Implementing lifetime performance index for the pareto lifetime businesses of the service industries

Process capability analysis is an effective means of measuring process performance and potential capability. In the service industries, process capability indices (PCIs) are utilized to assess whether business quality meets the required level. Hence, the performance index C _L is used as a means of measuring business performance, where L is the lower specification limit. In the technology of data transformation, this study constructs a uniformly minimum variance unbiased estimator (UMVUE) of C _L based on the right type II censored sample from the pareto distribution. The UMVUE of C _L is then utilized to develop a novel hypothesis testing procedure in the condition of known L. Finally, we give one practical example and the Monte Carlo simulation to assess the behavior of this test statistic for testing null hypothesis under given significance level. Moreover, the managers can then employ the new testing procedure to determine whether the business performance adheres to the required level.     

Distributional and Inferential Properties of the Estimated Precision Cp Based on Multiple Samples

Process precision index C_p has been widely used in the manufacturing industry to provide numerical measures on process potential. Pearn et al. (1998) considered an unbiased estimator of C_p for one single sample. They showed that the unbiased estimator is the UMVUE. They also proposed an efficient test for C_p based on one single sample, and showed that the test is the UMP test. In this paper, we consider an unbiased estimator of C_p for multiple samples. We show that the unbiased estimator is the UMVUE of C_p, which is asymptotically efficient. We also consider an efficient test for C_p, and show that the test is the UMPtest for multiple samples. The practitioners can use the proposed test on theirin-plant applications to obtain reliable decisions.     

Estimating a semi-parametric duration model without specifying heterogeneity

This paper presents a new estimator for the mixed proportional hazard model that allows for a nonparametric baseline hazard and time-varying regressors. In particular, this paper allows for discrete measurement of the durations as happens often in practice. The integrated baseline hazard and all parameters are estimated at the regular rate, $\sqrt{N}$, where $N$ is the number of individuals. A hazard model is a natural framework for time-varying regressors. In particular, if a flow or a transition probability depends on a regressor that changes with time, a hazard model avoids the curse of dimensionality that would arise from interacting the regressors at each point in time with one another. This paper also presents a new test to detect unobserved heterogeneity.     

A conditional distribution approach to uniform sampling on spheres and balls in Lp spaces

Liang and Ng (Metrika 68:83–98, 2008) proposed a componentwise conditional distribution method for L _p -uniform sampling on L _p -norm n-spheres. On the basis of properties of a special family of L _p -norm spherical distributions we suggest a wide class of algorithms for sampling uniformly distributed points on n-spheres and n-balls in L _p spaces, generalizing the approach of Harman and Lacko (J Multivar Anal 101:2297–2304, 2010), and including the method of Liang and Ng as a special case. We also present results of a numerical study proving that the choice of the best algorithm from the class significantly depends on the value of p.     

Akaike's Information Criterion,Cp and Estimators of Loss for Elliptically Symmetric Distributions

In this article, we develop a modern perspective on Akaike's information criterion and Mallows's C_p for model selection, and propose generalisations to spherically and elliptically symmetric distributions. Despite the differences in their respective motivation, C_p and Akaike's information criterion are equivalent in the special case of Gaussian linear regression. In this case, they are also equivalent to a third criterion, an unbiased estimator of the quadratic prediction loss, derived from loss estimation theory. We then show that the form of the unbiased estimator of the quadratic prediction loss under a Gaussian assumption still holds under a more general distributional assumption, the family of spherically symmetric distributions. One of the features of our results is that our criterion does not rely on the specificity of the distribution, but only on its spherical symmetry. The same kind of criterion can be derived for a family of elliptically contoured distribution, which allows correlations, when considering the invariant loss. More specifically, the unbiasedness property is relative to a distribution associated to the original density.     

Vergleich zweier Operationscharakteristiken

Summary We compare the OC-curvesL _n.c (p) (1) andL _n.c ^* (p) (2). The first is founded on the binomial distribution, the latter relates to the Poisson distribution and is often used as approximation. These OC-curves occur in Statistical Quality Control as probabilities for the acception of a lot as approximations for such probabilities; they are regarded as functions of the fraction defectivep. It is shown that the two OC-curves have exactly one intersection point between 0 and 1, if the acceptance numberc is 1 and the sample sizen is >c+1.Forp between 0 and the intersection pointp _s we have thenL _n.c.(p)>L _n.c ^* (p); from p _s <p1 followsL _n.c(p)n.c ^* (p).An interval is given which coversp _s and with an example it is shown how one might use the results of this paper for the construction of sampling plans.     

