Bivariate Uniform Distribution Research Articles

Estimation of a Scale Parameter of Morgenstern Type Bivariate Uniform Distribution by Ranked Set Sampling

In this paper, we obtain several estimators of a scale parameter of Morgenstern type bivariate uniform distribution (MTBUD) based on the observations made on the units of the ranked set sampling regarding the study variable Y which is correlated with the auxiliary variable X, when (X;Y ) follows a MTBUD. Eciency comparisons among these estimators are also made in this work. Finally, we illustrate the methods developed by using a real data set.

Parameter estimation of Cambanis-type bivariate uniform distribution with Ranked Set Sampling

The concept of ranked set sampling (RSS) is applicable whenever ranking on a set of sampling units can be done easily using a judgment method or based on an auxiliary variable. In this paper, we consider a study variable correlated with the auxiliary variable and use it to rank the sampling units. Further is assumed to have Cambanis-type bivariate uniform (CTBU) distribution. We obtain an unbiased estimator of a scale parameter associated with the study variable based on different RSS schemes. We perform the efficiency comparison of the proposed estimators numerically. We present the trends in the efficiency performance of estimators under various RSS schemes with respect to parameters through line and surface plots. Further, we develop a Matlab function to simulate data from CTBU distribution and present the performance of proposed estimators through a simulation study. The results developed are implemented to real-life data also.

Estimating Morgenstern Type Bivariate Association Parameter Using a Modified Maximum Likelihood Method

This paper investigates estimating the association parameter of Morgenstern type bivariate distribution using a modified maximum likelihood method where the regular maximum likelihood methods failed to achieve estimation. The simple random sampling, concomitant of ordered statistics and bivariate ranked set sampling methods are used and compared. Efficiency and bias of the produced estimators are compared for two specific examples, Morgenstern type bivariate uniform and exponential distributions.

A new bivariate beta distribution of Kind-1 of Type-A

This paper proposes a new form of bivariate beta distribution of Kind-1 of Type-A as well as the density function is visualized in terms of Generalised hypergeometric function and the marginals are univariate beta distribution of Kind-1. Moreover, conditional distributions and constants of the conditionals are also computed. Generating functions such as moment, Cumulant, characteristic functions and the Shannon's differential entropy are also determined. The Bivariate correlations between the random variables for different combination of shape parameters are tabulated and three dimensional probability surfaces are used to visualize the shape of the derived densities graphically. Finally, the special cases of the distribution such as Bivariate Beta distribution of Kind-2 of Type-A, Bivariate F-distribution of Kind-1, Bivariate uniform distribution, Different forms of Bivariate Kumaraswamy distribution, Bivariate exponential and exponentiated distribution, Bivariate Wigner distribution and Bivariate Arcsine distribution are also discussed.

On the maximum and minimum in a bivariate uniform distribution

We obtain means and variances of max {X, Y} and min {X, Y} in the underlying Morgenstern type bivariate uniform variables X and Y with same scale parameters and different scale parameters respectively. And we obtain the conditional expectations in the underlying Morgenstern type bivariate uniform variables. Here, we shall consider the conditional expectations to know the dependence of one variable on the other variable and we consider the behaviors of means and variances of max {X, Y} and min {X, Y} with respect to changes in means, variances, and the correlation coeffcient of the underlying Morgenstern type bivariate uniform variables.

Estimation of scale parameter of a Morgenstern type bivariate uniform distribution using censored ranked set samples

In this article, we have obtained several point estimators for the estimation of the scale parameters in Morgenstern type bivariate uniform distribution (MTBUD) using censored ranked set sampling. We have also suggested some linear shrinkage estimators of scale parameter of Morgenstern type bivaria te uniform distribution (MTBUD). Efficiency comparisons are also made in this work.

Improved estimation of scale parameters of morgenstern type bivariate uniform distribution using ranked set sampling

ABSTRACTIn this article we suggest some improved version of estimators of scale parameter of Morgenstern-type bivariate uniform distribution (MTBUD) based on the observations made on the units of the ranked set sampling regarding the study variable Y which is correlated with the auxiliary variable X, when (X, Y) follows a MTBUD. We also suggest some linear shrinkage estimators of scale parameter of Morgenstern type bivariate uniform distribution (MTBUD). Efficiency comparisons are also made in this work.

Generating Bivariate Uniform Data with a Full Range of Correlations and Connections to Bivariate Binary Data

In this article, random number generation algorithms for generating bivariate uniform data based on a known class of symmetric bivariate uniform distributions that allow the entire correlation range are given, and its previously unrecognized connection with bivariate binary data is established via matching the cumulative distribution functions.

