Assessment of Interjudge Reliability in the Open-Ended Questions Coding Process

In the process of coding open-ended questions, the evaluation of interjudge reliability is a critical issue. In this paper, using real data, the behavior of three coefficients of reliability among coders, Cohen’s K, Krippendorff’s α and Perreault and Leigh’s I _r are patterned, in terms of the number of judges involved and the categories of answer defined. The outcome underlines the importance of both variables in the valuations of interjudge reliability, as well as the higher adequacy of Perreault and Leigh’s I _r and Krippendorff’s α for marketing and opinion research.     

Generalized entropies in coding theory

Equivalence of the generalized entropyH ^β (P, Φ _t ) defined in this paper andKapur’s entropy of orderα and typeβ, ie.H _α ^β (P), is established. The results given recently byCampbell follow as special cases. International Conference on System Sciences, Honolulu, January 1968.     

One optional observation inflates α by $${100/\sqrt{n}}$$ per cent

For one-sample level α tests ψ _m based on independent observations X ₁, . . . , X _m , we prove an asymptotic formula for the actual level of the test rejecting if at least one of the tests ψ _n , . . . , ψ _n+k would reject. For k = 1 and usual tests at usual levels α, the result is approximately summarized by the title of this paper. Our method of proof, relying on some second order asymptotic statistics as developed by Pfanzagl and Wefelmeyer, might also be useful for proper sequential analysis. A simple and elementary alternative proof is given for k = 1 in the special case of the Gauss test.     

Verbesserung des nicht-randomisierten exakten Testes von R.A. Fisher

Summary The so-called Exact Test of R. A. Fisher for comparing two probabilitiesp ₁ andp ₂ in a Fourfold-Table with small cell frequencies is known as a UMPU-Test. But in practice the test is used in a nonrandomized, often tabulated version. Given a certain level of significanceα it is shown: the critical region of this nonrandomized test, referred to as “Fisher 1”, can be enlarged considerably. For instance for all sample-size-sums up to 20 andα=0.01 the total number of points in the critical regions of “Fisher 1” is 552 whereas the analogous number of the new version “Fisher 2” is 788. The size of tables for “Fisher 2” can be reduced considerably because the main parts of the critical regions can be described by the aid of some Chi-square-test versions. In particular Yates’ continuity-correction turns out to be always conservative in the above mentioned region relative to “Fisher 2” whereas this is not strictly true relative to “Fisher 1”.      

Testing outliers from an exponential population

Summary Likes obtains the distributions ofDixon’s statistics for an exponential population and tabulates upper 100α% points (α=0.1, 0.05, 0.01) of some of these distributions. The distribution of these statistics can be expressed in terms of finite series of beta functions and hence the probabilities of the rejection of suspected observed outliers may be easily calculated on a desk calculator. Thus we may avoid the difficult task of tabulating 100α% values of these statistics.     

Distribution of Dixon’s statistics in the case of an exponential population

Summary The distributions of Dixon’s statistics for the case of the sample from an exponential population are found. For some of these statistics the upper 100 α percentage points (α=0,1; 0,05 and 0,01) are tabulated.

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Zusammenfassung Die Verteilungen der Variablen von Dixon werden für den Fall der Stichproben aus einer exponentialen Grundgesamtheit gefunden. Für einige dieser Variablen werden die oberen 100 α Prozentpunkte (α=0,1; 0,05 und 0,01) tabelliert.

     

Hypothesis testing and parameter estimation based onM-statistics ink samples with unequal variances

Summary Ink samples with unequal variances,M-tests for homogeneity ofk location parameters are proposed. The asymptoticχ ²-distributions of the test statistics and the robustness of the tests are investigated. NextM-estimators (ME’s) of parameters are discussed. Furthermore positive-part shrinkage versions (PSME’s) of theM-estimators for the location parameters are considered along with modified James-Stein estimation rule. In asymptotic distributional risks based on a special feasible loss, it is shown that the PSME’s dominate the ME’s, and preliminary test and shrinkageM-versions fork≧4.     