A method for generating uniformly scattered points on the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-norm unit sphere and its applications

Tashiro (Ann Inst Stat Math 29:295–300, 1977) studied methods for generating unform points on the surface of the regular unit sphere. The L _p -norm unit sphere is a generalization of the regular unit sphere. In this paper we propose a method associated with an algorithm for generating uniformly scattered points on the L _p -norm unit sphere and discuss its applications in statistical simulation, representative points of a wide class of multivariate probability distributions and optimization problems. Some examples are illustrated for these applications. This research was supported by The University of Hong Kong Research Grant and a University of New Haven 2005 and 2006 Summer Faculty Fellowships.     

Preliminary test ridge regression estimators with student’s <Emphasis Type="Italic">t</Emphasis> errors and conflicting test-statistics

The preliminary test ridge regression estimators (PTRRE) based on the Wald (W), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests for estimating the regression parameters has been considered in this paper. Here we consider the multiple regression model with student t error distribution. The bias and the mean square errors (MSE) of the proposed estimators are derived under both null and alternative hypothesis. By studying the MSE criterion, the regions of optimality of the estimators are determined. Under the null hypothesis, the PTRRE based on LM test has the smallest risk followed by the estimators based on LR and W tests. However, the PTRRE based on W test performs the best followed by the LR and LM based estimators when the parameter moves away from the subspace of the restrictions. The conditions of superiority of the proposed estimators for both shrinkage parameter, k and the departure parameter, are provided. Some tables for the maximum and minimum guaranteed efficiency of the proposed estimators have been given, which allows us to determine the optimum level of significance corresponding to the optimum estimator. Finally, we conclude that the estimator based on Wald test dominates the other two estimators in the sense of having highest minimum guaranteed efficiency.     

The non-existence of smooth demand in general banach spaces

We study the questions of existence and smoothness of demand functions with an infinite number of commodities. The main result obtained, under some hypothesis, is: if a C¹ demand exists in a commodity space B, then B can be given an inner product structure. For example, if B is L_p, 1p∞, and if there exists a C¹ demand function defined on B then p must be 2. Another result is: if a demand function exists, defined for all prices p and income, then the commodity space must be reflexive. For example, if B is L_p and a demand function exists on B, defined for all prices and incomes then 1<p<∞. We also study the cases L_∞ and L₁ with weaker assumptions. We finish the paper proving that the demand function is always defined for a dense set of prices and convenient incomes.     

Characterization of some types of completeness resp. Total completeness and their conservation under direct products

Some notions ofL _p (μ)-completeness resp. totally L _p (μ)-completeness (1≦p≦∞) are characterized for families of probability distributions dominated by aσ-finite measureμ and their conservation with respect to direct products is proved. Furthermore, it is shown that totallyL _∞(μ)-completeness does not implyL ₁(μ)-completeness and that there are families of probability distributions in the i.i.d. case induced by the order statistic, which are L₁(μ)-complete but not totallyL _∞(μ)-complete.     

Using Complex Surveys to Estimate the L1‐Median of a Functional Variable: Application to Electricity Load Curves

Mean profiles are widely used as indicators of the electricity consumption habits of customers. Currently, in Électricité De France (EDF), class load profiles are estimated using point‐wise mean profiles. Unfortunately, it is well known that the mean is highly sensitive to the presence of outliers, such as one or more consumers with unusually high‐levels of consumption. In this paper, we propose an alternative to the mean profile: the L ₁ ‐ median profile which is more robust. When dealing with large data sets of functional data (load curves for example), survey sampling approaches are useful for estimating the median profile avoiding storing the whole data. We propose here several sampling strategies and estimators to estimate the median trajectory. A comparison between them is illustrated by means of a test population. We develop a stratification based on the linearized variable which substantially improves the accuracy of the estimator compared to simple random sampling without replacement. We suggest also an improved estimator that takes into account auxiliary information. Some potential areas for future research are also highlighted.     

A general class of estimators for estimating population mean using auxiliary information

Summary A general class of estimators for estimating the population mean of the character under study which make use of auxiliary information is proposed. Under simple random sampling without replacement (SRSWOR), the expressions of Bias and Mean Square Error (MSE), up to the first and the second degrees of approximation are derived. General conditions, up to the first order approximation, are also obtained under which any member of this class performs more efficiently than the mean per unit estimator, the ratio estimator and the product estimator. The class of estimators in its optimum case, under the first degree approximation, is discussed. It is shown that it is not possible to obtain optimum values of parameters “a”, “b” and “p”, that are independent of each other. However, the optimum relation among them is given by (b−a)p=ρ C _y/C _x. Under this condition, the expression of MSE of the class is that of the linear regression estimator.     

