Search

Let S 1 , S 2 , … , S k be survival functions of life distributions. They are said to be uniformly stochastically ordered, S 1 ≤ u s o S 2 ≤ u s o ⋯ ≤ u s o S k , if S i / S i + 1 is a survival function for 1 ≤ i ≤ k − 1 . The nonparametric maximum likelihood estimators of the survival functions subject to this ordering constraint are known to be inconsistent in general. Consistent estimators were developed only for the case of k = 2 . In this paper we provide consistent estimators in the k -sample case, with and without censoring. In proving consistency, we needed to develop a new algorithm for isotonic regression that may be of independent interest. [ABSTRACT FROM AUTHOR]

The mean residual life of a life distribution,X, with a finite mean is defined byM(t) =E[X−tX>t] fort⩾ 0. Kochar et al. (2000) provided an estimator ofMwhen it is assumed to be decreasing. They showed that its asymptotic distribution was the same as that of the empirical estimate, but only under very stringent analytic and distributional assumptions. We provide a more general asymptotic theory, and under much weaker conditions. We also provide improved asymptotic confidence bands. [ABSTRACT FROM AUTHOR]

Abstract: The peakedness of a random variable (RV) about a point is defined by . A RV is said to be less peaked about than a RV about , denoted by , if for all , i.e., is stochastically larger than . These generalize the original definitions of Birnbaum (1948) who considered the cases where and were symmetric about and , respectively. Statistical inferences about the distribution functions of continuous and under peakedness ordering in the symmetric case have been treated in the literature. Rojo et al. (2007) provided estimators of the distributions in the general case and analyzed their properties. We show that these estimators could have poor asymptotic properties relative to those of the empiricals. We provide improved estimators of the DFs, show that they are consistent, derive the weak convergence of the estimators, compare them with the empirical estimators, and provide formulas for statistical inferences. An example is also used to illustrate our theoretical results. [Copyright &y& Elsevier]

Abstract: There are many ways to measure the dispersion of a random variable. One such method uses the concept of peakedness. If the random variable is symmetric about a point , then Birnbaum [Z.W. Birnbaum, On random variables with comparable peakedness, The Annals of Mathematical Statistics 19 (1948) 76–81] defined the function , as the peakedness of . If two random variables, and , are symmetric about the points and , respectively, then is said to be less peaked than , denoted by , if for all , i.e.,  is stochastically larger than . For normal distributions this is equivalent to variance ordering. Peakedness ordering can be generalized to the case where and are arbitrary points. However, in this paper we study the comparison of dispersions in two continuous random variables, symmetric about their respective medians, using the peakedness concept where normality, and even moment assumptions are not necessary. We provide estimators of the distribution functions under the restriction of symmetry and peakedness ordering, show that they are consistent, derive the weak convergence of the estimators, compare them with the empirical estimators, and provide formulas for statistical inferences. An example is given to illustrate the theoretical results. [Copyright &y& Elsevier]

If X1 and X2 are random variables with distribution functions F1 and F2, then X1 is said to be stochastically larger than X2 if F1 ≤ F2. Statistical inferences under stochastic ordering for the two-sample case has a long and rich history. In this article we consider the k-sample case; that is, we have k populations with distribution functions F1, F2,..., Fk, k ≥ 2, and we assume that F1 ≤ F2 ≤,...,≤ Fk. For k = 2, the nonparametric maximum likelihood estimators of F1 and F2 under this order restriction have been known for a long time; their asymptotic distributions have been derived only recently. These results have very complicated forms and are hard to deal with when making statistical inferences. We provide simple estimators when k ≥ 2. These are strongly uniformly consistent, and their asymptotic distributions have simple forms. If &Fcirc;i and &Fcirci* are the empirical and our restricted estimators of Fi, then we show that, asymptotically, P(\ &Fcirci* (x) - Fi (x)\ ≤ u) &ge P (\ &Fcirc;i (x) \ ≤ u) for all x and all u > 0, with strict inequality in some cases. This clearly shows a uniform improvement of the restricted estimator over the unrestricted one. We consider simultaneous confidence bands and a test of hypothesis of homogeneity against the stochastic ordering of the k distributions. The results have also been extended to the case of censored observations. Examples of application to real life data are provided. [ABSTRACT FROM AUTHOR]

Presents a study that examined an algorithm which utilizes elementary methods and requires no knowledge of linear programming. Information on the tests for independence or homogeneity in categorical data analysis; Development of convex polytopes; Background on the Bland's rule or the least subscript rule.