Prediction of <Emphasis Type="Italic">k</Emphasis>-records from a general class of distributions under balanced type loss functions

We study the problem of predicting future k-records based on k-record data for a large class of distributions, which includes several well-known distributions such as: Exponential, Weibull (one parameter), Pareto, Burr type XII, among others. With both Bayesian and non-Bayesian approaches being investigated here, we pay more attention to Bayesian predictors under balanced type loss functions as introduced by Jafari Jozani et al. (Stat Probab Lett 76:773–780, 2006a). The results are presented under the balanced versions of some well-known loss functions, namely squared error loss, Varian’s linear-exponential loss and absolute error loss or L ₁ loss functions. Some of the previous results in the literatures such as Ahmadi et al. (Commun Stat Theory Methods 34:795–805, 2005), and Raqab et al. (Statistics 41:105–108, 2007) can be achieved as special cases of our results. Partial support from Ordered and Spatial Data Center of Excellence of Ferdowsi University of Mashhad is acknowledged by J. Ahmadi. M. J. Jozani’s research supported partially by a grant of Statistical Research and Training Center. é. Marchand’s research supported by NSERC of Canada. A. Parsian’s research supported by a grant of the Research Council of the University of Tehran.     

On characterizing discrete distributions via conditional expectations of weak record values

Let (W _n ,n ≥ 0) denote the sequence of weak records from a distribution with support S = { α₀,α₁,...,α _N }. In this paper, we consider regression functions of the form ψ _n (x) = E(h(W _n ) |W _n+1 = x), where h(·) is some strictly increasing function. We show that a single function ψ _n (·) determines F uniquely up to F(α₀). Then we derive an inversion formula which enables us to obtain F from knowledge of ψ _n (·), ψ _n-1(·), h(·) and F(α₀).     

Largest excess of boundary crossings for martingales

Summary LetX ₁,X ₂, ... form a sequence of martingale differences and denote byZ(a, α) = sup n (S _n −an ^α)⁺ the largest excess forS _n =X ₁ + ... +X _n crossing the boundaryan ^α. We give a sufficient condition for the finiteness ofEZ(a, α)^β which is formulated in terms of bounds forE(X _i ^{+ p} andE(|X _i |γ|X ₁, ...,X _i-1), whereα, β, γ, p are suitably related. This general result is then applied to the case of independent random variables.     

Measuring process capability index <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">pm</Emphasis></Subscript> with fuzzy data

The process capability index C _pm , which considers the process variance and departure of the process mean from the target value, is important in the manufacturing industry to measure process potential and performance. This paper extends its applications to calculate the process capability index [(C)\tilde]_pm{\tilde {C}_{pm} } of fuzzy numbers. In this paper, the α-cuts of fuzzy observations are first derived based on various values of α. The membership function of fuzzy process capability index [(C)\tilde]_pm{\tilde {C}_{pm} } is then constructed based on the α-cuts of fuzzy observations. An example is presented to demonstrate how the fuzzy process capability index [(C)\tilde]_pm{\tilde {C}_{pm} } is interpreted. When the quality characteristic cannot be precisely determined, the proposed method provides the most possible value and spread of fuzzy process capability index [(C)\tilde]_pm{\tilde {C}_{pm} }. With crisp data, the proposed method reduces to the classical method of process capability index C _pm .     

An approach to fuzzy hypothesis testing

In this paper an approach is presented how to test fuzzily formulated hypotheses with crisp data. The quantitiesα andβ, the probabilities of the errors of type I and of type II, are suitably generalized and the concept of a best test is introduced. Within the framework of a one-parameter exponential distribution family the search for a best test is considerably reduced. Furthermore, it is shown under very weak conditions thatα andβ can simultaneously be diminished by increasing the sample size even in the case of testingH ₀ against the omnibus alternativeH ₁: notH ₀, a result completely different from the case of crisp setsH ₀ andH ₁: notH ₀.     

