Weighted bivariate logarithmic series distributions

In this paper, we are interested in the weighted distributions of a bivariate three parameter logarithmic series distribution studied by Kocherlakota and Kocherlakota (1990). The weighted versions of the model are derived with weight W(x,y) = x^[r] y^[s]. Explicit expressions for the probability mass function and probability generating functions are derived in the case r = s = l. The marginal and conditional distributions are derived in the general case. The maximum likelihood estimation of the parameters, in both two parameter and three parameter cases, is studied. A procedure for computer generation of bivariate data from a discrete distribution is described. This enables us to present two examples, in order to illustrate the methods developed, for finding the maximum likelihood estimates.     

Probability inequalities for weighted distributions

In this paper, properties of weighted distributions for general weight functions are investigated. We establish the subadditivity [superadditivity] property of weighted distributions for log-concave [log-convex] weight functions in the sense of the usual stochastic order. The main result generalizes Lemma 2.3 in Brown (2006). Several interesting moment inequalities are presented.     

Progressively censored tests for clinical experiments and life testing problems based on weighted empirical distributions

In the context of time-sequential studies, progressively censored tests for a simple regression model based on weighted empirical distributions are considered for ungrouped as well as grouped data situations. Early decision rules based on such tests are formulated. The asymptotic theory of the proposed tests rests on a construction of suitable empirical processes and their convergence (in distribution) to appropriate Gaussian functions. Critical values of the proposed test statistics are obtained by simulation, For a hypothetical example (of practical interest), a comparative study is made for the empirical powers and stopping times for some rival tests.     

Fisher information in exponential-weighted location-scale distributions

In this article, the comparison between the Fisher information on parameters of the weighted distributions and the parent distributions is done. The most common family of distributions, location–scale family, is considered with the exponential weight function w(x) = e^βx where β is a constant. Conditions under which the weighted distributions are more (less) informative than the parent distribution are given. This was done for location, scale, and location–scale families when the scale parameter is considered as a nuisance parameter. Furthermore, using the transformation technique, we show that the results in location–scale family can be generalized to the broader classes of problems that studied the Fisher information of the weighted distributions such as Tzavelas and Economou (2014 Tzavelas, G., and P. Economou. 2014. Characterization properties based on the fisher information for weighted distributions. Statistics and Probability Letters 84:54–9.[Crossref], [Web of Science ®] , [Google Scholar]). As the exponential weight function can include some other weight functions, the obtained results in this article can be generalized for some other weight functions.     

Weighted similarity tests for location-scale families of stable distributions

ABSTRACT

The class of stable distributions plays a central role in the study of asymptotic behavior of normalized partial sums, the same role performed by normal distribution among those with finite second moment. In this note, by exploiting the connection between stable laws and regularly varying functions, we present weighted similarity tests for stable location-scale families. The proposed weight functions are based on the 2nd-order Mallows distance between the empirical distribution and the root stable distribution. And the resulting statistics converge weakly to functionals of Brownian bridge.     

The role of weighted distributions in stochastic modeling

C. R. Rao pointed out that “The role of statistical methodology is to extract the relevant information from a given sample to answer specific questions about the parent population” and raised the question “What population does a sample represent”? Wrong specification can lead to invalid inference giving rise to a third kind of error. Rao introduced the concept of weighted distributions as a method of adjustment applicable to many situations.

In this paper, we study the relationship between the weighted distributions and the parent distributions in the context of reliability and life testing. These relationships depend on the nature of the weight function and give rise to interesting connections between the different ageing criteria of the two distributions. As special cases, the length biased distribution, the equilibrium distribution of the backward and forward recurrence times and the residual life distribution, which frequently arise in practice, are studied and their relationships with the original distribution are examined. Their survival functions, failure rates and mean residual life functions are compared and some characterization results are established.     

On asymptotic optimality of bayes empirical bayes estimators

In an empirical Bayes decision problem, a prior distribution ? is placed on a one-dimensfonal family G of priors G_w, wεΩ, to produce a Bayes empirical Bayes estimator, The asymptotic optimaiity of the Bayes estimator is established when the support of ? is Ω and the marginal distributions H_w have monotone likelihood ratio and continuous Kullback-Leibler information number.     

