On the Fisher information in record values

In this article, we establish some general results concerning the comparison of the amount of the Fisher information contained in n record values with the Fisher information contained in n iid observations from the original distribution. Some common distributions are classified according to this criterion. We also propose some methods of estimation based on record values. The results may be of interest in some life testing problems. Received: September 1999     

Recurrence relations for moment and conditional moment generating functions of generalized order statistics

In this paper, we obtain recurrence relations for moment and conditional moment generating functions of generalized order statistics (gos) based on random samples drawn from a population whose distribution is a member of a doubly truncated class of distributions denoted by . Members of the class are characterized in Section (2) based on recurrence relations for moment generating functions (moments) of gos. In Section (3), we shall characterize members of the class based on recurrence relations for conditional moment generating functions (conditional moments) of gos. These results are specialized to the left, right and non-truncated cases. Ordinary order statistics and ordinary record values are also obtained as special cases of the gos. Characterizations of some members of class such as the Weibull, compound Weibull, Pareto, power function (beta is a special case), Gompertz and compound Gompertz distributions are given as illustrative examples.     

On characterizations of distributions by expected values of order statistics and record values with gap

In this paper, we obtain some recurrence relationships for conditional expectations of nonadjacent order statistics and record values when the distribution function is absolutely continuous, and we prove that the distribution function is uniquely determined by the distribution of conditioned record values and by the expected values of these records. Further, different distributions are characterized by these relationships.     

On maximum likelihood prediction based on Type II doubly censored exponential data

In this paper, the maximum likelihood predictor (MLP) of the kth ordered observation, t _k, in a sample of size n from a two-parameter exponential distribution as well as the predictive maximum likelihood estimators (PMLE's) of the location and scale parameters, θ and β, based on the observed values t _r, …, t _s (1≤r≤s<k≤n), are obtained in closed forms, contrary to the belief they cannot be so expressed. When θ is known, however, the PMLE of β and MLP of t _k do not admit explicit expressions. It is shown here that they exist and are unique; sharp lower and upper bounds are also provided. The derived predictors and estimators are reasonable and also have good asymptotic properties. As applications, the total duration time in a life test and the failure time of a k-out-of-n system may be predicted. Finally, an illustrative example is included. Received: August 1999     

Fisher information in order statistics and their concomitants in bivariate censored samples

We evaluate the Fisher information (FI) contained in a collection of order statistics and their concomitants from a bivariate random sample. Special attention is given to Type II censored samples. We present a general decomposition result and recurrence relations that are useful in finding the FI in all types of censored samples. We also obtain some asymptotic results for the FI. For the bivariate normal parent, we obtain explicit and asymptotic expressions for the elements of the FI matrix for Type II censored samples. We discuss implications of our findings on inference on the bivariate normal parameters, especially on the correlation. The first author’s research was supported in part by National Institutes of Health, USA, Grant # M01 RR00034 and the second author’s research was supported by a training grant from the Egyptian government     

Characterizations of probability distributions via bivariate regression of record values

Bairamov et al. (Aust N Z J Stat 47:543–547, 2005) characterize the exponential distribution in terms of the regression of a function of a record value with its adjacent record values as covariates. We extend these results to the case of non-adjacent covariates. We also consider a more general setting involving monotone transformations. As special cases, we present characterizations involving weighted arithmetic, geometric, and harmonic means.     

On the independence of integer and fractional parts

We use the independence of the integer and fractional parts of exponentially distributed random variables to obtain expressions for the order statistics from a geometric distribution. As our main result we show that a strong form of this independence characterizes the exponential distribution.     

Decision theoretic estimation using record statistics

We consider the problem of estimating the scale parameter θ of the shifted exponential distribution with unknown shift based on a set of observed records drawn from a sequential sample of independent and identically distributed random variables. Under a large class of bowl-shaped loss functions, the best affine equivariant estimator (BAEE) of θ is shown to be inadmissible. Two dominating procedures are proposed. A numerical study is performed to show the extent of risk reduction that the improved estimators provide over the BAEE.     

