameter and then provide a way to extend the construction to the n-dimensional framework. We can thus construct families of copulas in dimension n and parameterized by n ? 1 parameters, implying possibly asymmetric relations. Such “periodic” copulas can be simulated easily.     

The Distribution of Sums of Certain I.I.D. Pareto Variates

ABSTRACT

Though the Pareto distribution is important to actuaries and economists, an exact expression for the distribution of the sum of n i.i.d. Pareto variates has been difficult to obtain in general. This article considers Pareto random variables with common probability density function (pdf) f(x) = (α/β) (1 + x/β)^α+1 for x > 0, where α = 1,2,… and β > 0 is a scale parameter. To date, explicit expressions are known only for a few special cases: (i) α = 1 and n = 1,2,3; (ii) 0 < α < 1 and n = 1,2,…; and (iii) 1 < α < 2 and n = 1,2,…. New expressions are provided for the more general case where β > 0, and α and n are positive integers. Laplace transforms and generalized exponential integrals are used to derive these expressions, which involve integrals of real valued functions on the positive real line. An important attribute of these expressions is that the integrands involved are non oscillating.     

On the Exponential Decay Rate of the Tail of a Discrete Probability Distribution

Abstract

We give a sufficient condition for the exponential decay of the tail of a discrete probability distribution π = (π _n ) _n≥0 in the sense that lim _n→∞(1/n) log∑ _i>n π _i  = ?θ with 0 < θ < ∞. We focus on analytic properties of the probability generating function of a discrete probability distribution, especially, the radius of convergence and the number of poles on the circle of convergence. Furthermore, we give an example of an M/G/1 type Markov chain such that the tail of its stationary distribution does not decay exponentially.     

Matching Three Moments with Minimal Acyclic Phase Type Distributions

Abstract

A number of approximate analysis techniques are based on matching moments of continuous time phase type (PH) distributions. This paper presents an explicit method to compose minimal order continuous time acyclic phase type (APH) distributions with a given first three moments. To this end we also evaluate the bounds for the first three moments of order n APH distributions (APH(n)). The investigations of these properties are based on a basic transformation, which extends the APH(n ? 1) class with an additional phase in order to describe the APH(n) class.     

On a Confidence Interval for the Mean of an Asymmetric Distribution

A clarification is given of the main result (1.1) in Communications in Statistics: Theory and Methods 34:753–766. The term {1 + 6a(r ? a)}^1/3 is to be understood as sgn(1 + 6a(r ? a)) | 1 + 6a(r ? a)|^1/3. The result is expressed in a more user-friendly form. An issue is raised regarding the common usage of the expression x ^1/n when n is even.     

A new weighted rank coefficient of concordance

There are many situations where n objects are ranked by b>2 independent sources or observers and in which the interest is focused on agreement on the top rankings. Kendall's coefficient of concordance [10 M. Kendall and B. Smith, The problem of m rankings, Ann. Math. Stat. 10 (1939), pp. 275–287. doi: 10.1214/aoms/1177732186[Crossref] , [Google Scholar]] assigns equal weights to all rankings. In this paper, a new coefficient of concordance is introduced which is more sensitive to agreement on the top rankings. The limiting distribution of the new concordance coefficient under the null hypothesis of no association among the rankings is presented, and a summary of the exact and approximate quantiles for this coefficient is provided. A simulation study is carried out to compare the performance of Kendall's, the top-down and the new concordance coefficients in detecting the agreement on the top rankings. Finally, examples are given for illustration purposes, including a real data set from financial market indices.     

Some Properties of Order Statistics When the Sample Size Is Random

ABSTRACT

The study of r-out-of-n systems is of utmost importance in reliability theory. In this note, we study closure of different partial orders under the formation of r-out-of-N and (N ? s)-out-of-N systems when the number of components N, forming the system, is a random variable having support {k, k + 1,…}, where k is a fixed positive integer, r ∈ {1,…, k} and s ∈ {0, 1,…, k ? 1}. This generalizes quite a few results already known in the literature. We also study the closure of different partial orders when two systems are formed out of different random number of components.     

