Approximation of the Fisher price index by using Lowe,Young, and AG Mean indices

The Consumer Price Index (CPI) approximates changes in the costs of household consumption assuming the constant utility (COLI, Cost of Living Index). In practice, the Laspeyres price index is used to measure the CPI despite the fact that many economists consider the superlative indices to be the best approximation of COLI. The Fisher index is one of the superlative indices and additionally it satisfies most of tests from the axiomatic price index theory. Nevertheless, the Fisher price index makes use of current-period expenditure data and its usefulness in CPI measurement is limited. In this article, we verify the utility of using the Lowe, Young, and AG Mean indices for Fisher price index approximation. We confirm this utility in a simulation study and we provide an empirical proof.     

Characterizations,length-biasing,and infinite divisibility

SupposeL(X) is the law of a positive random variableX, andZ is positive and independent ofX. Admissible solution pairs (L(X),L(Z)) are sought for the in-law equation $\hat X \cong X o Z$ °Z, where $L\left( {\hat X} \right)$ is a weighted law constructed fromL(X), and ° is a binary operation which in some sense is increasing. The class of weights includes length biasing of arbitrary order. When ° is addition and the weighting is ordinary length biasing, the class of admissibleL(X) comprises the positive infinitely divisible laws. Examples are given subsuming all known specific cases. Some extensions for general order of length-biasing are discussed.     

The variance and covariance of a generalized index of similarity especially for a generalizaition on an index of hellinger and bhattacharyya

Formulae are derived for approximating the variance or a general similarity measlire and the covariance between two similarity measures for multinomial populations.     

Characterizations of the poisson process using,the variance

Let γ(t) be the residual life at time t of the renewal process {A(t), t > 0}, which has F as the common distribution function of the inter-arrival times. In this article we prove that if Var(γ(t)) is constant, then F will be exponentially or geometrically distributed under the assumption F is continuous or discrete respectively. An application and a related example also are given.     

Effect of outliers on the GFI quality adjustment index in structural equation model and proposal of alternative indices

This work is intended to suggest modifications in the construction of the GFI index using robust methods for estimating the unrestricted sample covariance matrix, leading to new indices called GFI_(MCD) and GFI_(MVE). The validation of this proposal was made using Monte Carlo simulation methods, considering differences between the unrestricted sample covariance matrix and those imposed by the structural model, and different numbers of outliers generated by distributions with deviations from symmetry and excess kurtosis. It was concluded that for larger samples size (n ? 100), given that the outliers are from distributions that are symmetrical, the GFI_(MCD) and GFI_(MVE) present similar results, including samples with high percentages of outliers.     

Characterizations and o-statistic representations of lu-statistics

The LU-statistics, that is, the class of all L-statistics that are also U-statistics, are characterised both via conditions on their coefficients and as linear combinations of the 0-statistics, which generalize the usual sample order statistics. The explicit computational formulas provided can be used to decide whether a given L-statistic is an LU-statistic and, if so, to find its kernel and its simplest 0-statistic representation.     

Characterizations of the General Multivariate Weibull Distributions

For studying and modeling the time to failure of a system or component, many reliability practitioners used the hazard rate and its monotone behaviors. However, nowadays, there are two problems. First, the modern components have high reliability and, second, their distributions are usually have non monotone hazard rate, such as, the truncated normal, Burr XII, and inverse Gaussian distributions. So, modeling these data based on the hazard rate models seems to be too stringent. Zimmer et al. (1998 Zimmer , W. J. , Wang , Y. , Pathak , P. K. ( 1998 ). Log-odds rate and monotone log-odds rate distributions . J. Qual. Technol. 30 ( 4 ): 376 – 385 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Wang et al. (2003 Wang , Y. , Hossain , A. M. , Zimmer , W. J. ( 2003 ). Monotone log-odds rate distributions in reliability analysis . Commun. Statist. Theor. Meth. 32 ( 11 ): 2227 – 2244 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 2008 Wang , Y. , Hossain , A. M. , Zimmer , W. J. ( 2008 ). Useful properties of the three-parameter Burr XII distribution. In: Ahsanullah M., Applied Statistics Research Progress. pp. 11–20 . [Google Scholar]) introduced and studied a new time to failure model in continuous distributions based on log-odds rate (LOR) which is comparable to the model based on the hazard rate.