Regression analysis constitutes a large portion of the statistical repertoire in applications. In cases where such analysis is used for exploratory purposes with no previous knowledge of the structure, one would not wish to impose any constraints on the problem. But in many applications we are interested in curve fitting with a simple parametric model to describe the structure of a system with some prior knowledge about the structure. An important example of this occurs when the experimenter has a strong belief that the regression function changes monotonically with some or all of the predictor variables in a region of interest. The analyses needed for statistical inferences under such constraints are nonstandard. Considering the present body of knowledge developed for unconstrained regression, it will be an enormous task to derive the analogs of even a small fraction of this for the restricted case. In this article we initiate the study with simple linear regression on a single variable. The estimators of the regression parameters may be intuitively obvious in this case, but, as discussed, very little else is. [ABSTRACT FROM AUTHOR]

A problem frequently encountered in the practice of statistics is the comparison of several treatments with a control or standard. We consider an experimental situation where prior knowledge indicates that all of the treatments are at least as effective as the control and the problem is to determine if any are significantly better than the control. A number of statistical procedures have been proposed for this situation, of which the best known is Dunnett's (1955) multiple comparison procedure. Dunnett's test rejects equality of the treatments and the control for a large value of the maximum contrast of the data vector with several vectors that are located "symmetrically" within the alternative region. We study a large class of such tests, which includes Dunnett's test as a particular case. One of these tests, which is based on the maximum contrast of the data vector with several orthogonal vectors, is very easy to implement and has an uncomplicated, good, and fairly uniform power function over the entire alternative region. In fact, a small Monte Carlo power study suggests that this orthogonal contrast test is approximately "maximin" within this class of tests. Moreover, the simplicity of the power function of the orthogonal contrast test enables the experimenter to determine sample sizes for designed experiments with specific power characteristics. Such sample size determinations can be difficult, if not impossible, using other procedures. Abelson and Tukey (1963) suggested tests for a large class of restricted problems that are based on contrasts of the data vector with a single vector that is "centrally" located within the alternative region. These single contrast tests have the advantage that their distributions under the null hypotheses (a t distribution) and under the alternative (a... [ABSTRACT FROM AUTHOR]

A life distribution function F is said to have an increasing failure rate average if H (x) ∕ x is nondecreasing where H (x) is the corresponding cumulative hazard function. In this paper we provide a uniformly strongly consistent estimator of F and derive the convergence in distribution of the estimator at a point where H (x) ∕ x is increasing using the arg max theorem. We also show using simulations that, unlike other estimators of the past, this new estimator outperforms the empirical distribution in terms of mean square error at all quantiles. An example is also discussed to illustrate the theoretical results • Uniformly consistent estimation of an IFRA distribution function. • Convergence in distribution of an estimate of an IFRA distribution. • Use of the arg max theorem for asymptotic distribution. [ABSTRACT FROM AUTHOR]

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X − t|X > t] for t ⩾ 0. One model of aging is the decreasing mean residual life (DMRL): M is decreasing (non increasing) in time. It vastly generalizes the more stringent model of increasing failure rate (IFR). The exponential distribution lies at the boundary of both of these classes. There is a large literature on testing exponentiality against DMRL alternatives which are all of the integral type. Because most parametric families of DMRL distributions are IFR, their relative merits have been compared only at some IFR alternatives. We introduce a new Kolmogorov-Smirnov type sup-test and derive its asymptotic properties. We compare the powers of this test with some integral tests by simulations using a class of DMRL, but not IFR alternatives, as well as some popular IFR alternatives. The results show that the sup-test is much more powerful than the integral tests in all cases. [ABSTRACT FROM AUTHOR]

