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 Title
 BAYESIAN SOLUTIONS TO SOME CLASSICAL PROBLEMS OF STATISTICS.
 Creator

PEREIRA, CARLOS ALBERTO DE BRAGANCA., Florida State University
 Abstract/Description

Three of the basic questions of Statistics may be stated as follows: (A) Which portion of the data X is actually informative about the parameter of interest (theta)? (B) How can all the relevant information about (theta) provided by the data X be extracted? (C) What kind of information about (theta) do the data X possess?, The perspective of this dissertation is that of a Bayesian., Chapter I is essentially concerned with question A. The theory of conditional independence is explained and the...
Show moreThree of the basic questions of Statistics may be stated as follows: (A) Which portion of the data X is actually informative about the parameter of interest (theta)? (B) How can all the relevant information about (theta) provided by the data X be extracted? (C) What kind of information about (theta) do the data X possess?, The perspective of this dissertation is that of a Bayesian., Chapter I is essentially concerned with question A. The theory of conditional independence is explained and the relations between ancillarity, sufficiency, and statistical independence are discussed in depth. Some related concepts like specific sufficiency, bounded completeness, and splitting sets are also studied in some details. The language of conditional independence is used in the remaining Chapters., Chapter II deals with question B for the particular problem of analysing categorical data with missing entries. It is demonstrated how a suitably chosen prior for the frequency parameters can streamline the analysis in the presence of missing entries due to nonresponse or other causes. The two cases where the data follow the Multinomial or the Multivariate Hypergeometric model are treated separately. In the first case it is adequate to restrict the prior (for the cell probabilities) to the class of Dirichlet distributions. In the Hypergeometric case it is convenient to select a prior (for the cell population frequencies) from the class of DirichletMultinomial (DM) distributions. The DM distributions are studied in detail., Chapter III is directly related to question C. Conditions on the likelihood function and on the prior distribution are presented in order to assess the effect of the sample on the posterior distribution. More specifically, it is shown that under certain conditions, the larger the observations obtained, the larger (stochastically in terms of the posterior distribution) is the appropriate parameter., Finally, Chapter IV deals with the characterization of distributions in terms of Blackwell comparison of experiments. It is shown that a result (for the Hypergeometric model) obtained in Chapter II is actually a consequence of a property of complete families of distributions.
Show less  Date Issued
 1980, 1980
 Identifier
 AAI8108380, 3084857, FSDT3084857, fsu:74358
 Format
 Document (PDF)
 Title
 THE COMPARISON OF SENSITIVITIES OF EXPERIMENTS (MAXIMUM LIKELIHOOD, RANDOM, FIXED, ANALYSIS OF VARIANCE).
 Creator

YOUNG, BARBARA NELSON., Florida State University
 Abstract/Description

The sensitivity of a measurement technique is defined to be its ability to detect differences among the treatments in a fixed effects design, or the presence of a between treatments component of variance in a random effects design. Consider an experiment, consisting of two identical subexperiments, designed specifically for the purpose of comparing two measurement techniques. It is assumed that the techniques of analysis of variance are applicable in analyzing the data obtained from the two...
Show moreThe sensitivity of a measurement technique is defined to be its ability to detect differences among the treatments in a fixed effects design, or the presence of a between treatments component of variance in a random effects design. Consider an experiment, consisting of two identical subexperiments, designed specifically for the purpose of comparing two measurement techniques. It is assumed that the techniques of analysis of variance are applicable in analyzing the data obtained from the two measurement techniques. The subexperiments may have either fixed or random treatment effects in either oneway or general block designs. It is assumed that the experiment yields bivariate observations from the two measurement methods which may or may not be independent. Likelihood ratio tests are used in the various settings of this dissertation to both extend current techniques and provide alternative methods for comparing the sensitivities of experiments.
Show less  Date Issued
 1985, 1985
 Identifier
 AAI8524629, 3086182, FSDT3086182, fsu:75665
 Format
 Document (PDF)
 Title
 DETERMINING A SUFFICIENT LEVEL OF INTERRATER RELIABILITY (POWER ANALYSIS, MISCLASSIFICATION, SAMPLE SIZE).
 Creator