Linearity of regression for non-adjacent order statistics

Let X ₁,X ₂,…,X _n be a random sample from a continuous distribution with the corresponding order statistics X _1:n≤X _2:n≤…≤X _n:n. All the distributions for which E(X _{k+r: n}|X _k:n)=a X _k:n+b are identified, which solves the problem stated in Ferguson (1967). Received February 1998     

Conditional distributions of generalized order statistics and some characterizations

Generalized order statistics have been introduced in Kamps (1995a). They enable a unified approach to several models of ordered random variables, e.g. (ordinary) order statistics, record values, sequential order statistics, record values from non-identical distributions. The purpose of this paper is to develop conditional distributions of one generalized order statistic given another and to characterize the underlying continuous distribution by different conditional expectations. Well-known results for ordinary order statistics and record values are extended to generalized order statistics. Received: July 1997     

Concomitants of Record Values Arising from Morgenstern Type Bivariate Logistic Distribution and Some of their Applications in Parameter Estimation

In this paper we have obtained the joint probability density function of concomitants of two record values and hence obtained an explicit expression for the product moment of concomitants of two record values arising from Morgenstern family of distributions. Appling this expression for the product moments of concomitants of record values we have derived the best linear unbiased estimators based on concomitants of record values of some parameters involved in Morgenstern type bivariate logistic distribution which is a subfamily of the Morgenstern family of distributions. The efficiencies of these estimators based on the first n concomitants of record values for n≤10 are also obtained.     

Random variables generated by ranks in dependent schemes

Some nonstationary sequences having independent vector of ranks and vector of order statistics are under consideration. We extend some characterizations in a class of independent r.v.'s to a class of Archimedean copula processes and construct the interpretation which gives us a simple way for simulating Archimedean copula processes. Received November 1997     

Analysis of Hybrid Life-tests in Presence of Competing Risks

The mixture of type-I and type-II censoring schemes, called the hybrid censoring scheme is quite common in life-testing or reliability experiments. In this paper, we consider the competing risks model in presence of hybrid censored data. Under this set up, it is assumed that the item may fail due to various causes and the corresponding lifetime distributions are independent and exponentially distributed with different scale parameters. We obtain the maximum likelihood estimators of the mean life of the different causes and derive their exact distributions. Using the exact distributions, all the moments can be obtained. Asymptotic confidence intervals and two bootstrap confidence intervals are also proposed. Bayes estimates and credible intervals of the unknown parameters are obtained under the assumptions of independent inverted gamma priors of the mean life of the different causes. Different methods have been compared using Monte Carlo simulations. Onereal data set has been analyzed for illustrative purposes. Part of the work was supported by a grant from the Natural Sciences and Engineering Research Council.     

Analysis of Instantaneous and Early Failures in Weibull Distribution

There are many situations in life testing experiment where an item fail instantaneously and hence the observed lifetime is reported as zero. The items that fail prematurely are called early failures. We propose a modified Weibull distribution as a suitable model to represent such situations by mixture of a singular distribution at zero and a two parameter Weibull distribution. We obtain the maximum likelihood estimates of the parameters and their asymptotic distributions. The methods are illustrated on drying of woods under different experiments and schedules reported by Vanmann (Research report, 1991:2).     

Healthcare information systems: data mining methods in the creation of a clinical recommender system

Recommender systems have been extensively studied to present items, such as movies, music and books that are likely of interest to the user. Researchers have indicated that integrated medical information systems are becoming an essential part of the modern healthcare systems. Such systems have evolved to an integrated enterprise-wide system. In particular, such systems are considered as a type of enterprise information systems or ERP system addressing healthcare industry sector needs. As part of efforts, nursing care plan recommender systems can provide clinical decision support, nursing education, clinical quality control, and serve as a complement to existing practice guidelines. We propose to use correlations among nursing diagnoses, outcomes and interventions to create a recommender system for constructing nursing care plans. In the current study, we used nursing diagnosis data to develop the methodology. Our system utilises a prefix-tree structure common in itemset mining to construct a ranked list of suggested care plan items based on previously-entered items. Unlike common commercial systems, our system makes sequential recommendations based on user interaction, modifying a ranked list of suggested items at each step in care plan construction. We rank items based on traditional association-rule measures such as support and confidence, as well as a novel measure that anticipates which selections might improve the quality of future rankings. Since the multi-step nature of our recommendations presents problems for traditional evaluation measures, we also present a new evaluation method based on average ranking position and use it to test the effectiveness of different recommendation strategies.     

Record values from generalized Pareto distribution and associated inference

In this paper, we derive exact explicit expressions for the single, double, triple and quadruple moments of the upper record values from a generalized Pareto distribution. We then use these expressions to compute the mean, variance, and the coefficients of skewness and kurtosis of certain linear functions of record values. Finally, we develop approximate confidence intervals for the location and scale parameters of the generalized Pareto distribution using the Edgeworth approximation and compare them with the intervals constructed through Monte Carlo simulations. Received: June 1999