Classes of power semicircle laws that are randomly weighted average distributions

We give an affirmative answer to the conjecture raised in Soltani and Roozegar [On distribution of randomly ordered uniform incremental weighted averages: divided difference approach. Statist Probab Lett. 2012;82(5):1012–1020] that a certain class of power semicircle distributions, parameterized by n, gives the distributions of the average of n independent and identically Arcsine random variables weighted by the cuts of (0,1) by the order statistics of a uniform (0, 1) sample of size n?1, for each n. Then we establish the central limit theorem for this class of distributions. We also use the Demni [On generalized Cauchy–Stieltjes transforms of some beta distributions. Comm Stoch Anal. 2009;3:197–210] results on the connection between the ordinary and generalized Cauchy or Stieltjes transforms, and introduce new classes of randomly weighted average distributions.     

Bivariate distributions of some ratios of independent noncentral chi-square random variables

The bivariate distributions of three pairs of ratios of in¬dependent noncentral chi-square random variables are considered. These ratios arise in the problem of computing the joint power function of simultaneous F-tests in balanced ANOVA and ANCOVA. The distributions obtained are generalizations to the noncentral case of existing results in the literature. Of particular note is the bivariate noncentral F distribution, which generalizes a special case of Krishnaiah*s (1964,1965) bivariate central F distribution. Explicit formulae for the cdf's of these distribu¬tions are given, along with computational procedures     

Parameter Estimation Through Weighted Least-Squares Rank Regression with Specific Reference to the Weibull and Gumbel Distributions

Probability plots are often used to estimate the parameters of distributions. Using large sample properties of the empirical distribution function and order statistics, weights to stabilize the variance in order to perform weighted least squares regression are derived. Weighted least squares regression is then applied to the estimation of the parameters of the Weibull, and the Gumbel distribution. The weights are independent of the parameters of the distributions considered. Monte Carlo simulation shows that the weighted least-squares estimators outperform the usual least-squares estimators totally, especially in small samples.     

Random weighting estimation of sampling distributions via importance resampling

This paper presents a new random weighting-based adaptive importance resampling method to estimate the sampling distribution of a statistic. A random weighting-based cross-entropy procedure is developed to iteratively calculate the optimal resampling probability weights by minimizing the Kullback-Leibler distance between the optimal importance resampling distribution and a family of parameterized distributions. Subsequently, the random weighting estimation of the sampling distribution is constructed from the obtained optimal importance resampling distribution. The convergence of the proposed method is rigorously proved. Simulation and experimental results demonstrate that the proposed method can effectively estimate the sampling distribution of a statistic.     

Censored Kullback-Leibler Information and Goodness-of-Fit Test with Type II Censored Data

The Kulback-Leibler information has been considered for establishing goodness-of-fit test statistics, which have been shown to perform very well (Arizono & Ohta, 1989; Ebrahimi et al., 1992, etc). In this paper, we propose censored Kullback-Leibler information to generalize the discussion of the Kullback-Leibler information to the censored case. Then we establish a goodness-of-fit test statistic based on the censored Kullback-Leibler information with the type 2 censored data, and compare the test statistics with some existing test statistics for the exponential and normal distributions.     

Relationship between the weighted distributions and some inequality measures

In this paper, we study the relationships between the weighted distributions and the parent distributions in the context of Lorenz curve, Lorenz ordering and inequality measures. These relationships depend on the nature of the weight functions and give rise to interesting connections. The properties of weighted distributions for general weight functions are also investigated. It is shown how to derive and to determine characterizations related to Lorenz curve and other inequality measures for the cases weight functions are increasing or decreasing. Some of the results are applied for special cases of the weighted distributions. We represent the reliability measures of weighted distributions by the inequality measures to obtain some results. Length-biased and equilibrium distributions have been discussed as weighted distributions in the reliability context by concentration curves. We also review and extend the problem of stochastic orderings and aging classes under weighting. Finally, the relationships between the weighted distribution and transformations are discussed.     

