The UMVUE ofP(X<Y) based on type-II censored samples from Weinman multivariate exponential distributions

Based on two independent samples from Weinman multivariate exponential distributions with unknown scale parameters, uniformly minimum variance unbiased estimators ofP(X<Y) are obtained for both, unknown and known common location parameter. The samples are permitted to be Type-II censored with possibly different numbers of observations. Since sampling from two-parameter exponential distributions is contained in the model as a particular case, known results for complete and censored samples are generalized. In the case of an unknown common location parameter with a certain restriction of the model, the UMVUE is shown to have a Gauss hypergeometric distribution, which is further examined. Moreover, explicit expressions for the variances of the estimators are derived and used to calculate the relative efficiency.     

Forecasting intermittent demand by hyperbolic-exponential smoothing

Croston’s method is generally viewed as being superior to exponential smoothing when the demand is intermittent, but it has the drawbacks of bias and an inability to deal with obsolescence, where the demand for an item ceases altogether. Several variants have been reported, some of which are unbiased on certain types of demand, but only one recent variant addresses the problem of obsolescence. We describe a new hybrid of Croston’s method and Bayesian inference called Hyperbolic-Exponential Smoothing, which is unbiased on non-intermittent and stochastic intermittent demand, decays hyperbolically when obsolescence occurs, and performs well in experiments.     

UMP unbiased tests for multiparameter testing problems with restricted alternatives

Applications of the standard theory of UMP unbiased tests depends on conditions which in general are difficult to verify. In the present paper, however, we suggest more simple rules for applying this theory for regular exponential families of distributions. This approach leads to UMP unbiased tests for various multiparameter testing problems with restricted alternatives, and is shown to give justification for conditional tests for testing symmetry, diagonals-parameter symmetry and independence in two-way contingency tables. The derived tests are shown to possess attractive small sample properties. Received: June 1998     

A note on the assumed distributions in stochastic frontier models

Stochastic frontier models all need an assumption on the distributional form of the (in)efficiency component. Generally this efficiency component is assumed to be half normally, truncated normally, or exponentially distributed. This paper shows that the exponential distribution is, just like the half normal distribution, a special case of the truncated normal distribution. Moreover, this paper discusses the implications that this finding has on estimation.     

Reparameterized inverse gamma regression models with varying precision

In this article, we propose a mean linear regression model where the response variable is inverse gamma distributed using a new parameterization of this distribution that is indexed by mean and precision parameters. The main advantage of our new parametrization is the straightforward interpretation of the regression coefficients in terms of the expectation of the positive response variable, as usual in the context of generalized linear models. The variance function of the proposed model has a quadratic form. The inverse gamma distribution is a member of the exponential family of distributions and has some distributions commonly used for parametric models in survival analysis as special cases. We compare the proposed model to several alternatives and illustrate its advantages and usefulness. With a generalized linear model approach that takes advantage of exponential family properties, we discuss model estimation (by maximum likelihood), black further inferential quantities and diagnostic tools. A Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples with a discussion of the obtained results. A real application using minerals data set collected by Department of Mines of the University of Atacama, Chile, is considered to demonstrate the practical potential of the proposed model.     

On the existence of locally mvu estimators

It is proved that there exists an unbiased estimator for some real parameter of a class of distributions, which has minimal variance for some fixed distribution among all corresponding unbiased estimators, if and. only if the corresponding minimal variances for all related unbiased estimation problems concerning finite subsets of the underlying family of distributions are bounded. As an application it is shown that there does not exist some unbiased estimator for θ^k+c(ε≥0) with minimal variance for θ =0 among all corresponding unbiased estimators on the base of k i.i.d. random variables with a Cauchy-distribution, where θ denotes some location parameter.     