RASP, JOHN M., Florida State University
 Abstract/Description

The reliability of a test or measurement procedure is, generally speaking, an index of the consistency of its results. Interrater reliability assesses the consistency of judgements among a set of raters. We model the observation taken on a subject by an unreliable procedure as the sum of a true score with mean (mu) and variance (sigma)(,T)('2) and an error term with mean 0 and variance (sigma)(,E)('2). The reliability coefficient then is (rho) = (sigma)(,T)('2)/((sigma)(,T)('2) + (sigma)(,E)...
Show moreThe reliability of a test or measurement procedure is, generally speaking, an index of the consistency of its results. Interrater reliability assesses the consistency of judgements among a set of raters. We model the observation taken on a subject by an unreliable procedure as the sum of a true score with mean (mu) and variance (sigma)(,T)('2) and an error term with mean 0 and variance (sigma)(,E)('2). The reliability coefficient then is (rho) = (sigma)(,T)('2)/((sigma)(,T)('2) + (sigma)(,E)('2))., The reliability of an instrument or rating procedure is generally evaluated in an initial experiment (or series of experiments) known as a "reliability study." Once an instrument is established as having some degree of reliability, it is then used as a measurement tool in subsequent research, known as "decision studies.", An unreliable procedure measures imperfectly. The impact of the error in measurement is investigated as it relates to three broad areas of statistical procedures: estimation, hypothesis testing, and decisionmaking., An unreliable measurement decreases the precision of estimates. The effect of an unreliable measurement on the width of a confidence interval for the population mean is examined. Also, an expression is developed to facilitate estimation of the reliability of a test or measurement in a decision study when the populations of interest may differ from those in the reliability study., An unreliable instrument weakens hypothesis tests. The extent to which lack of reliability attenuates the power of the twosample ttest, the Ftest in the analysis of variance, and the ttest for statistically significant correlation between two variables is investigated., An unreliable measurement engenders false classifications. A dichotomous decision is considered, and expressions for the probability of misclassifying a subject by a rating procedure with a given reliability are developed. Overall as well as directional misclassification rates are found under the model of true scores and errors distributed as independent normals. Effects of departures from this model, by heavytailed and skewed true score and error distributions, and by errors whose variance is a function of the true score, are considered. A general expression for this misclassification probability is found. A confidence interval for the misclassification probability is developed., These results provide tools for a researcher better to make decisions concerning the design of an experiment. They permit the costs of increased reliability to be more knowledgeably compared with the consequences of using an unreliable measurement procedure in a given situation.
Show less  Date Issued
 1984, 1984
 Identifier
 AAI8416723, 3085837, FSDT3085837, fsu:75324
 Format
 Document (PDF)
 Title
 ESTIMATING JOINTLY SYSTEM AND COMPONENT RELIABILITIES USING A MUTUAL CENSORSHIP APPROACH (SURVIVAL ANALYSIS, COUNTING PROCESSES, MARTINGALES, KAPLANMEIER, RELIABILITY FUNCTION).
 Creator

FREITAG, STEVEN ARTHUR., Florida State University
 Abstract/Description

Let F denote the life distribution of a coherent structure of independent components. Suppose that we have a sample of independent systems, each having the structure (phi). Each system is continuously observed until it fails. For every component in each system, either a failure time or a censoring time is recorded. A failure time is recorded if the component fails before or at the time of system failure; otherwise a censoring time is recorded. We introduce a method for finding estimates for F...
Show moreLet F denote the life distribution of a coherent structure of independent components. Suppose that we have a sample of independent systems, each having the structure (phi). Each system is continuously observed until it fails. For every component in each system, either a failure time or a censoring time is recorded. A failure time is recorded if the component fails before or at the time of system failure; otherwise a censoring time is recorded. We introduce a method for finding estimates for F(t), quantiles, and other functionals of F, based on the censorship of the component lives by system failure. We present limit theorems that enable the construction of confidence intervals for large samples.
Show less  Date Issued
 1986, 1986
 Identifier
 AAI8609671, 3086298, FSDT3086298, fsu:75781
 Format
 Document (PDF)
 Title
 ESTIMATING MULTIDIMENSIONAL TABLES FROM SURVEY DATA: PREDICTING MAGAZINE AUDIENCES.
 Creator

