Extensions of linear diagonal-parameter symmetry and quasi-symmetry models for cumulative probabilities in square contingency tables

For square contingency tables with ordered categories, CAUSSINUS [Annales de la Faculté des Sciences de l'Université de Toulouse (1965) Vol. 29, pp. 77–182] and AGRESTI [Statistics and Probability Letters (1983) Vol. 1, pp. 313–316] considered the quasi-symmetry and the linear diagonal-parameter symmetry models, respectively, which have multiplicative forms for cell probabilities. This paper proposes two kinds of models that have the similar multiplicative forms for cumulative probabilities that an observation will fall in row (column) category i or below and column (row) category j (> i ) or above. The endometrial cancer data are analyzed using these models.     

The exact density of a statistic related to the shape parameter of a gamma variate

The exact density of the statistic ln , where and denote, respectively, the arithmetic and the geometric means of a random sample from a two-parameter gamma distribution, is obtained in a computable form using the technique of the inverse Mellin transform. This statistic is related to the maximum likelihood estimator of the shape parameter of a gamma distribution.     

The discrepancy distribution of stationary multiplier rules for rounding probabilities

The problem of rounding finitely many (continuous) probabilities to (discrete) proportions N _i/n is considered, for some fixed rounding accuracy n. It is well known that the rounded proportions need not sum to unity, and instead may leave a nonzero discrepancy D=(∑N _i) −n. We determine the distribution of D, assuming that the rounding function used is stationary and that the original probabilities follow a uniform distribution. Received: April 1999     

The depth function of a population distribution

Tukey (1975) introduced the notion of halfspace depth in a data analytic context, as a multivariate analog of rank relative to a finite data set. Here we focus on the depth function of an arbitrary probability distribution on Â^p, and even of a non-probability measure. The halfspace depth of any point / in Â^p is the smallest measure of a closed halfspace that contains /. We review the properties of halfspace depth, enriched with some new results. For various measures, uniform as well as non-uniform, we derive an expression for the depth function. We also compute the Tukey median, which is the / in which the depth function attains its maximal value. Various interesting phenomena occur. For the uniform distribution on a triangle, a square or any regular polygon, the depth function has ridges that correspond to an 'inversion' of depth contours. And for a product of Cauchy distributions, the depth contours are squares. We also consider an application of the depth function to voting theory.     

Minimax invariant estimator of a continuous distribution function under a general loss function

The problem of invariant estimation of a continuous distribution function is considered under a general loss function. Minimaxity of the minimum risk invariant estimator of a continuous distribution function is proved for any sample size n ≥ 2.     

Estimating a distribution function in the presence of auxiliary information

For estimating the distribution functionF of a population, the empirical or sample distribution functionF _n has been studied extensively. Qin and Lawless (1994) have proposed an alternative estimator for estimatingF in the presence of auxiliary information under a semiparametric model. They have also proved the point-wise asymptotic normality of . In this paper, we establish the weak convergence of to a Gaussian process and show that the asymptotic variance function of is uniformly smaller than that ofF _n . As an application of , we propose to employ the mean and varianceŜ _n ² of to estimate the population mean and variance in the presence of auxiliary information. A simulation study is presented to assess the finite sample performance of the proposed estimators , andŜ _n ² .     

Income distribution and inequality measurement: The problem of extreme values

We examine the statistical performance of inequality indices in the presence of extreme values in the data and show that these indices are very sensitive to the properties of the income distribution. Estimation and inference can be dramatically affected, especially when the tail of the income distribution is heavy, even when standard bootstrap methods are employed. However, use of appropriate semiparametric methods for modelling the upper tail can greatly improve the performance of even those inequality indices that are normally considered particularly sensitive to extreme values.     

On the distribution of a distance function on the sphere

Summary The distribution of the square of the distance between a random point and a fixed point on ap-dimensional unit sphere when (i) the two points lie on the whole sphere and (ii) the two points lie in the positive quadrant, has been derived, assuming that the random point is distributed proportionally to exp (ky ₁), wherek is a concentration parameter. Then-th order moment in both cases is also obtained.     

The distribution function of a statistic for testing the equality of scale parameters in two gamma populations

Summary The exact distribution function of the ratio of two sums of gamma variates is derived in this paper. The result applies to ratios of quadratic forms and to a statistic used for testing the equality of scale parameters in two gamma populations.     

Minimax invariant estimation of a continuous distribution function under entropy loss

For the invariant decision problem of estimating a continuous distribution function F with two entropy loss functions, it is proved that the best invariant estimators d ₀ exist and are the same as the best invariant estimator of a continuous distribution function under the squared error loss function L (F, d)=∫|F (t) −d (t) |² dF (t). They are minimax for any sample size n≥1.     

Models for the probability of concordance in cross-classification tables

For cross-classification tables having an ordinal response variable, logit and probit models are formulated for the probability that a pair of subjects is concordant. For multidimensional tables, generalized models are given for the probability that the response at one setting of explanatory variables exceeds the response at another setting. Related measures of association are discussed for two-way tables.     

Kernel estimation of a smooth distribution function based on censored data

The problem of estimating a smooth distribution functionF at a pointτ based on randomly right censored data is treated under certain smoothness conditions onF. The asymptotic performance of a certain class of kernel estimators is compared to that of the Kaplan-Meier estimator ofF(τ). It is shown that the relative deficiency of the Kaplan-Meier estimator ofF(τ) with respect to the appropriately chosen kernel type estimator tends to infinity as the sample sizen increases to infinity. Strong uniform consistency and the weak convergence of the normalized process are also proved. Research Surported in part by NIH grant 1R01GM28405.     

Minimax estimation of a cumulative distribution function by converting to a parametric problem

Let X = (X ₁,...,X _n ) be a sample from an unknown cumulative distribution function F defined on the real line . The problem of estimating the cumulative distribution function F is considered using a decision theoretic approach. No assumptions are imposed on the unknown function F. A general method of finding a minimax estimator d(t;X) of F under the loss function of a general form is presented. The method of solution is based on converting the nonparametric problem of searching for minimax estimators of a distribution function to the parametric problem of searching for minimax estimators of the probability of success for a binomial distribution. The solution uses also the completeness property of the class of monotone decision procedures in a monotone decision problem. Some special cases of the underlying problem are considered in the situation when the loss function in the nonparametric problem is defined by a weighted squared, LINEX or a weighted absolute error.     

The impacts of education spending and finance reform on manufacturing property shadow values: a cost function approach

The effects of court-ordered education finance reform on property values and residential choice have received increasing attention in recent years (Fischel 2001). However, little attention has been focused on the effects of education finance reform on manufacturing sector property values within an optimizing framework. This is pursued here by modeling education expenditures and education finance reform as “free” variables to manufacturing firms in a cost function model together with input demand equations. This framework is applied to panel data on manufacturers’ capital (building and structures) stocks for the 48 continental US for 1982–1996 to estimate implicit (shadow) values to the manufacturing sector of education spending and school finance reform. On average, school finance reform lowers the implicit value of manufacturing firms’ stock of buildings and structures capital, while greater education spending lowers manufacturing variable costs.

Lindahl's solution and values for a public-goods example

An example of an economy with a public good is presented with a non-atomic measure space of agents and money (transferable utility medium). Three solutions are computed: the Lindahl solution, the Shapley value, and the Harsanyi-Selten value. The three are found to differ significantly in their assignment of the societal benefits attributable to the presence of the public good.