There are many components and devices in industry, that have discrete distributions with non monotone hazard rate, so, in this article, we introduce the discrete log-odds rate which is different from its analog in continuous case. Also, an alternative discrete reversed hazard rate which we called it the second reversed rate of failure in discrete times is also defined here. It is shown that the failure time distributions can be characterized by the discrete LOR. Moreover, we show that the discrete logistic and log logistics distributions have property of a constant discrete LOR with respect to t and ln t, respectively. Furthermore, properties of some distributions with monotone discrete LOR, such as the discrete Burr XII, discrete Weibull, and discrete truncated normal are obtained.     

Characterizations of exponentiality within the HNBUE class and related tests

A harmonic new better than used in expectation (HNBUE) variable is a random variable which is dominated by an exponential distribution in the convex stochastic order. We use a recently obtained condition on stochastic equality under convex domination to derive characterizations of the exponential distribution and bounds for HNBUE variables based on the mean values of the order statistics of the variable. We apply the results to generate discrepancy measures to test if a random variable is exponential against the alternative that is HNBUE, but not exponential.     

Characterizations of gamma,inverse Gaussian,and negative binomial distributions via their length-biased distributions

A general form for characterizing inverse Gaussian and Wald distributions, based on their respective length-biased distributions, is introduced. Further results for characterizations of the gamma distribution, the negative binomial distribution and some mixtures of them by using their lengthbiased distributions are establised.     

Characterizations of a discrete normal distribution

The paper obtains a discrete analogue of the normal distribution as the distribution that is characterized by maximum entropy, specified mean and variance, and integer support on (− ∞, ∞). Two alternative characterizations are given, firstly as the distribution of the difference of two related Heine distributions, and secondly as a weighted distribution.     

Characterizations based on record values and order statistics

Consider a sequence of independent and identically distributed random variables {_Xi,i?1}$\{X_i,i?1\}$ with a common absolutely continuous distribution function F  . Let _X1:n?_X2:n???_Xn:n$X_{1:n}?X_{2:n}???X_{n:n}$ be the order statistics of {_X1,_X2,…,_Xn}$\{X_1,X_2,\dots ,X_n\}$ and {_Yl,l?1}$\{Y_l,l?1\}$ be the sequence of record values generated by {_Xi,i?1}$\{X_i,i?1\}$. In this work, the conditional distribution of _Yl$Y_l$ given _Xn:n$X_{n:n}$ is established. Some characterizations of F   based on record values and _Xn:n$X_{n:n}$ are then given.     

Characterizations and ordering properties based on log-odds functions

In this paper, we obtain some general results on characterizations of probability distributions from relationships between conditional moment, failure rate, and log-odds rate functions. We also study stochastic orders and classes based on the log-odds rate function and some relationships with usual stochastic orderings and classes. Some characterizations and ordering properties are obtained by using weighted distributions.     

Characterizations via regression of generalized order statistics

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.     

Characterizations of the exponential distribution by some properties of the record values

A sequence {X_n, n≥1} of independent and identically distributed random variables with continuous cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j>max {X₁, X₂, ..., X_j?1}. We define L(n)=min {j|j>L(n?1), X_j>X_L(n?1)}, with L(0)=1. Let Z_n,m=X_L(n)?X_L(m), n>m≥0. Some characterizations of the exponential distribution are considered in terms of Z_n,m and X_L(m).     

Inequality indices and the starshaped principle of transfers

The evaluation of income distributions is usually based on the Pigou-Dalton (PD) principle which says that a transfer from any people to people who have less decreases economic inequality, i.e., increases the social evaluation index. We introduce two weaker principles of transfers which refer to a parameter θ. With the new principles, only those PD transfers increase the social evaluation index which take from the class of incomes above θ and give to the class below θ. The relative positions of individuals remain unchanged, and either no individual may cross the line θ (principle of transfers about θ) or some may do who have been situated next to it (starshaped principle of transfers at θ). θ may be a given constant, a function of mean income, or a quantile of the income distribution. The classes of indices which are consistent with these transfers are completely characterized, and examples are given.