De gustibus non est disputandum? A new approach to the estimation of equivalence scales

We argue that when the household composition changes, consumption patterns vary not only because of the cost effect that equivalence scales try to measure, but also because of a “taste” or “style” effect. This effect can be identified and measured, under a few assumptions, with the use of a new methodology, calledDM ² (Decomposition Model of the effects of Demographic Metamorphosis), that can be viewed as a generalisation of Ray's (1983) price-scaling approach to the construction of equivalent scales. An empirical application to data drawn from the Istat 1995 Italian Household Budget Survey suggests that the proposed method improves our understanding of households' consumption patterns and the reliability of the equivalence scales that we derive. We gratefully acknowledge helpful comments from Giorgio Calzolari, Franco Polverini, Ugo Trivellato and an anonymous referee, but retain full responsibility for all errors, and for the processing of Istat (Italian Institute of Statistics) microdata on Household Budgets. Financial support for this research was provided by the italian MURST (Research project on “Equivalence scales” directed by Prof. Guido Ferrari, University of Firenze, Ref. No. 9913105354; and Research project on “Low fertility in Italy: between economic constraints and value changes”, directed by Prof. Massimo Livi Bacci, University of Firenze, Ref. No. MM13107238). Preliminary findings on this research topic have been presented in a few seminars and conferences: cf., e.g., De Santis and Maltagliati (2000 and 2001).     

Explicit formulas for the smoother weights of the Whittaker–Henderson graduation of order 1

The Whittaker–Henderson (WH) graduation is a widely applied smoothing method. This paper contributes to the literature by providing explicit formulas for the smoother weights of the WH graduation of order 1 along with some related results, which leads to a richer understanding of the filter.     

On the Behrens–Fisher problem from the spatial median point of view

We present a non-parametric affine-invariant test for the multivariate Behrens–Fisher problem. The proposed method based on the spatial medians is asymptotic and does not require normality of the data. To improve its finite sample performance, we apply a correction of the type which was already used in a similar test based on trimmed means, however, our simulations show that in the case of heavy-tailed distributions our method performs better. Also in a simulation comparison with a recently published rank-based test our test yields satisfactory results.     

A simplified proof of a theorem on the difference of the moore–penrose inverses of two positive semi–difinte matrices

Simplified proofs are given of a standard result that establishes positive semi–definiteness of the difference of the inverses of two non–singular matrices, and of the extension of this result by Milliken and Akdeniz (1977) to the difference of the Moore–Penrose inverse of two singular matrices.     

What is the effective sample size of a spatial point process?

Point process models are a natural approach for modelling data that arise as point events. In the case of Poisson counts, these may be fitted easily as a weighted Poisson regression. Point processes lack the notion of sample size. This is problematic for model selection, because various classical criteria such as the Bayesian information criterion (BIC) are a function of the sample size, n, and are derived in an asymptotic framework where n tends to infinity. In this paper, we develop an asymptotic result for Poisson point process models in which the observed number of point events, m, plays the role that sample size does in the classical regression context. Following from this result, we derive a version of BIC for point process models, and when fitted via penalised likelihood, conditions for the LASSO penalty that ensure consistency in estimation and the oracle property. We discuss challenges extending these results to the wider class of Gibbs models, of which the Poisson point process model is a special case.     

Moments of order statistics of Topp–Leone distribution

We derive explicit algebraic expressions for both of the single and product moments of order statistics from Topp–Leone distribution. We also give an identity about single moments of order statistics. These expressions will be useful for computational purposes.     

A non parametric CUSUM control chart based on the Mann–Whitney statistic

We consider a novel univariate non parametric cumulative sum (CUSUM) control chart for detecting the small shifts in the mean of a process, where the nominal value of the mean is unknown but some historical data are available. This chart is established based on the Mann–Whitney statistic as well as the change-point model, where any assumption for the underlying distribution of the process is not required. The performance comparisons based on simulations show that the proposed control chart is slightly more effective than some other related non parametric control charts.     

One–sided simulataneous prediction intervals for the order statistics of a future samples from an exponential distribution

Assume that a sample is available from a population having an exponential distribution, and that l Future sample are to be taken from the same population. This paper provides a formula for the same population. This paper provides a formula for computing a one–sided lower simulataneous prediction limit which is to be below the (k_i ? m_i + 1) –st order statistics of a future sample of size k_i for the i = 1,…,2, hased on the sample mean of a past sample. Tables for factors for one–sided lower simultaneous predicition limits are provided. Such limits are of practical importance in determining acceptance criteria and predicting system survival times.     

On the maximization of likelihoods belonging to the exponential family using a Levenberg–Marquardt approach

The Levenberg–Marquardt algorithm is a flexible iterative procedure used to solve non-linear least-squares problems. In this work, we study how a class of possible adaptations of this procedure can be used to solve maximum-likelihood problems when the underlying distributions are in the exponential family. We formally demonstrate a local convergence property and discuss a possible implementation of the penalization involved in this class of algorithms. Applications to real and simulated compositional data show the stability and efficiency of this approach.     

A functional central limit theorem for Markov additive processes with an application to the closed Lu–Kumar network

A functional central limit theorem for a class of time-homogeneous continuous-time Markov processes (X,Y) is proved. The process X is a positive recurrent Markov process on a countable-state space and the process Y has conditionally independent increments given X. The pair (X,Y) is called a Markov additive process. This paper unifies and generalizes several functional central limit theorems for Markov additive processes. An explicit expression for the variance parameter of the limit process is calculated using the local characteristics of the X process. The functional central limit theorem is then used to prove a heavy traffic limit theorem for the closed Lu–Kumar network.     