DANAHER, PETER JOSEPH., Florida State University
 Abstract/Description

Suppose an advertiser constructs an advertising campaign by placing k advertisements in a magazine. He now estimates the proportion of the population which sees none, one, or up to all k advertisements (called the exposure distribution). Several criteria for evaluating the effectiveness of the campaign can be obtained directly from the exposure distribution. Two of them are reach, the proportion of the population which is exposed to at least one of the advertisements and effective reach, the...
Show moreSuppose an advertiser constructs an advertising campaign by placing k advertisements in a magazine. He now estimates the proportion of the population which sees none, one, or up to all k advertisements (called the exposure distribution). Several criteria for evaluating the effectiveness of the campaign can be obtained directly from the exposure distribution. Two of them are reach, the proportion of the population which is exposed to at least one of the advertisements and effective reach, the mean of the exposure distribution., We develop three exposure distribution models for the cases where advertising campaigns are comprised of one, two, or three or more magazines. The models build on each other in that the model for one magazine is used to improve the fit of the model for two magazines and the model for two magazines is used to estimate the parameters of the model for three or more magazines., A thorough empirical test, using the AGB:McNair "National Media Survey", shows that each of our models outperforms the best currentlyavailable models. In addition, the three models are proved to have optimal asymptotic properties., The models are used to select a media schedule which maximizes either reach or effective reach subject to a budget constraint. A monotonicity property of reach and effective reach yields an algorithm for optimizing both reach and effective reach that greatly reduces computation time over conventional methods used to solve integer programming problems., It is more useful to estimate the proportion of the population which sees the advertisements in a magazine rather than the proportion which sees the magazine. Often, however, no advertisement recall data is available so we are forced to estimate the proportion which is exposed to just the magazines. If advertisement recall data is available we give a natural and simple adjustment of the original magazine exposure data to get advertisement exposure data. Our models also give an excellent fit to these adjusted exposure data.
Show less  Date Issued
 1987, 1987
 Identifier
 AAI8721837, 3086665, FSDT3086665, fsu:76140
 Format
 Document (PDF)
 Title
 ESTIMATION AND PREDICTION FOR EXPONENTIAL TIME SERIES MODELS.
 Creator

MOHAMED, FOUAD YEHIA., Florida State University
 Abstract/Description

This work is concerned with the study of stationary time series models in which the marginal distribution of the observations follows an exponential distribution. This is in contrast to the standard models in the literature where the error sequence and hence the marginal distributions of the o
 Date Issued
 1981, 1981
 Identifier
 AAI8205698, 3085176, FSDT3085176, fsu:74671
 Format
 Document (PDF)
 Title
 AN INCREASING FAILURE RATE APPROACH TO CONSERVATIVE LOW DOSE EXTRAPOLATION (SAFE DOSE).
 Creator

SCHELL, MICHAEL J., Florida State University
 Abstract/Description

This dissertation provides a new method of treating the conservative low dose extrapolation problem. One wishes to determine the largest dose d, called the "safe" dose, for which P(F(d) (LESSTHEQ) r) (GREATERTHEQ) 1  (eta) where F(d) is the proportion of failures, say cancers induced, at dose d by time T. F is a life distribution function, presumed to come from some class of functions F, T is prespecified, r () {0,1}, denotes the proportion of failures at doses (x,y) by fixed time T. Four...
Show moreThis dissertation provides a new method of treating the conservative low dose extrapolation problem. One wishes to determine the largest dose d, called the "safe" dose, for which P(F(d) (LESSTHEQ) r) (GREATERTHEQ) 1  (eta) where F(d) is the proportion of failures, say cancers induced, at dose d by time T. F is a life distribution function, presumed to come from some class of functions F, T is prespecified, r () {0,1}, denotes the proportion of failures at doses (x,y) by fixed time T. Four extensions of the univariate class of IFR functions are introduced, differing in the way that convexity of the hazard function, H(x,y) = ln(1F(x,y)) is posited. The notion of dependent action is considered and a hypothesis test for its existence given., Conservative low dose extrapolation techniques for the two most prominent classes are given. An upper bound for the hazard function is established for low doses with proofs that the bounds are sharp.
Show less  Date Issued
 1984, 1984
 Identifier
 AAI8427325, 3085936, FSDT3085936, fsu:75422
 Format
 Document (PDF)
 Title
 INFORMATION IN CENSORED MODELS.
 Creator