A life distribution function (DF)Fis said to be star-shaped ifis nondecreasing on its support. This generalises the model of a convex DF, even allowing for jumps. The nonparametric maximum likelihood estimation is known to be inconsistent. We provide a uniformly strongly consistent least-squares estimator. We also derive the convergence in distribution of the estimator at a point whereis increasing using the arg max theorem. [ABSTRACT FROM AUTHOR]

The mean residual life of a non negative random variableXwith a finite mean is defined byM(t) =E[X−t|X>t] fort⩾ 0. A popular nonparametric model of aging is new better than used in expectation (NBUE), whenM(t) ⩽M(0) for allt⩾ 0. The exponential distribution lies at the boundary. There is a large literature on testing exponentiality against NBUE alternatives. However, comparisons of tests have been made only for alternatives much stronger than NBUE. We show that a new Kolmogorov-Smirnov type test is much more powerful than its competitors in most cases. [ABSTRACT FROM AUTHOR]

Abstract: In competing risks studies, the Kaplan–Meier estimators of the distribution functions (DFs) of lifetimes and the corresponding estimators of cumulative incidence functions (CIFs) are used widely when no prior information is available for these distributions. In some cases better estimators of the DFs of lifetimes are available when they obey some inequality constraints, e.g., if two lifetimes are stochastically or uniformly stochastically ordered, or some functional of a DF obeys an inequality in an empirical likelihood estimation procedure. If the restricted estimator of a lifetime differs from the unrestricted one, then the usual estimators of the CIFs will not add up to the lifetime estimator. In this paper we show how to estimate the CIFs in this case. These estimators are shown to be strongly uniformly consistent. In all cases we consider, when the inequality constraints are strict the asymptotic properties of the restricted and the unrestricted estimators are the same, thus providing the asymptotic properties of the restricted estimators essentially “free of charge”. We give an example to illustrate our procedure. [Copyright &y& Elsevier]

In the competing risks problem an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t for a particular type of failure in the presence of other risks. Its estimation and asymptotic distribution theory have been studied by many. In some cases there are reasons to believe that the CIFs due to two types of failure are order restricted. Several procedures have appeared in the literature for testing for such orders. In this paper we initiate the study of estimation of two CIFs subject to a type of stochastic ordering, both when there are just two causes of failure and when there are more than two causes of failure, treating those other than the two of interest as a censoring mechanism. We do not assume independence of the two types of failure of interest; however, these are assumed to be independent of the other causes in the censored case. Weak convergence results for the estimators have been derived. It is shown that when the order restriction is strict, the asymptotic distributions are the same as those for the empirical estimators without the order restriction. Thus we get the restricted estimators “free of charge”, at least in the asymptotic sense. When the two CIFs are equal, the asymptotic MSE is reduced by using the order restriction. For finite sample sizes simulations seem to indicate that the restricted estimators have uniformly smaller MSEs than the unrestricted ones in all cases. [Copyright &y& Elsevier]

If X is a life distribution with finite mean then its mean residual life function (MRLF) is defined by M(x)=E[X−x|X>x]. It has been found to be a very intuitive way of describing the aging process. Suppose that M1 and M2 are two MRLFs, e.g., those corresponding to the control and the experimental groups in a clinical trial. It may be reasonable to assume that the remaining life expectancy for the experimental group is higher than that of the control group at all times in the future, i.e., M1(x)⩽M2(x) for all x. Randomness of data will frequently show reversals of this order restriction in the empirical observations. In this paper we propose estimators of M1 and M2 subject to this order restriction. They are shown to be strongly uniformly consistent and asymptotically unbiased. We have also developed the weak convergence theory for these estimators. Simulations seem to indicate that, even when M1=M2, both of the restricted estimators improve on the empirical (unrestricted) estimators in terms of mean squared error, uniformly at all quantiles, and for a variety of distributions. [Copyright &y& Elsevier]