Strong consistency of the MLE under random censoring

LetX ₁, ...,X _n be an i.i.d. sample from some parametric family {_θ :θ (Θ} of densities. In the random censorship model one observesZ _i =min (X _i ,Y _i ) andδ _i =1_{ x _i ≤Y _i}, whereY _i is a censoring variable being independent ofX _i . In this paper we investigate the strong consistency ofθ _n maximizing the modified likelihood function based on (Z _i ,δ _i , 1≤i≤n. The main result constitutes an extension of Wald’s theorem for complete data to censored data. Work partially supported by the “Deutsche Forschungsgemeinschaft”.     

Ein kombiniertes test & klassifikations-problem

Summary A decision process is considered which consists of two steps: First, a nullhypothesis H₀ is to be tested. If H₀ is rejected, a decision is to be made as to which of the alternative hypotheses H₁, H₂, ... H _k is valid. This second step is called "classification". It is assumed, that in case H₀ is not valid, each of the alternative hypotheses H₁, H₂, ... H _k has the same probability. Starting with this assumption, an optimal decision process is developed which has a specified level of significance α (i.e. by which the nullhypothesis H₀ is accepted with probability α, if it is valid), and for which the probability of a correct classification is a maximum in the case where the nullhypothesis is not valid. This decision process rests on a generalisation of the fundamental lemma of Neyman and Pearson, similar to that used in discriminant analysis.     

Social networks and labour productivity in Europe: an empirical investigation

This paper uses firm-level data recorded in the Amadeus database to investigate the distribution of labour productivity in different European countries. We find that the upper tail of the empirical productivity distributions follows a decaying power-law, whose exponent α is obtained by a semi-parametric estimation technique recently developed by Clementi et al. [Physica A 370(1):49–53, 2006]. The emergence of “fat tails” in productivity distribution has already been detected in Di Matteo et al. [Eur Phys J B 47(3):459–466, 2005] and explained by means of a model of social network. Here we show that this model is tested on a broader sample of countries having different patterns of social network structure. These different social attitudes, measured using a social capital indicator, reflect in the power-law exponent estimates, verifying in this way the existence of linkages among firms’ productivity performance and social network.     

Su Una Estensione Bidimensionale del Teorema di Scomposizione di Peccati

Facendo ricorso all’introduzione di una opportuna condizione di sincronia fra due coppie di tassi (i, j), esterna, e (i*, j*), interna, si dimostra che condizione necessaria e sufficiente per la validità finanzíaria di una estensione hidimensionale del teorema di scomposizione di Peccati è che le due coppie (i, j) e (i*, j*) soddisfino tale proprietà di sincronia.

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A two-dimensional extension of peccati’s decomposition theorem

Summary This paper keeps as starting poing the Peccati’s decomposition of the d.c.f. of an enterpreneural project seen as a present value of single-period results, recently given by the author in an unidimensional framework. The goal of the paper is twofold: the first is to provide a bidimensional extension of the decomposition in order to keep account of mixed projects, that is projects that according to any internal couple (i*, j*) reveal alternance of investment and respectively collection periods. A straightforward extension is given on the basis of a simple analogic argument. But a more careful evaluation of the problem suggests that if we require that the decomposition be meaningful from a financial point of view. things are no more simple either in one or in two dimensions. To solve the problem we introduce a proper synchrony condition: precisely, the pair of couples (i, j) and (i*, j*) are said to satisfy the synchrony condition if and only if the sequences of the signs of the partial balancesM _t (i, j) andM _t (i*, j*) are the same. After that we are able to show that, with reference to a couple (i, j) of external rates, a bidimensional decomposition centered on an internal couple (i*, j*) is meaningful if and only if the pair (i, j) and (i*, j*) satisfy the synchrony condition. Formally, the result comes as a nice byproduct of a rule on the factorization of particular bivariate polynomials recently given by Stucchi. In particular going back to a one-dimensional world, it turns out that for pure investment (Soper) projects the synchrony condition works as a condition of applicability of the Peccati’s decomposition.

