Characterization theorems for some discrete distributions based on conditional structure

The joint distribution of (X,Y) is determined if the conditional expectation E {g(X)|Y = y} is given and the conditional distribution of Y|(X = x) is a conditional power series distribution, where g(·) is a function satisfying some minor conditions.     

A characterization based on conditional expectations

We obtain the necessary and sufficient conditions so that any real function (x) is the conditional expectation E(h(X)/X≥x) of a random variable X with continuous distribution function, where h is a given real, continuous and strictly monotonic function.     

Measures of biomarker dependence using a copula-based multivariate epsilon–skew–normal family of distributions

In this note we develop a new multivariate copula model based on epsilon–skew–normal marginal densities for the purpose of examining biomarker dependency structures. We illustrate the flexibility and utility of this model via a variety of graphical tools and a data analysis example pertaining to salivary biomarker. The multivariate normal model is a sub-model of the multivariate epsilon–skew–normal distribution.     

A new futility test approach in clinical interim data monitoring

Sequential monitoring of efficacy and safety data has become a vital component of modern clinical trials. It affords companies the opportunity to stop studies early in cases when it appears as if the primary objective will not be achieved or when there is clear evidence that the primary objective has already been met. This paper introduces a new concept of the backward conditional hypothesis test (BCHT) to evaluate clinical trial success. Unlike the regular conditional power approach that relies on the probability that the final study result will be statistically significant based on the current interim look, the BCHT was constructed based on the hypothesis test framework. The framework comprises a significant test level as opposed to the arbitrary fixed futility index utilized in the conditional power method. Additionally, the BCHT has proven to be a uniformly most powerful test. Noteworthy features of the BCHT method compared with the conditional power method will be presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Characterizations based on conditional expectations

For given real functionsg andh, first we give necessary and sufficient conditions such that there exists a random variableX satisfying thatE(g(X)|X≥y)=h(y)r _x (y),∀y ∈ C _x , whereC _x andT _X are the support and the failure rate function ofX, respectively. These extend the results of Ruiz and Navarro (1994) and Ghitany et al. (1995). Next we investigate necessary and sufficient conditions such thath(y)=E(g(X)|X≥y), for a given functionh. Support for this research was provided in part by the National Science Council of the Republic of China, Grant No. NSC 86-2115-M-110-014 and NSC 88-2118-M-110-001     

Excursion and meander in random walk

Bernoulli bridge, excursion and meander are defined on the symmetric random walk similarly to Brownian bridge, excursion and meander (cf. Chung 1976). Distributions of certain characteristics defined on these Bernoulli processes, which are of a combinatorial nature, and their limits are obtained. Using weak convergence, these derivations give a verification of some of the earlier results on Brownian excursion and Brownian meander, as well as some new results.     

位置参数变点的统计推断

关于随机变量序列中是否存在变点的问题一直不断地有人研究,对变点进行估计具有很重要的应用价值。为此,研究至多一个变点的位置参数变点问题,对变点是否存在进行假设检验,论证相关统计量的渐进性质,在此基础上提出变点的一种点估计,并证明估计的强相合性。     

Characterizations of discrete distributions based on conditional expectations of record values

r -th record values subject to (r + 1)-th record values, record mean function, from a distribution of discrete type. We give some properties of the record mean function and an explicit expression for the distribution function based on its record mean function, which allows us to characterize particular discrete distributions using the record mean functions. Received: January 4, 1999; revised version: September 27, 1999     

On conditional sampling strategies

Estimation of the population average by means of a conditional strategy has been considered e.g. in [2–6,9] and [10]. Let us assume that the sampling design depends on a function of an auxiliary variable called an auxiliary statistic like: the sample mean or the sample variance. Under the conditional versions of these designs several estimators of the population mean are considered: the Horvitz-Thompson statistic, ratio and regression type estimators. The unbiasedness and accuracy of the strategies are considered. The empirical example of the accuracy comparisons of strategies is developed. The conditional strategies could improve the accuracy of estimation and protect it against outliers observations.     

A NOTE ON SAMPLE QUANTILE PROCESSES OF FINITE POPULATIONS

A representation is established for sample quantile processes of simple random samples drawn without replacement from finite populations. The result is then used to obtain weak limits of properly normalized quantile processes and their linear functions.     

