Hypothesis testing and parameter estimation based onM-statistics ink samples with unequal variances

Summary Ink samples with unequal variances,M-tests for homogeneity ofk location parameters are proposed. The asymptoticχ ²-distributions of the test statistics and the robustness of the tests are investigated. NextM-estimators (ME’s) of parameters are discussed. Furthermore positive-part shrinkage versions (PSME’s) of theM-estimators for the location parameters are considered along with modified James-Stein estimation rule. In asymptotic distributional risks based on a special feasible loss, it is shown that the PSME’s dominate the ME’s, and preliminary test and shrinkageM-versions fork≧4.     

On the distribution of city sizes

The city size distribution in many countries is remarkably well described by a Pareto distribution. We derive conditions that standard urban models must satisfy in order to explain this regularity. We show that under general conditions urban models must have (i) a balanced growth path and (ii) a Pareto distribution for the underlying source of randomness. In particular, one of the following combinations can induce a Pareto distribution of city sizes: (i) preferences for different goods follow reflected random walks, and the elasticity of substitution between goods is 1; or (ii) total factor productivities of different goods follow reflected random walks, and increasing returns are equal across goods.     

On two properties of an unequal probability sampling scheme

Summary For an inclusion probability proportional to size (IPPS) sampling scheme recently proposed by Saxena, Singh and Srivastava (1986), it is shown that under certain simple verifiable conditions (1) the Horvitz-Thompson (1952) estimator based on it has a smaller variance than the variance of the Hansen-Hurwitz (1943) estimator based on probability proportional to size (PPS) sampling with replacement (WR) both involving the same size-measures and the expected sample size in the former being equal to the number of draws in the latter and (2) the Yates-Grundy (1953) estimator for the variance of the Horvitz-Thompson estimator based on this IPPS scheme is uniformly non-negative.     

A two-stage sampling by unequal size samples from two normal populations with unknown variances

A two-stage procedure based on the statistic introduced by Scheffé (1943) to solve the Behrens-Fisher problem is considered in estimating the difference between the mean values of two normal distributions having unequal and unknown variances. Samples of unequal size are considered at each stage. A numerical example is given where the two-stage procedure is compared with the Satterthwaite (1946) approximate method.     

Variance estimation for unequal probability sampling

In this paper, we discuss the optimality of the variance estimator of the Horvitz-Thompson estimator proposed by Kott (1988) in the class of model-unbiased quadratic estimators. We also propose some improved estimators over Kott's estimator in the class of general quadratic estimators. Received: February 1999     

M-estimation in the presence of unequal scale

Abstract  The class of weighted M-estimators is defined. The ratio of the asymptotic variance of the weighted estimator to the asymptotic variance of the optimally weighted estimator is defined as the inefficiency. A K antorovich inequality is proved, its implications are investigated for the misweighted mean and misweighted median, and the results are applied to a batch of demographic data.     

Unequal jobs or unequal pay?

This article draws on new data to consider whether women would benefit most from upgrading the skill levels of their jobs or from achieving equal pay for work of equal value. It looks in detail at skill, gender, and whether work is full-time or part-time, concluding that reassessment of the value of women's jobs would be of greater immediate financial benefit to women than upgrading the skill level of their jobs unless this upgrading gave them access to men's jobs at men's pay and benefit rates.     

Model-based variance estimation under unequal probability sampling

This paper deals with a model-based variance estimation of the Horvitz–Thompson (HT) estimator when auxiliary information is available. A small simulation study is carried out to illustrate and establish some of the findings.     

Scheduling independent jobs on nonuniform, unequal processors

The problem of scheduling n independent jobs on a single processor to minimize the total tardiness of the assignment has attracted much attention. Solution algorithms, both exact and approximate, have been reported, but no polynomial time exact algorithm has yet been found, nor has the problem been proven NP-complete.In this paper we consider the more general case of scheduling n independent jobs on m unequal processors to minimize total tardiness. Since this problem is more complex than the corresponding single-processor problem, no polynomial-time algorithm is in sight. For problems of this nature, approximate algorithms may be more valuable than exact algorithms in terms of applications. A heuristic algorithm is developed to solve the multiple-processor problem. Computational experiments show that the heuristic algorithm is efficient, fast, and easy to implement.     

