A wider class of lagrange distributions of the second kind

Starting with the second Lagrange expansion, with f(z) and g(z) as two probability generating functions defined on non-negative integers, Janardan and Rao( 1983) introduced a new class of discrete distributions called the Lagrange Distributions (LD₂) of the second kind. In this note, this class of LD₂ distributions is widened by removing the restriction that the functions f(z) and g(z) be probability generation functions. It is also shown that the class of modified power series distributions is a subclass of LD₂.     

Testing equality of several exponential distributions

Roy's union-intersection principle is used to develop a test procedure to test the equality of scale parameters of several exponential distributions. Upper five and one percent values of the test statistic for two and three exponential distributions are tabulated and an illustrative simulated example is qiven.     

Selecting a power-series distribution for goodness of fit

The binomial, Poisson, logarithmic, negative binomial, and extended negative binomial distributions are characterized in the class of power-series distributions (1) through a differential equation based on the ratio of two successive derivatives of the series function, (2) through the ratio of two probabilities associated with two successive values in the range of the random variable, and (3) through the ratio of two consecutive factorial moments. The ratios referred to in (2) and (3) above can be utilized to discriminate between the five power-series distributions mentioned at the beginning.     

The exact distribution of the ratio of a linear combination of chi-square variables over the root of a product of chi-square variables

This article gives the exact distribution of a statistic whose numerator and denominator are independent, the former being a linear combination of independent chi-square variables and the latter being the kth root of a product of k independent chi-square variables. This structure appears in the study of multivariate linear functional relationship models. The technique of the inverse Mellin transform is used in order to obtain the density of this test statistic in a computable form.     

The asymptotic behavior of monotone percentile regression estimates

The least-absolute-deviation estimate of a monotone regression function on an interval has been studied in the literature. If the observation points become dense in the interval, the almost sure rate of convergence has been shown to be O(n^1/4). Applying the techniques used by Brunk (1970, Nonparametric, Techniques in Statistical Inference. Cambridge Univ. Press), the asymptotic distribution of the l₁ estimator at a point is obtained. If the underlying regression function has positive slope at the point, the rate of convergence is seen to be O(n^1/3). Monotone percentile regression estimates are also considered.     

General affine transform families: why is the Pareto an exponential transform?

A slight extension of the transform method by Efron (1982) is used to analyze general affine transforms of arbitrary random variables. Our application concerns the simple question “Why is the Pareto an exponential transform?”. This result, already known to Gumbel (1958), finds a satisfactory mathematical answer in the framework of general affine transforms, and holds for several other distributions including heavy-tailed ones.     

The left-truncated Weibull distribution: theory and computation

This paper studies the two-parameter, left-truncated Weibull distribution (LTWD) with known, fixed, positive truncation pointT. Important hitherto unknown statistical properties of the LTWD are derived. The asymptotic theory of the maximum likelihood estimates (MLEs) is invoked to develop parameter confidence intervals and regions. Numerical methods are described for computing the MLEs and for evaluating the exact, asymptotic variances and covariances of the MLEs. An illustrative example is given.     

寿命分布检验和参数估计的线性回归法及微机实现

本文介绍了一种用线性回归原理来检验产品寿命分布类型和估计特征量的方法及其自动实现寿命试验数据处理的通用计算机程序。此程序所涉及的分布函数有指数分布、双参数威布尔分布、三参数威布尔分布、正态分布、对数正态分布等五种,并具有自动判别最佳分布的功能。     

Sequential and two-stage procedures for selecting the better exponential population covering the case of unknown and unequal scale parameters

The problems of selecting the larger location parameter of two exponential distributions are discussed. When the scale parameters are the same but unknown, we consider the procedure of Desu et al. (1977) in detail, and study some of its exact and asymptotic properties. We indicate how this procedure can be modified along the lines of Mukhopadhyay (1979, 1980) to achieve first-order asymptotic efficiency. We then propose a sequential procedure for this set-up and show that it is asymptotically second-order efficient according to Ghosh and Mukhopadhyay (1981). In case the scale parameters are completely unknown and unequal, we propose a two-stage procedure that guarantees the probability of correct selection to exceed the prescribed nominal level in the preference zone. We do not need any new tables to implement this particular procedure other than those in Krishnaiah and Armitage (1964), Gupta and Sobel (1962), Guttman and Milton (1969). We also propose a sequential method in this case and derive some of its asymptotic properties.     

Moment inequalities for truncated multivariate distributions

In this note some properties of the absolute moments of a doubly truncated arbitrary multivariate distribution are studied and several moment inequalities are derived.