KENDALL'S TAU AND CONTINGENCY TABLES

Kendall's tau is a coefficient of concordance between two rankings of n objects. Its definition and large sample normal approximation are easily extended to the case where one of the rankings contains ties. In this paper, definition and normal approximation are extended further to the case where both rankings contain ties. The results are applied to give a fully distribution-free test for two-way contingency tables with ordered categories.     

Tables of the exact quantiles of the top-down correlation coefficient for n - 3(1)14

Iman and Connver (1985, 1987) have suggested the top-down correlation coefficient as a measure of association when n objects are ranked by two or more independent sources and interest centers primarily on agreement in the top rankings, with disagreements on items at the bottom of the rankings being of little or no importance. The top-down correlation coefficient results from computing the ordinary Pearson correlation coefficient on Savage scores. Quantiles of the exact distribution of the top-down correlation coefficient based on the assumption of independent rankings are provided for n = 3(1)14.     

Optimal Nonparametric Quantile Estimators. Towards a General Theory. A Survey

“Nonparametric” in the title is used to say that observations X ₁,…,X _n come from an unknown distribution F ∈ ? with ? being the class of all continuous and strictly increasing distribution functions. The problem is to estimate the quantile of a given order q ∈ (0,1) of the distribution F. The class ? of distributions is very large; it is so large that even X _nq:n , where nq is an integer, may be very poor estimator of the qth quantile. To assess the performance of estimators no properties based on moments may be used: expected values of estimators should be replaced by their medians, their variances—by some characteristics of concentration of distributions around the median. If an estimator is median-biased for one of distributions, the bias of the estimator may be infinitely large for other distributions. In the note optimal estimators with respect to various criteria of optimality are presented. The pivotal function F(T) of the estimator T is introduced which enables us to apply the classical statistical approach.     

On Thresholds of Moving Averages with Given On-Target Significant Levels

Let X ₁, X ₂,… be a sequence of independent and identically distributed random variables, and let Y _n , n = K, K + 1, K + 2,… be the corresponding backward moving average of order K. At epoch n ≥ K, the process Y _n will be off target by the input X _n if it exceeds a threshold. By introducing a two-state Markov chain, we define a level of significance (1 ? a)% to be the percentage of times that the moving average process stays on target. We establish a technique to evaluate, or estimate, a threshold, to guarantee that {Y _n } will stay (1 ? a)% of times on target, for a given (1 ? a)%. It is proved that if the distribution of the inputs is exponential or normal, then the threshold will be a linear function in the mean of the distribution of inputs μ _X . The slope and intercept of the line, in each case, are specified. It is also observed that for the gamma inputs, the threshold is merely linear in the reciprocal of the scale parameter. These linear relationships can be easily applied to estimate the desired thresholds by samples from the inputs.     

A Test for Slope Heterogeneity in Fixed Effects Models

Typical panel data models make use of the assumption that the regression parameters are the same for each individual cross-sectional unit. We propose tests for slope heterogeneity in panel data models. Our tests are based on the conditional Gaussian likelihood function in order to avoid the incidental parameters problem induced by the inclusion of individual fixed effects for each cross-sectional unit. We derive the Conditional Lagrange Multiplier test that is valid in cases where N → ∞ and T is fixed. The test applies to both balanced and unbalanced panels. We expand the test to account for general heteroskedasticity where each cross-sectional unit has its own form of heteroskedasticity. The modification is possible if T is large enough to estimate regression coefficients for each cross-sectional unit by using the MINQUE unbiased estimator for regression variances under heteroskedasticity. All versions of the test have a standard Normal distribution under general assumptions on the error distribution as N → ∞. A Monte Carlo experiment shows that the test has very good size properties under all specifications considered, including heteroskedastic errors. In addition, power of our test is very good relative to existing tests, particularly when T is not large.