Nonparametric approach to rank global petroleum business opportunities

Crude oil continues to be one of the significant energy sources. Several countries do not have enough indigenous oil and gas resources. These countries resort to overseas business of Exploration and Production (E&P) of oil to secure a stable supply. Profitability, risk and growth guide overseas investment decisions. Selection of overseas investment opportunities are critical for a firm because of uncertainty in identifying and quantifying the attendant geological, commercial, social and political risks as well as return on investment. To secure overseas oil acreage, business entities intend to invest in overseas E&P destination having reasonable petroleum reserve, favorable contract terms (fiscal terms), well-developed infrastructure, sound legal system, minimum country risk (CR) (economic, social and political) and facilitate relative ease to do business in that country. The countries have varied mix of these parameters, and it leads to growing concern to screen and rank overseas investment opportunities. Methodologies to rank global opportunities should take into consideration the risk factors such as petroleum potential, infrastructure, geo-political scenario, contract terms, etc. We coin the term for the numerical rank as Globalization Index (GI), which is a function of the factors considered to affect the decision of a business entity in screening the global destinations for venturing in to E&P business of crude oil. This paper is an attempt to model these factors by invoking Alternating Conditional Expectation methodology to find GI.     

A month carlo approximation of the distributions of the maximum of various brownjan bridges

A Brownian bridge of order a is the weak limit of a residual partial sum obtained from regression fitting. When q=0, the process is teh usual Brownian bridge, and the distribution of the maximum is known analytically. For q>1, the reflection principle does not easily apply. The supremum distributions are approximated by a Monte Carlo technique. Tables of these distributions, as well as a finite sample size correction are given.     

CONDITIONAL EXPECTATION FORMULAE FOR COPULAS

Not only are copula functions joint distribution functions in their own right, they also provide a link between multivariate distributions and their lower‐dimensional marginal distributions. Copulas have a structure that allows us to characterize all possible multivariate distributions, and therefore they have the potential to be a very useful statistical tool. Although copulas can be traced back to 1959, there is still much scope for new results, as most of the early work was theoretical rather than practical. We focus on simple practical tools based on conditional expectation, because such tools are not widely available. When dealing with data sets in which the dependence throughout the sample is variable, we suggest that copula‐based regression curves may be more accurate predictors of specific outcomes than linear models. We derive simple conditional expectation formulae in terms of copulas and apply them to a combination of simulated and real data.     

Heuristic approach to some laws for Brownian motion

Csàki and Vincze have shown that for an elementary tied-down random walk, the pair (maximum, instant of maximum) has the same law as (time spent in (0, 1/2), time spent above 1/2). Formal passage to the limit indicates that the former pair has for a Brownian bridge the same law as (local time at 0, duration of positivity). A quadrivariate density of Karatzas and Shreve and an equivalence for Brownian motion with drift follow.     

Uniform asymptotic linearity in a regression parameter of a process based on a rank statistic

For X₁, …, X_N a random sample from a distribution F, let the process S^δ_N(t) be defined as where K²_N = σ^N_i=1(c_i ? c?)² and R x_i, + Δd, is the rank of X_i + Δd_i, among X₁ + Δd₁, …, X_N + Δd_N. The purpose of this note is to prove that, under certain regularity conditions on F and on the constants c_i and d_i, S^Δ_N (t) is asymptotically approximately a linear function of Δ, uniformly in t and in Δ, |Δ| ≤ C. The special case of two samples is considered.     

Tabulating weighted sup-norm functionals used in change-point analysis

We introduce a number of weighted partial sum processes of which certain sup-norm functionals may be used, for example, to detect changes in the mean of independent observations. Their limiting distributions are derived mathematically, simulated and then tabulated. With this information, a detailed numerical investigation of the power of these functionals is carried out.* ^*Research mainly done while at Carleton University, partially supported by an NSERC Canada Grant of Miklós Csörg? at Carleton University, Ottawa. Also supported by the Austrian Science Foundation (FWF) under grant SFB#010 (‘Adaptive Information Systems and Modelling in Economics and Management Science’) while at Technische Universität Wien. ? ^?Research partially supported by an NSERC Canada Grant of Miklós Csörg? at Carleton University, Ottawa.      

Functional principal component analysis in age–period–cohort analysis of body mass index data by gender and ethnicity

In this paper, we propose a two-stage functional principal component analysis method in age–period–cohort (APC) analysis. The first stage of the method considers the age–period effect with the fitted values treated as an offset; and the second stage of the method considers the residual age–cohort effect conditional on the already estimated age-period effect. An APC version of the model in functional data analysis provides an improved fit to the data, especially when the data are sparse and irregularly spaced. We demonstrate the effectiveness of the proposed method using body mass index data stratified by gender and ethnicity.