On a test of homogeneity for samples from a negative binomial distribution

Summary An exact test is proposed for the equality of probabilities of an event in a series of independent trials, each drawn from a negative binomial distribution with a specified number of successes. The power function of this conditional test is also derived, as well as an approximate test for homogeneity.     

Estimation from two-stage unequal probability sampling with missing units

A formula is presented for an unbiased estimator for the variance of an unbiased estimator of a survey population total as well as for an unbiased estimator of its variance based on sampling in two-stages following Rao et al. J Roy Stat Soc B 24: 482–491 (1962) scheme in both stages when the originally selected units in both stages cannot be fully covered in the survey but are to be randomly sub-sampled. The development is helpful to tackle non-responses if assumed to have occurred at random in either or both the stages     

An indirect test of efficient city sizes

This paper develops a model which demonstrates necessary conditions for efficient distribution of labor across areas given the presence of agglomeration economies and congestion externalities. An indirect empirical test for efficient allocation is formulated based on a comparison of estimated agglomeration economies with the compensating variation in occupation-specific wages needed to attract workers to larger cities. The results of this test suggest that there are specific city size ranges where necessary conditions for efficient allocation of resources are not met, particularly for cities in the 1.5 to 2.5 million population range. Current claims that continued growth of the largest U. S. cities is inefficient are not confirmed by the empirical results.     

Repeated two-person zero-sum games with unequal discounting and private monitoring

We consider discounted repeated two-person zero-sum games with private monitoring. We show that even when players have different and time-varying discount factors, each player’s payoff is equal to his stage-game minmax payoff in every sequential equilibrium. Furthermore, we show that: (a) in every history on the equilibrium path, the pair formed by each player’s conjecture about his opponent’s action must be a Nash equilibrium of the stage game, and (b) the distribution of action profiles in every period is a correlated equilibrium of the stage game. In the particular case of public strategies in public monitoring games, players must play a Nash equilibrium after any public history.     

Multiproject scheduling: Analysis of scheduling strategies under unequal delay penalties

The multiproject scheduling problem is investigated under the assumption that delays corresponding to different projects carry different penalties. Five penalty functions are introduced that simulate typical business behavior. These are: 1) assigning the highest penalty to the project requiring the greatest amount of resources; 2) assigning the highest priority to the longest project; 3) assigning the highest priority to the project requiring the least amount of resources; 4) assigning the highest priority to the shortest project; and 5) random assignment. Justification for each case is provided.Two powerful project summary measures were used to generate the test problems. The first measure, average resource load factor (ARLF), identifies whether the location of the peak requirement of all (i.e., combined) resources is in the first or second half of the project's critical path. If each resource is expressed in terms of dollars, then ARLF identifies the location of the peak cash requirement. Since in practice the choice of the scheduling strategy is influenced by the location of the peak cash requirement, performance of the scheduling strategies is analyzed with respect to values of this measure. The second measure, average utilization factor (AUF), calculates the ratio of resource requirements to availabilities. It is shown that problems tested must have different AUF values in order to be classified different for purposes of experimental design. Thus a factorial model of the form, y = Strategy + Penalty + ARLF + AUF + ? was used to generate 385 problems, each requiring 2 to 4 resources, containing 3 to 5 projects and 34 to 63 activities.The computer program used in the study is based on a parallel method of scheduling in which priorities of the activities are determined when the activity is considered for scheduling. Ten scheduling strategies are tested. Some important ones are: 1) to schedule the shortest activity first; 2) to schedule the activity with minimum slack first; 3) to schedule the activity with maximum work content first (i.e., from the project with highest work content); 4) to schedule the shortest activity from the shortest project first; and 5) to schedule the activity with maximum penalty first. Shortest activity first and minimum slack first are popular strategies introduced by other researchers.Using number of times ranked first as our criterion, scheduling the activity with maximum penalty first provides the best results followed by the strategy of scheduling the activity with maximum work content. But when these results are analyzed with respect to existence of a very expensive project (i.e., dominance) in the problem, performance of the maximum penalty strategy improves. When the overall results are analyzed with respect to values of ARLF, a different picture emerges. Then the strategy of scheduling the activity with the highest penalty first provides the best results if the peak requirement is early. When the peak requirement is toward the middle of a project's unconstrained critical path, scheduling the activity with the highest work content provides the best results. When the peak requirement is late in a project's life, scheduling the shortest activity from the shortest project is the best strategy to adopt. These findings were tested for statistical significance by using nonparametric testing procedures and were found to be significant.