Estimation Optimality of Corrected AIC and Modified Cp in Linear Regression

Model selection criteria often arise by constructing unbiased or approximately unbiased estimators of measures known as expected overall discrepancies (Linhart & Zucchini, 1986, p. 19). Such measures quantify the disparity between the true model (i.e., the model which generated the observed data) and a fitted candidate model. For linear regression with normally distributed error terms, the "corrected" Akaike information criterion and the "modified" conceptual predictive statistic have been proposed as exactly unbiased estimators of their respective target discrepancies. We expand on previous work to additionally show that these criteria achieve minimum variance within the class of unbiased estimators.     

Median unbiased estimates for M.L.R.-families

Summary Lehmann [p. 83] has shown that some families of probability measures with monotone likelihood ratios (m.l.r.) admit median unbiased estimates which are optimum in the sense that among all median unbiased estimates they minimize the expected loss for any loss function which assumes its minimal value zero for the “true” parameter value and is nondecreasing as the parameter moves away from the true value in either direction. This very strong optimum property was proved under the assumption that all probability measures of the m.l.r.-family have continuous distribution functions, that they are mutually absolutely continuous and that each element of the support is the median of somep-measure of the family. This result does therefore not cover important cases such as the binomial families or thePoisson family. The purpose of the present paper is to show the existence ofrandomized median unbiased estimates with the same optimum property for m.l.r.-families which are closed and connected with respect to the strong topology. Such families are always dominated. We do, however, neither assume that thep-measures are mutually absolutely continuous nor that the distribution functions are continuous. We remark that the use of randomized estimates is indispensable here because nonrandomized median unbiased estimates do not always exist in the general case.     

Estimation procedures for a family of density functions representing various life-testing models

A family of density functions is considered which contains several life-testing models as specific cases. Uniformly minimum variance unbiased estimators are obtained for the positive and negative powers of the parameter, moments and reliability function. These general results provide the estimators for the specific models.     

Zur Parameter- und Prozentpunktschätzung von Lebensdauerverteilungen bei kleinem Stichprobenumfang

Summary Some of the many methods developed for estimating parameters or percentage points of the Weibull distribution are compared. It is shown that the known estimation of the reciprocal shape parameter with the aid of a straight line in the extremal probability paper is rather biased for small sample sizes. To avoid the bias, correction factors are given, and the efficiency of the resulting unbiased estimator is calculated for sample sizesn=2, 3, …, 9. Results ofJ. Lieblein concerning the double exponential distribution are slightly modified in order to get best linear unbiased estimators for parameters and for the logarithms of percentage points of the Weibull distribution. Other methods are shortly discussed and a median-unbiased estimator for the shape parameter is derived.      

Percentile estimators in location-scale parameter families under absolute loss

Estimators of percentiles of location-scale parameter families are optimized based on median unbiasedness and absolute risk. Median unbiased estimators and minimum absolute risk estimators are shown to exist within a class of equivariant estimators and depend upon medians of two completely specified distributions. This work extends earlier findings to a larger class of equivariant estimators. These estimators are illustrated in the normal and exponential distributions.     

On the estimation ofPr{Y<X} for the two-parameter exponential distribution

In this paper the Blackwell-Rao and Lehmann-Scheffé theorems are used to derive the minimum variance unbiased estimator ofP=Pr{Y when the independent random variablesX andY follow the two-parameter exponential distribution. Following a Bayesian approach, an estimator ofP is also obtained for this distribution. These results are extended for the case of censored samples.     

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.     

Tests with correct size when instruments can be arbitrarily weak

This paper applies classical exponential-family statistical theory to develop a unifying framework for testing structural parameters in the simultaneous equations model under the assumption of normal errors with known reduced-form variance matrix. The results can be divided into the limited-information and full-information categories. In the limited-information model, it is possible to characterize the entire class of similar tests in a model with only one endogenous explanatory variable. In the full-information framework, this paper proposes a family of similar tests for subsets of endogenous variables’ coefficients. For both limited- and full-information models, there exist power upper bounds for unbiased tests. When the model is just-identified, the Anderson–Rubin, score, and (pseudo) conditional likelihood ratio tests are optimal. When the model is over-identified, the (pseudo) conditional likelihood ratio test has power close to the power envelope when identification is strong.     