SCONING, JAMES., Florida State University
 Abstract/Description

Criteria are developed for measuring information in the randomly rightcensored model. Measures which are appropriate include an extension of Shannon's entropy. The measures are seen to satisfy some fundamental properties including (1) information decreases as censoring increases stochastically, (2) the uncensored case is always at least as informative as any censored model, and (3) the information gain is marginally decreasing., Measures of information in censored models can also be...
Show moreCriteria are developed for measuring information in the randomly rightcensored model. Measures which are appropriate include an extension of Shannon's entropy. The measures are seen to satisfy some fundamental properties including (1) information decreases as censoring increases stochastically, (2) the uncensored case is always at least as informative as any censored model, and (3) the information gain is marginally decreasing., Measures of information in censored models can also be developed by adapting measures of dependence between the lifetime variable and the observed variable. Some common notions of bivariate dependence enjoy property (1) cited above. An exception occurs when dependence is defined in terms of association. Conditions under which the coefficients of divergence satisfy (1) and (2) are established., Information is also studied in terms of asymptotic efficiency. We consider the proportional hazards model where the distribution G of the censoring random variable is related to the distribution F of the lifetime variable via (1G) = (1F)(beta). Nonparametric estimators of F are developed for the case where (beta) is unknown and the case where (beta) is known. Of interest in their own right, these estimators also enable us to study the robustness of the KaplanMeier estimator (KME) in a nonparametric model for which it is not the preferred estimator. Comparisons are based on asymptotic efficiencies and exact mean square errors. We also compare the KME to the empirical survival function thereby providing, in a nonparametric setting, a measure of the loss in efficiency due to censoring.
Show less  Date Issued
 1986, 1986
 Identifier
 AAI8605791, 3086279, FSDT3086279, fsu:75762
 Format
 Document (PDF)
 Title
 AN INVESTIGATION OF THE EFFECT OF THE SWAMPING PHENOMENON ON SEVERAL BLOCK PROCEDURES FOR MULTIPLE OUTLIERS IN UNIVARIATE SAMPLES.
 Creator

WOOLLEY, THOMAS WILLIAM, JR., Florida State University
 Abstract/Description

Statistical outliers have been an issue of concern to researchers for over two centuries, and are the focus of this study. Sources of outliers, and various means for dealing with them are discussed. Also presented are general descriptions of univariate outlier tests as well as the two approaches to handling multiple outlier situations, consecutive and block testing. The major problems inherent in these latter methods, masking and swamping, respectively, are recounted., Specifically, the...
Show moreStatistical outliers have been an issue of concern to researchers for over two centuries, and are the focus of this study. Sources of outliers, and various means for dealing with them are discussed. Also presented are general descriptions of univariate outlier tests as well as the two approaches to handling multiple outlier situations, consecutive and block testing. The major problems inherent in these latter methods, masking and swamping, respectively, are recounted., Specifically, the primary aim of this study is to assess the susceptibility to swamping of four block procedures for multiple outliers in univariate samples., Pseudorandom samples are generated from a unit normal distribution, and varying numbers of upper outliers are placed in them according to specified criteria. A swamping index is created which reflects the relative vulnerability of each test to declare a block of outliers and the most extreme upper nonoutlier discordant, as a unit., The results of this investigation reveal that the four block tests disagree in their respective susceptibilities to swamping depending upon sample size and the prespecified number of outliers assumed to be present. Rank orderings of these four tests based upon their vulnerability to swamping under varying circumstances are presented. In addition, alternate approaches to calculating the swamping index when four or more outliers exist are described., Recommendations concerning the appropriate application of the four block procedures under differing situations, and proposals for further research, are advanced.
Show less  Date Issued
 1981, 1981
 Identifier
 AAI8113272, 3084903, FSDT3084903, fsu:74401
 Format
 Document (PDF)
 Title
 LARGE DEVIATION LOCAL LIMIT THEOREMS, WITH APPLICATIONS.
 Creator