Estimation of a change point in the variance function based on the χ2-distribution

ABSTRACT

Let us consider that the variance function or its νth derivative in a regression model has a change/discontinuity point at an unknown location. To use the local polynomial fits, the log-variance function which break the positivity is targeted. The location and the jump size of the change point are estimated based on a one-sided kernel-weighted local-likelihood function which is provided by the χ²-distribution. The whole structure of the log-variance function is then estimated using the data sets split by the estimated location. Asymptotic results of the proposed estimators are described. Numerical works demonstrate the performances of the methods with simulated and real examples.     

How far from identifiability? A systematic overview of the statistical matching problem in a non parametric framework

Statistical matching consists in estimating the joint characteristics of two variables observed in two distinct and independent sample surveys, respectively. In a parametric setup, ranges of estimates for non identifiable parameters are the only estimable items, unless restrictive assumptions on the probabilistic relationship between the non jointly observed variables are imposed. These ranges correspond to the uncertainty due to the absence of joint observations on the pair of variables of interest. The aim of this paper is to analyze the uncertainty in statistical matching in a non parametric setting. A measure of uncertainty is introduced, and its properties studied: this measure studies the “intrinsic” association between the pair of variables, which is constant and equal to 1/6 whatever the form of the marginal distribution functions of the two variables when knowledge on the pair of variables is the only one available in the two samples. This measure becomes useful in the context of the reduction of uncertainty due to further knowledge than data themselves, as in the case of structural zeros. In this case the proposed measure detects how the introduction of further knowledge shrinks the intrinsic uncertainty from 1/6 to smaller values, zero being the case of no uncertainty. Sampling properties of the uncertainty measure and of the bounds of the uncertainty intervals are also proved.     

Recursive computation of the single and product moments of order statistics from the complementary exponential–geometric distribution

The complementary exponential–geometric distribution has been proposed recently as a simple and useful reliability model for analysing lifetime data. For this distribution, some recurrence relations are established for the single and product moments of order statistics. These recurrence relations enable the computation of the means, variances and covariances of all order statistics for all sample sizes in a simple and efficient recursive manner. By using these relations, we have tabulated the means, variances and covariances of order statistics from samples of sizes up to 10 for various values of the shape parameter θ. These values are in turn used to determine the best linear unbiased estimator of the scale parameter β based on complete and Type-II right-censored samples.     

A Revisit to the Behrens–Fisher Problem: Comparison of Five Test Methods

We revisit the well-known Behrens–Fisher problem and apply a newly developed ‘Computational Approach Test’ (CAT) to test the equality of two population means where the populations are assumed to be normal with unknown and possibly unequal variances. An advantage of the CAT is that it does not require the explicit knowledge of the sampling distribution of the test statistic. The CAT is then compared with three widely accepted tests—Welch–Satterthwaite test (WST), Cochran–Cox test (CCT), ‘Generalized p-value’ test (GPT)—and a recently suggested test based on the jackknife procedure, called Singh–Saxena–Srivastava test (SSST). Further, model robustness of these five tests are studied when the data actually came from t-distributions, but wrongly perceived as normal ones. Our detailed study based on a comprehensive simulation indicate some interesting results including the facts that the GPT is quite conservative, and the SSST is not as good as it has been claimed in the literature. To the best of our knowledge, the trends observed in our study have not been reported earlier in the existing literature.     

An application of Lévy's inversion formula for higher order asymptotic expansion

The asymptotic expansions for the distribution of statistics are, in general, given by applying Lévy's inversion formula to the characteristic function. This paper shows an inversion formula for higher order asymptotic expansion of the distribution of a scalar valued function which contains dependent statistics. The usage of the formula is illustrated by derivation of third order asymptotic expansion of the distribution of Hotelling's T²-statistic under the elliptical distribution as an example.     

A modification of the Whittaker–Henderson method of graduation

This paper presents a modified Whittaker–Henderson (WH) Method of Graduation. After giving a closed-form solution, we show that it is of practical use because it provides not only a smoothed series identical to that of the WH graduation, but also an extrapolation beyond the sample limit of current data. In addition, we introduce two other penalized least squares problems and show that they provide the same results as those of the modified WH graduation.     

Investigating the role of orthogonal and non – orthogonal rotation in multivariate factor analysis,in regard to the repeatability of the extracted factors: A simulation study

ABSTRACT

Factor analysis (FA) is the most commonly used pattern recognition methodology in social and health research. A technique that may help to better retrieve true information from FA is the rotation of the information axes. The main goal is to test the reliability of the results derived through FA and to reveal the best rotation method under various scenarios. Based on the results of the simulations, it was observed that when applying non-orthogonal rotation, the results were more repeatable as compared to the orthogonal rotation, and, when no rotation was applied.     

Concomitants of generalized order statistics from Farlie–Gumbel–Morgenstern distributions

Generalized order statistics constitute a unified model for ordered random variables that includes order statistics and record values among others. Here, we consider concomitants of generalized order statistics for the Farlie–Gumbel–Morgenstern bivariate distributions and study recurrence relations between their moments. We derive the joint distribution of concomitants of two generalized order statistics and obtain their product moments. Application of these results is seen in establishing some well known results given separately for order statistics and record values and obtaining some new results.     

Analytic approach to coset–leader decoding leads to extension of the Welch–Berlekamp theorem

This paper provides a complete proof of the Welch–Berlekamp theorem on which the Welch–Berlekamp algorithm was founded. By introducing an analytic approach to coset–leader decoders for Reed–Solomon codes, the Welch–Berlekamp key-equation of error corrections is enlarged and a complete proof of the Welch–Berlekamp theorem is derived in a natural way, and the theorem is extended such that the BCH-bound constraint is moved.