Location models with stochastic demand points

This paper presents two approaches for solving both the minimax and minisum location problems and as well as a bi-objective location problem. The bi-objective location problem is a combination of both minimax and minisum location problems with recti-linear distances and randomly distributed destinations. Wesolowsky. G.O. [Journal of Regional Science 18: 53–60, 1977] has considered the stochastic extension of Weber problem with rectilinear distance norm and suggested some iterative method after dividing the problem into two parts, one for the X coordinates and other for the Y coordinates, where these co-ordinates may be correlated. In the present investigation exact approaches for both Minisum and Minimax objectives with randomly distributed destinations have been considered separately with both bi-variate exponential and bi-variate uniform distributions and formulated as a non-linear programming problem. Also the area restriction concept has been introduced so that the facility to be located should be within certain restricted area. The consideration of area restrictions has been implemented by incorporating a convex polygon as the constraint set. It has been proved that the solution obtained by this method will give the global optimal solution. The second part of this investigation is the bi-objective problem. The aim of the bi-objective problem is to find the satisfying solution which is desired in many realistic situations. Situation in which the proposed model is applicable is the location of a new warehouse where the sources of goods shipped to the warehouse and also the supply points to which the goods will be shipped is not known in advance. Due to the volatile and competitive nature of the market the supply and the demand points or in other words the variable points are not known in advance but some probability distribution can be predicted. It is required to find the location of a new warehouse which is closer to the variable points and simultaneously the total shipping cost per month from the warehouse to the variable points is minimum. For both single as well as bi-objective problems rectilinear distance norm has been considered as it is more appropriate to the different realistic situations. Two types of compromise solution methods have been suggested to get a satisfactory solution of the bi-objective problem.

Some multivariate goodness-of-fit tests based on data depth

Based on data depth, three types of nonparametric goodness-of-fit tests for multivariate distribution are proposed in this paper. They are Pearson’s chi-square test, tests based on EDF and tests based on spacings, respectively. The Anderson–Darling (AD) test and the Greenwood test for bivariate normal distribution and uniform distribution are simulated. The results of simulation show that these two tests have low type I error rates and become more efficient with the increase in sample size. The AD-type test performs more powerfully than the Greenwood type test.

Optimal Solutions for Mixed Bundling of Two Independently-Valued Goods

This paper develops analytical results and insights for the mixed bundling problem of pricing a product line consisting of two component goods and a bundle of the two goods. Consumer valuations for the two goods follow a symmetric bivariate uniform distribution, a setting which has previously been considered analytically intractable. Prior analytical work has demonstrated the superiority of mixed bundling, but failed to provide an explicit specification of the optimal solution or prices. Alternately, researchers have employed numerical solution techniques to solve instances of the mixed bundling problem. I reduce the multi-product pricing problem to a univariate nonlinear optimization problem in the bundle price with a global maximum that can be identified uniquely. I derive tight lower and upper bound functions for the optimal bundle price. The average of these two functions can be used as an alternate, and extremely simple, analytical expression for the optimal bundle price. Another alternative formula for the optimal bundle price is obtained by applying statistical estimation over solutions for a large space of problem instances. Both approximations are highly accurate, producing a pro t that is within half percent of the exact optimal pro t. The optimal bundle price has an elegant analytical expression (a linear function of the marginal costs of the two goods) and prices for component goods are an algebraic function of the bundle price. With a bivariate distribution in the unit square and constant unit marginal costs w1 and w2, the optimal bundle price can be written as 0:86032 0:55929(w1 w2)/2.

Stochastic models for transfer point location problem

The transfer point location problem has been introduced recently and for the case of minimax objective and planar topology, has only been studied for situations in which demand points are not weighted and have known coordinates. In this paper, we consider the case in which demand points are weighted and their coordinates have bivariate uniform distribution. Also, the problem is developed from a conceptual view and different distance measures are used to make models more applicable in real world situations. The problem is to find the best location for the transfer point such that the maximum expected weighted distance to all demand points through the transfer point is minimized. Depending on assumptions for uniform distributions, two models are considered, convexity conditions are discussed, properties of the optimal solution are obtained and methods to solve the problems are proposed. Finally, numerical examples are given.

ON THE LIMIT BEHAVIOR OF EXTENDED NEGATIVE QUADRANT DEPENDENCE

We discuss in this paper the notions of extended negative quadrant dependence and its properties. We study a class of bivariate uniform distributions having extended negative quadrant dependence, which is derived by generalizing the uniform representation of a well-known Farlie-Gumbel-Morgenstern distribution. Finally, we also study the limit behavior on the extended negative quadrant dependence.