Il testo è frutto di una riflessione congiunta degli autori che ne condividono l’impostazione generale e i risultati. In dettaglio, la stesura dell’introduzione e delle conclusioni deriva da una claborazione comune, quella dei paragrafi 4, 5 e 7 è dovuta a Pressacco mentre quella dei rimanenti è dovuta a Stucchi. Il lavoro è stato svolto nell’ambito della ricerca nazionale “Modelli per la Finanza Matematica”.     

Using an approximate kiefer-weiss solution for testing insect population densities

Summary A sequential testing procedure called 2-SPRT for the meanμ of the negative binomial distribution with known exponentk is presented. For any fixedμ ₀(μ ₁<μ ₀<μ ₂), the 2-SPRT provides an asymptotic solution to the modified Kiefer-Weiss problem. Determination ofμ ₀ such that the test provides an approximate solution to the Kiefer-Weiss problem is described. The behavior of the 2-SPRT and the Sequential Probability Ratio Test is investigated using Monte Carlo methods. The error probabilities and the average sample numbers are compared. All the computations were carried out on the Alabama Supercomputer.     

Multi-stage point and interval estimation of the largest mean ofK normal populations and the associated second-order properties

Summary We havek independent normal populations with unknown meansμ ₁, …,μ _k and a common unknown varianceσ ². Both point and interval estimation procedures for the largest mean are proposed by means of sequential and three-stage procedures. For the point estimation problem, we require that the maximal risk be at mostW, a preassigned positive number. For the other problem, we wish to construct a fixed-width confidence interval having the confidence coefficient at least 1-α, a preassigned number between zero and one. Asymptotic second order expansions are provided for various characteristics, such as average sample size, associated risks etc., for the suggested multi-stage estimation procedures.     

One-Sample Bayesian Predictive Interval of Future Ordered Observations for the Pareto Distribution

Nigm et al. (2003, statistics 37: 527–536) proposed Bayesian method to obtain predictive interval of future ordered observation Y _(j) (r < j≤ n ) based on the right type II censored samples Y ₍₁₎ < Y ₍₂₎ < ... < Y _(r) from the Pareto distribution. If some of Y ₍₁₎ < ... < Y _(r-1) are missing or false due to artificial negligence of typist or recorder, then Nigm et al.’s method may not be an appropriate choice. Moreover, the conditional probability density function (p.d.f.) of the ordered observation Y _(j) (r < j ≤ n ) given Y ₍₁₎ <Y ₍₂₎ < ... < Y _(r) is equivalent to the conditional p.d.f. of Y _(j) (r < j ≤ n ) given Y _(r). Therefore, we propose another Bayesian method to obtain predictive interval of future ordered observations based on the only ordered observation Y _(r), then compares the length of the predictive intervals when using the method of Nigm et al. (2003, statistics 37: 527–536) and our proposed method. Numerical examples are provided to illustrate these results.     

Efficiency of the Indian leather firms: some results obtained using the two conventional methods

Indian leather industry has massive potential for generating employment and achieving high export-oriented growth. However, its economic performance has not been assessed much till date. The present paper attempts to fill in this gap by examining technical efficiency (TE) of individual leather producing firms for some years since the mid-1980’s. Analyzing the industry’s firm-level data through the two conventional tools, viz., data envelopment analysis and stochastic frontier analysis, the paper observes a significant positive association between a firm’s size and its TE, but no such clear relation between a firm’s age and TE. It also finds significant variation in TE across firms in different groups of states as well as under different organizational structures and observes some technological heterogeneity across states. Although, non-availability of panel data does not allow one to assess effects of economic reforms on the performance of the Indian leather firms, the average firm-level TE, however, seems to be on an increasing path, except for downswing in the immediate post-reform years.