Test of fit for Marshall–Olkin distributions with applications

Marshall and Olkin [1967. A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44], introduced a bivariate distribution with exponential marginals, which generalizes the simple case of a bivariate random variable with independent exponential components. The distribution is popular under the name ‘Marshall–Olkin distribution’, and has been extended to the multivariate case. L2-type statistics are constructed for testing the composite null hypothesis of the Marshall–Olkin distribution with unspecified parameters. The test statistics utilize the empirical Laplace transform with consistently estimated parameters. Asymptotic properties pertaining to the null distribution of the test statistic and the consistency of the test are investigated. Theoretical results are accompanied by a simulation study, and real-data applications.     

Asymptotic biases of information and cross-validation criteria under canonical parametrization

An asymptotic expansion of the cross-validation criterion (CVC) using the Kullback-Leibler distance is derived when the leave-k-out method is used and when parameters are estimated by the weighted score method. By this expansion, the asymptotic bias of the Takeuchi information criterion (TIC) is derived as well as that of the CVC. Under canonical parametrization in the exponential family of distributions when maximum likelihood estimation is used, the magnitudes of the asymptotic biases of the Akaike information criterion (AIC) and CVC are shown to be smaller than that of the TIC. Examples in typical statistical distributions are shown.     

Adaptive methods for sequential importance sampling with application to state space models

In this paper we discuss new adaptive proposal strategies for sequential Monte Carlo algorithms—also known as particle filters—relying on criteria evaluating the quality of the proposed particles. The choice of the proposal distribution is a major concern and can dramatically influence the quality of the estimates. Thus, we show how the long-used coefficient of variation (suggested by Kong et al. in J. Am. Stat. Assoc. 89(278–288):590–599, 1994) of the weights can be used for estimating the chi-square distance between the target and instrumental distributions of the auxiliary particle filter. As a by-product of this analysis we obtain an auxiliary adjustment multiplier weight type for which this chi-square distance is minimal. Moreover, we establish an empirical estimate of linear complexity of the Kullback-Leibler divergence between the involved distributions. Guided by these results, we discuss adaptive designing of the particle filter proposal distribution and illustrate the methods on a numerical example. This work was partly supported by the National Research Agency (ANR) under the program “ANR-05-BLAN-0299”.     

Minimum and Maximum Information Censoring Plans in Progressive Censoring

Expressions for the entropy, the Kullback-Leibler information, and the I _α-information are established for distributions of progressively Type-II censored order statistics. These results are used to identify minimum and maximum information censoring plans. In particular, we find minimum and maximum entropy plans for DFR, exponential, Pareto, reflected power, and Weibull distributions. The results for Kullback-Leibler and I _α-information hold for any continuous distribution.     

A nonparametric assessment of model adequacy based on Kullback-Leibler divergence

A discrepancy measure to assess model fitness against a nonparametric alternative is proposed. First, a Polya tree prior is constructed so that the centering distribution is the null. Second, the prior is updated in the light of data to obtain the posterior centering distribution as the alternative. Third, a Kullback-Leibler divergence type of test statistic is derived to assess the discrepancy between the two centering distributions. The properties of the test statistic are derived, and a power comparison with several well-known test statistics is conducted. The use of the test statistic is illustrated using network traffic data.     

A maximum likelihood prediction function for the linear model with consistency results

An alternative technique to current methods for constructing a prediction function for the normal linear regression model is proposed based on the concept of maximum likelihood. The form of this prediction function is evaluated and normalized to produce a multivariate Student's t-density. Consistency properties are established under regularity conditions, and an empirical comparison, based on the Kullback-Leibler information divergence, is made with some other prediction functions.     

On some properties of bivariate weighted distributions

Let X^w and Y^w be weighted random variables arising from the distribution of (X,Y). We explore implications of independence of X and Y on the dependence structure of (X^w, Y^w). We also show that when X and Y are independent and the weight function is symmetric, identical distribution of X^w and Y^w implies that of X and Y. We discuss application of these results to the study of a renewal process.