Seasonal exponential smoothing with damped trends : An application for production planning

This paper reviews a spreadsheet-based forecasting approach which a process industry manufacturer developed and implemented to link annual corporate forecasts with its manufacturing/distribution operations. First, we consider how this forecasting system supports overall production planning and why it must be compatible with corporate forecasts. We then review the results of substantial testing of variations on the Winters three-parameter exponential smoothing model on 28 actual product family time series. In particular, we evaluate whether the use of damping parameters improves forecast accuracy. The paper concludes that a Winters four-parameter model (i.e. the standard Winters three-parameter model augmented by a fourth parameter to damp the trend) provides the most accurate forecasts of the models evaluated. Our application confirms the fact that there are situations where the use of damped trend parameters in short-run exponential smoothing based forecasting models is beneficial.     

Mean squared error of the empirical best linear unbiased predictor in an orthogonal finite discrete spectrum linear regression model

The mean squared error (MSE) of the empirical best linear unbiased predictor in an orthogonal finite discrete spectrum linear regression model is derived and a comparison with the MSE of the best linear unbiased predictor in this model is made. It is shown that under weak conditions these two mean square errors are asymptotically the same.     

Hedging and pricing early-exercise options with complex fourier series expansion

We introduce a new numerical method called the complex Fourier series (CFS) method proposed by Chan (2017) to price options with an early-exercise feature—American, Bermudan and discretely monitored barrier options—under exponential Lévy asset dynamics. This new method allows us to quickly and accurately compute the values of early-exercise options and their Greeks. We also provide an error analysis to demonstrate that, in many cases, we can achieve an exponential convergence rate in the pricing method as long as we choose the correct truncated computational interval. Our numerical analysis indicates that the CFS method is computationally more comparable or favourable than the methods currently available. Finally, the superiority of the CFS method is illustrated with real financial data by considering Standard & Poor’s depositary receipts (SPDR) exchange-traded fund (ETF) on the S&P 500® index options, which are American options traded from November 2017 to February 2018 and from 30 January 2019 to 21 June 2019.     

A test for distributional assumptions for the stochastic frontier functions

Some asymptotic tests for testing distributional assumptions, namely, the half-normal and truncated normal distributions for the stochastic frontier functions have been proposed. The tests are Lagrangean multiplier tests based on the Pearson family of truncated distributions. The statistics can be easily computed. Simple interpretations of the statistics and two empirical examples are provided.     

Estimation of Fisher information using model selection

In the paper the problem of estimation of Fisher information I _f for a univariate density supported on [0, 1] is discussed. A starting point is an observation that when the density belongs to an exponential family of a known dimension, an explicit formula for I _f there allows for its simple estimation. In a general case, for a given random sample, a dimension of an exponential family which approximates it best is sought and then estimator of I _f is constructed for the chosen family. As a measure of quality of fit a modified Bayes Information Criterion is used. The estimator, which is an instance of Post Model Selection Estimation method is proved to be consistent and asymptotically normal when the density belongs to the exponential family. Its consistency is also proved under misspecification when the number of exponential models under consideration increases in a suitable way. Moreover we provide evidence that in most of considered parametric cases the small sample performance of proposed estimator is superior to that of kernel estimators.     

Exact inference for the difference of Laplace location parameters

We consider exact procedures for testing the equality of means (location parameters) of two Laplace populations with equal scale parameters based on corresponding independent random samples. The test statistics are based on either the maximum likelihood estimators or the best linear unbiased estimators of the Laplace parameters. By conditioning on certain quantities we manage to express their exact distributions as mixtures of ratios of linear combinations of standard exponential random variables. This allows us to find their exact quantiles and tabulate them for several sample sizes. The powers of the tests are compared either numerically or by simulation. Exact confidence intervals for the difference of the means corresponding to those tests are also constructed. The exact procedures are illustrated via a real data example.