Dennis Gilliland

A Note on Confidence Interval Estimation and Margin of Error

Confidence interval estimation is a fundamental technique in statistical inference. Margin of error is used to delimit the error in estimation. Dispelling misinterpretations that teachers and students give to these terms is important. In this note, we give examples of the confusion that can arise in regard to confidence interval estimation and margin of error.

Integral of the Bivariate Normal Distribution over an Offset Circle

Abstract In problems where guidance to the proximity of a point is required, the probable success of the mission often is described by the probability of hitting within a given radius of the point. In this paper is presented a method of determining the probability of a hit within a given circle when it is assumed that the guidance error is distributed according to a bivariate normal distribution. This problem can be solved readily if the random error variables are independently distributed along two orthogonal axes with equal standard deviations and if they are not biased relative to the target location. Since these assumptions often are not valid, a series solution to the general problem is provided with an analysis of the error introduced by considering only a finite number of the terms.

Identification of the Ordered Bivariate Normal Distribution by Minimum Variate

Abstract For full-rank covariance matrix and other subfamilies of the ordered bivariate normal distribution, the identifiability of parameters by the minimum variate is demonstrated. For the general family, the distribution is not identified. These identifiability questions arise in connection with estimation within certain econometric models, for example, Tobin and Fair-Jaffee.

Asymptotic risk stability resulting from play against the past in a sequence of decision problems

When a given statistical decision problem occurs repeatedly, there is interest in the strategy that plays Bayes versus the empirical distribution of past parameter values. We point out some implications that some previously investigated continuity conditions have on the stability of average loss, risk, and average risk across the sequence of decision problems. This is done first for arbitrary parameter sequences and then in the context of independent and identically distributed parameters.

Quantitative Literacy at Michigan State University, 2: Connection to Financial Literacy

The lack of capability of making financial decisions has been recently described for the adult United States population. A concerted effort to increase awareness of this crisis, to improve education in quantitative and financial literacy, and to simplify financial decision-making processes is critical to the solution. This paper describes a study that was undertaken to explore the relationship between quantitative literacy and financial literacy for entering college freshmen. In summer 2010, incoming freshmen to Michigan State University were assessed. Well-tested financial literacy items and validated quantitative literacy assessment instruments were administered to 531 subjects. Logistic regression models were used to assess the relationship between level of financial literacy and independent variables including quantitative literacy score, ACT mathematics score, and demographic variables including gender. The study establishes a strong positive association between quantitative literacy and financial literacy on top of the effects of the other independent variables. Adding one percent to the performance on a quantitative literacy assessment changes the odds for being at the highest level of financial literacy by a factor estimated to be 1.05. Gender is found to have a large, statistically significant effect as well with being female changing the odds by a factor estimated to be 0.49.

Quantitative Literacy at Michigan State University, 1: Development and Initial Evaluation of the Assessment

Development, psychometric testing, and the results of the administration of a quantitative literacy (QL) assessment to undergraduate students are described. Three forms were developed covering a wide range of skills, contexts, and quantitative information presentation formats. Following item generation and revision based on preliminary testing and cognitive interviewing, a total of 3,701 consented undergraduate students at Michigan State University completed one of the three forms. Two of the forms contained 14 multiple-choice items, and one form contained 17 multiple-choice items. All forms were completed by students in less than 30 minutes. Evidence of validity and reliability were obtained for the three forms. Unidimensionality of the underlying construct was established using confirmatory factor analysis. Correlations with ACT and university mathematics placement test ranged from .41 to .67, and correlations with the Lipkus numeracy scale ranged from .40 to .45. Cronbach’s alphas for the three forms were near or exceeded .70. Comparison of student QL performance according to demographic characteristics revealed gender differences, with males scoring higher than females. These gender differences persisted even after controlling for ACT composite scores. Race/ethnicity differences were significant in unadjusted analysis, but did not persist over and above ACT composite scores in the adjusted analyses. The three newly developed forms of QL assessment will need to be further tested in the future to determine if they capture the effects of interventions that aim to improve

Bounds for robust maximum likelihood and posterior consistency in compound mixture state experiments

Uniform bounds on rates of L1-consistency for the empiric mean of state sequences in a family of probability models (compound with finite-mixture-state component) are obtained for MLEs (Section sec2) and posterior means for quasi-uniform hyperpriors (Section sec3), both determined in the iid mixture (empirical Bayes) sub-models. Qualitative aspects of results of this type were described by Robbins (1951). Application to the Gilliland and Hannan (1974/86) restricted-risk-finite-state-component compound decision problem (Section sec4) yields uniform bounds on rates of asymptotic regret of Bayes solutions therein (with extension to mixture-state by expectation), giving strong affirmation to an asymptotic form of a Robbins (1951) conjecture. The general extension to mixture-state components (Remark rem4.1) strengthens much of the existing compound literature.

On empirical Bayes with sequential component

Laippala (1979, Scand. J. Statist., 6, 113–118, correction note, 7, 105; 1985, Ann. Inst. Statist. Math., 37, 315–327) has defined a concept within the empirical Bayes framework that he calls “floating optimal sample size”. We examine this concept and show that it is one of many possibilities resulting from restricting the class of component sampling procedures in the empirical Bayes decision problem with a sequential component. All ideas are illustrated with the finite state component.

Votes and a half-binomial

Abstract In two party elections with popular vote ratio p q , 1 2 ≤p=1 −q , a theoretical model suggests replacing the so-called MacMahon cube law approximation ( p q ) 3 , for the ratio P Q of candidates elected, by the ratio ƒ k (p) ƒ k (q) of the two half sums in the binomial expansion of (p+q)2k+1 for some k. This ratio is nearly ( p q ) 3 when k = 6. The success probability g k (p)=( p a (p a +q a ) for the power law ( p q ) a ≐ P Q is shown to so closely approximate ƒ k (p)=Σ 0 k ( r 2k+1 )p 2k+1−r q r , if we choose a = a k = (2k+1)! 4 k k!k! , that 1≤ ƒ k (p) g k (p) ≤1.01884086 for k≥1 if 1 2 ≤p≤1 . Computationally, we avoid large binomial coefficients in computing ƒ k (p) for k>22 by expressing 2ƒ k (p)−1 as the sum (p−q) Σ 0 k (4pq) s a s (2s+1) , whose terms decrease by the factors (4pq)(1− 1 2s ) . Setting K = 4k+3, we compute ak for the large k using a continued fraction πa k 2 = K+1 2 ( 2K+3 2 ( 2K+5 2 (2K+…) ) ) derived from the ratio of π to the finite Wallis product approximation.

Using Randomized Confidence Limits to Balance Risk: An Application to Medicare Investigations

Consider a population of N payments by Medicare to a health care provider, each payment for