A Survey of L1 Regression

L₁ regularization, or regularization with an L₁ penalty, is a popular idea in statistics and machine learning. This paper reviews the concept and application of L₁ regularization for regression. It is not our aim to present a comprehensive list of the utilities of the L₁ penalty in the regression setting. Rather, we focus on what we believe is the set of most representative uses of this regularization technique, which we describe in some detail. Thus, we deal with a number of L₁‐regularized methods for linear regression, generalized linear models, and time series analysis. Although this review targets practice rather than theory, we do give some theoretical details about L₁‐penalized linear regression, usually referred to as the least absolute shrinkage and selection operator (lasso).     

Nonparametric estimation for a current status right‐censored data model

We consider the Case 1 interval censoring approach for right‐censored survival data. An important feature of the model is that right‐censored event times are not observed exactly, but at some inspection times. The model covers as particular cases right‐censored data, current status data, and life table survival data with a single inspection time. We discuss the nonparametric estimation approach and consider three nonparametric estimators for the survival function of failure time: maximum likelihood, pseudolikelihood, and the naïve estimator. We establish strong consistency of the estimators with the L₁ rate of convergence. Simulation results confirm consistency of the estimators.     

Steiner minimal trees and urban service networks

The optimal configuration of urban service networks has recently been shown to be a computationally difficult problem. However, there are efficient and effective techniques by which this optimal configuration of urban service networks can be approximated. In this paper, we analyze the L_p Steiner Network problem in the plane R² and demonstrate its applicability to the urban service network problem. We present a simple algorithm for estimating the L_p metric parameter for random points in the plane, then utilize it to find the L_p values for four different American cities. Finally, we apply the L_pSMT algorithm described within the text to one of the cities in order to demonstrate the effectiveness of our algorithm for determining optimal network configurations.     

Minimax estimators andΓ-minimax estimators for a bounded normal mean under the lossl p (θ, d)=|θ-d|p

Summary Let the random variableX be normal distributed with known varianceσ ²>0. It is supposed that the unknown meanθ is an element of a bounded intervalΘ. The problem of estimatingθ under the loss functionl _p (θ, d)=|θ-d| ^p p≥2 is considered. In case the length of the intervalθ is sufficiently small the minimax estimator and theΓ(β, τ)-minimax estimator, whereΓ(β, τ) represents special vague prior information, are given.     

Adaptive Estimation of a Conditional Density

The authors consider the problem of estimating a conditional density by a conditional kernel density estimate when the error associated with the estimate is measured by the L₁‐norm. On the basis of the combinatorial method of Devroye and Lugosi ( 1996 ), they propose a method for selecting the bandwidths adaptively and for providing a theoretical justification of the approach. They use simulated data to illustrate the finite‐sample performance of their estimator.     

The adequacy of asymptotic approximations in the near-integrated autoregressive model with dependent errors

We consider the normalized least squares estimator of the parameter in a nearly integrated first-order autoregressive model with dependent errors. In a first step we consider its asymptotic distribution as well as asymptotic expansion up to order O_p(T⁻¹). We derive a limiting moment generating function which enables us to calculate various distributional quantities by numerical integration. A simulation study is performed to assess the adequacy of the asymptotic distribution when the errors are correlated. We focus our attention on two leading cases: MA(1) errors and AR(1) errors. The asymptotic approximations are shown to be inadequate as the MA root gets close to −1 and as the AR root approaches either −1 or 1. Our theoretical analysis helps to explain and understand the simulation results of Schwert (1989) and DeJong, Nankervis, Savin, and Whiteman (1992) concerning the size and power of Phillips and Perron's (1988) unit root test. A companion paper, Nabeya and Perron (1994), presents alternative asymptotic frameworks in the cases where the usual asymptotic distribution fails to provide an adequate approximation to the finite-sample distribution.     

Semiparametric tests of conditional moment restrictions under weak or partial identification

We propose two new semiparametric specification tests which test whether a vector of conditional moment conditions is satisfied for any vector of parameter values θ₀. Unlike most existing tests, our tests are asymptotically valid under weak and/or partial identification and can accommodate discontinuities in the conditional moment functions. Our tests are moreover consistent provided that identification is not too weak. We do not require the availability of a consistent first step estimator. Like Robinson [Robinson, Peter M., 1987. Asymptotically efficient estimation in the presence of heteroskedasticity of unknown form. Econometrica 55, 875–891] and many others in similar problems subsequently, we use k-nearest neighbor (knn) weights instead of kernel weights. The advantage of using knn weights is that local power is invariant to transformations of the instruments and that under strong point identification computation of the test statistic yields an efficient estimator of θ₀ as a byproduct.