Vertical Density Representation Research Articles

Randomized statistical inference: A unified statistical inference frame of frequentist, fiducial, and Bayesian inference

We propose randomized inference (RI), a new statistical inference approach. RI may be realized through a randomized estimate (RE) of a parameter vector,which is a random vector that takes values in the parameter space with a probability density function (PDF) that depends on the sample or sufficient statistics, such asthe posterior distributions in Bayesian inference.Based on the PDF of an RE of an unknown parameter, we propose a framework for both the vertical density representation (VDR) test and the construction of a confidence region.This approach is explained with the aid of examples. For the equality hypothesis of multiple normal means without the condition of variance homogeneity, we present an exact VDR test, which is shown as an extension of one-way analysis of variance (ANOVA). In the case of two populations, the PDF of the Welch statistics is given by using the RE. Furthermore, through simulations,we show that the empirical distribution function, the approximated $t$, and the RE distribution function of Welch statistics are almost equal. The VDR test of the homogeneity of variance is shown to be more efficient than both the Bartlett test and the revised Bartlett test.Finally, we discuss the prospects of RI.

On the Student's t-distribution and the t-statistic

In this paper we provide rather weak conditions on a distribution which would guarantee that the t-statistic of a random vector of order n follows the t-distribution with n - 1 degrees of freedom. The results sharpen the earlier conclusions of Mauldon [Characterizing properties of statistical distributions, Quart. J. Math. 2(7) (1956) 155–160] and the more recent advances due to Bondesson [When is the t-statistic t-distributed, Sankhyā, Ser. A 45 (1983) 338–345]. The basic tool involved in the derivations is the vertical density representation originally suggested by Troutt [A theorem on the density of the density ordinate and an alternative interpretation of the Box–Muller method, Statistics 22(3) (1991) 463–466; Vertical density representation and a further remark on the Box–Muller method, Statistics 24 (1993) 81–83]. Several illustrative examples are presented.

Center-similar distributions with applications in multivariate analysis

In this paper, we introduce center-similar multivariate distributions (CSDs) based on the vertical density representation theory. A number of families of CSDs are constructed and related statistical problems are briefly examined.

On Khintchine's Theorem and its Place in Random Variate Generation

It is well known that if X and Y are uniformly distributed over the region between the horizontal axis and a density function f, then X is distributed according to density f. The algorithm “generate Y from its marginal distribution, then X from its uniform conditional distribution given Y == y” follows. The main point made in this article is that for monotone and unimodal distributions, this construction reduces to Khintchine's theorem, thereby yielding a simple explication thereof. This observation is followed up with further consideration of the general, nonunimodal, case for both univariate and multivariate distributions, and parallels are drawn with an alternative random variate generation method called vertical density representation.

Generation of multivariate distributions by vertical density representation

Troutt (1991,1993) proposed the idea of the vertical density representation (VDR) based on Box-Millar method. Kotz, Fang and Liang (1997) provided a systematic study on the multivariate vertical density representation (MVDR). Suppose that we want to generate a random vector X[d]Rnthat has a density function ƒ(x). The key point of using the MVDR is to generate the uniform distribution on [D]ƒ(v) = {x :ƒ(x) = v} for any v > 0 which is the surface in RnIn this paper we use the conditional distribution method to generate the uniform distribution on a domain or on some surface and based on it we proposed an alternative version of the MVDR(type 2 MVDR), by which one can transfer the problem of generating a random vector X with given density f to one of generating (X, Xn+i) that follows the uniform distribution on a region in Rn+1defined by ƒ. Several examples indicate that the proposed method is quite practical.

Further results on multivariate vertical density representation and an application to random vector generation*

In this paper we further develop the theory of vertical density representation (VDR) in the multivariate case and provide a formula for the calculation of the conditional probability density of a random vector when its density value is given. An application to random vector generation is also given.

On Multivariate Vertical Density Representation and its Application to Random Number Generation

Motivated by the works of Troutt (1991, 1993) and Kotz and Troutt (1996), we provide a multivariate definition of the vertical density representation (vdr) and calculate its value for some basic multivariate distributions presented in Fang et al. (1990) and Johnson (1987). Utilizing the multivariate vdr and the theorem of Troutt (1993) we generalize the well-known Box-Muller method to generate the basic multivariate distributions.

A Further VDR-type Density Representation Based on the Box-muller Method

In recent papers a concept called vertical density representation was derived from the well-known Box-Muller method for generating normally distributed deviates. In this paper we show that this method may be regarded as one particular solution to a density composition equation. We then derive a second such solution. We also illustrate an extension of the method to certain normal-like densities.

On Vertical Density Representation and Ordering of Distributions

Use of the density of the ordinate of a given density has previously been called vertical density representation (vdr). This paper uses vdr to study the heaviness of tails of univariate densities. Twelve common and widely used distributions, and a newly proposed one due to Johnson, are ordered in terms of heaviness of tails by this approach.

Vertical density representation and a further remark on the box-muller method

Summary. In an earlier nsner a theorem on the density of the density ordinate itself was obtained and used to provide an alternative proof of the Box-Muller method for the generation of Normal deviates. A more general theorem on what may be called vertical density representation is given here, The Box-Muller method requires using two uniform random numbers to produce two Normal deviates, If a method using only one uniform random number may be regarded as an ultimate method, then evidence is provided here that this limit cannot be achieved via such methods