Generalized Diagonal Band Copulas with Two-Sided Generating Densities

Copulas are joint continuous distributions with uniform marginals and have been proposed to capture probabilistic dependence between random variables. Maximum-entropy copulas introduced by Bedford and Meeuwissen (Bedford, T., A. M. H. Meeuwissen. 1997. Minimally informative distributions with given rank correlations for use in uncertainty analysis. J. Statist. Comput. Simulation 57(1–4) 143–175) provide the option of making minimally informative assumptions given a degree-of-dependence constraint between two random variables. Unfortunately, their distribution functions are not available in a closed form, and their application requires the use of numerical methods. In this paper, we study a subfamily of generalized diagonal band (GDB) copulas, separately introduced by Ferguson (Ferguson, T. F. 1995. A class of symmetric bivariate uniform distributions. Statist. Papers 36(1) 31–40) and Bojarski (Bojarski, J. 2001. A new class of band copulas—Distributions with uniform marginals. Technical report, Institute of Mathematics, Technical University of Zielona Góra, Zielona Góra, Poland). Similar to Archimedean copulas, GDB copula construction requires a generator function. Bojarski's GDB copula generator functions are symmetric probability density functions. In this paper, symmetric members of a two-sided framework of distributions introduced by van Dorp and Kotz (van Dorp, J. R., S. Kotz. 2003. Generalizations of two-sided power distributions and their convolution. Comm. Statist.: Theory and Methods 32(9) 1703–1723) shall be considered. This flexible setup allows for derivations of GDB copula properties resulting in novel convenient expressions. A straightforward elicitation procedure for the GDB copula dependence parameter is proposed. Closed-form expressions for specific examples in the subfamily of GDB copulas are presented, which enhance their transparency and facilitate their application. These examples closely approximate the entropy of maximum-entropy copulas. Application of GDB copulas is illustrated via a value-of-information decision analysis example.

Distribution of extremes ofr th concomitant from the Morgenstern family

In the present paper the distribution theory of maximum and minimum of ther th concomitants from k independent subgroups each of same size m from the Morgenstern family is investigated. Some applications of the results in estimation of the scale parameter of a marginal variable in the bivariate uniform distribution and a selection problem are discussed.

Generalized diagonal band copulas

This paper investigates the class of generalized diagonal band (GDB) copulas, which form an extension of diagonal band (DB) copulas originally introduced by Cooke and Waij [Cooke, R.N., Waij, R., 1986. Monte Carlo sampling for generalized knowledge dependence, with application to human reliability. Risk Anal. 6, 335–343] for application in simulating high-dimensional distributions for dependent random variables. Ferguson [Ferguson, T.S., 1995. A class of bivariate uniform distributions. Stat. Pap. 36, 31–40] provided a method for constructing copulas which include the convex closure of DB copulas. Bojarski [Bojarski, J., 2001. A New Class of Band Copulas—Distributions with Uniform Marginals, Technical Publication. Institute of Mathematics, Technical University of Zielona Góra] provided a different extension of the original DB construction and obtained (independently) the same class as Ferguson, which we now call the GDB. Our interest in GDB copulas results from their possible application in the vine–copula method for simulating high-dimensional dependent vectors; see Cooke [Cooke, R.M., 1997. Markov and entropy properties of tree- and vine-dependent variables. Proceedings of the ASA Section on Bayesian Statistical Science], Bedford and Cooke [Ann. Math. Artif. Intel. 32 (2001) 245–268], and Kurowicka and Cooke [Linear Algebra Appl. 372 (2003) 225–251]. This method requires using copulas with certain properties which GDB copulas often possess. We list these requirements and construct distributions that meet them using the class of GDB copulas.

On the computing of the moments of bivariate order statistics

In this paper, we establish general recurrence relations satisfied by the product moments (of any order) of bivariate order statistics from any arbitrary bivariate uniform distribution function. Moreover, we present formulae to easily compute the product moments (of any order) of bivariate order statistics from any arbitrary bivariate distribution function, with positive left endpoints, or with negative right endpoints.

A class of symmetric bivariate uniform distributions

A class of symmetric bivariate uniform distributions is proposed for use in statistical modeling. The distributions may be constructed to be absolutely continuous with correlations as close to±1 as desired. Expressions for the correlations, regressions and copulas are found. An extension to three dimensions is proposed.

Interpoint squared distance as a measure of spatial clustering

The expectation and variance for the mean interpoint squared distance are presented. In order to evaluate these expressions it is necessary to calculate the moments of a bivariate uniform distribution defined over an arbitrary polygon. Expressions for these moments are presented, allowing the mean interpoint squared distance to be used as a measure of spatial clustering. The distribution and power of this test statistic is explored on the unit square, and the spatial distribution of 11 cases of non-Hodgkin's lymphoma is investigated to illustrate an application of the approach.

On the monotone regression dependence for archimedian bivariate uniform

For a family of archimedian bivariate uniform distributions indexed by an association parameter a, a necessary and sufficient condition is derived for which the family has the ‘monotone regression dependence’ property. Using the condition, the laws of Frank (1979) and Cook and Johnson (1981) are shown to have this ‘monotone regression dependence’. The importance of this condition is that tests of independence based on measures of association which preserve the ‘more regression dependent’ ordering will have a monotone power function when the condition is satisfied.