CHAGANTY, NARASINGA RAO., Florida State University
 Abstract/Description

Let {X(,n), n (GREATERTHEQ) 1} be a sequence of i.i.d. random variables withE(X(,1)) = 0, Var(X(,1)) = 1. Let (psi)(s) be the cumulant generating function (c.g.f.) and, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI), be the large deviation rate of X(,1). Let S(,n) = X(,1) + ... + X(,n). Under some mild conditions on (psi), Richter (Theory Prob. Appl. (1957) 2, 206219) showed that the probability density function f(,n) of(' )S(,n)/SQRT.(n has the asymptotic expression, (DIAGRAM, TABLE...
Show moreLet {X(,n), n (GREATERTHEQ) 1} be a sequence of i.i.d. random variables withE(X(,1)) = 0, Var(X(,1)) = 1. Let (psi)(s) be the cumulant generating function (c.g.f.) and, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI), be the large deviation rate of X(,1). Let S(,n) = X(,1) + ... + X(,n). Under some mild conditions on (psi), Richter (Theory Prob. Appl. (1957) 2, 206219) showed that the probability density function f(,n) of(' )S(,n)/SQRT.(n has the asymptotic expression, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI), whenever x(,n) = o(SQRT.(n) and SQRT.(n x(,n) > 1. In this dissertation we obtain similar large deviation local limit theorems for arbitrary sequences of random variables, not necessarily sums of i.i.d. random variables, thereby increasing the applicability of Richter's theorem. Let {T(,n), n (GREATERTHEQ) 1} be an arbitrary sequence of nonlattice random variables with characteristic function (c.f.) (phi)(,n). Let (psi)(,n), (gamma)(,n) be the c.g.f. and the large deviation rate of T(,n)/n. The main theorem in Chapter II shows that under some standard conditions on (psi)(,n), which imply that T(,n)/n converges to a constant in probability, the density function K(,n) of T(,n)/n has the asymptotic expression, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI), where m(,n) is any sequence of real numbers and (tau)(,n) is defined by(psi)(,n)'((tau)(,n)) = m(,n). When T(,n) is the sum of n i.i.d. random variables our result reduces to Richter's theorem. Similar theorems for lattice valued random variables are also presented which are useful in obtaining asymptotic probabilities for Wilcoxon signedrank test statistic and Kendall's tau., In Chapter III we use the results of Chapter II to obtain central limit theorem for sums of a triangular array of dependent random variables X(,j)('(n)), j = 1, ..., n with joint distribution given by z(,n)('1)exp{H(,n)(x(,1), ..., x(,n))}(PI)dP(x(,j)), where x(,i) (ELEM) R (FOR ALL) i (GREATERTHEQ) 1. The function H(,n)(x(,1), ..., x(,n)) is known as the Hamiltonian. Here P is a probability measure on R. When H(,n)(x(,1), ..., x(,n)) = log (phi)(,n)(s(,n)/n), where s(,n) = x(,1) + ... + x(,n) and the probability measure P satisfies appropriate conditions, we show that there exists an integer r (GREATERTHEQ) 1 and a sequence (tau)(,n) such that (S(,n)  n(tau)(,n))/n('1 1/2r) has a limiting distribution which is nonGaussian if r (GREATERTHEQ) 2. This result generalizes the theorems of JongWoo Jeon (Ph.D. Thesis, Dept. of Stat., F.S.U. (1979)) and Ellis and Newman (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete. (1978) 44, 117139). Chapters IV and V extend the above to the multivariate case.
Show less  Date Issued
 1982, 1982
 Identifier
 AAI8225279, 3085419, FSDT3085419, fsu:74914
 Format
 Document (PDF)
 Title
 LUMPABILITY AND WEAK LUMPABILITY IN FINITE MARKOV CHAINS.
 Creator

ABDELMONEIM, ATEF MOHAMED., Florida State University
 Abstract/Description

Consider a Markov chain x(t), t = 0, 1, 2, ..., with a finite state space, N = {1, 2, ..., n}, transition probability matrix P = (p(,ij)) i, j (epsilon) N, and an initial probability vector V = (v(,i)) i (epsilon) N. For m (LESSTHEQ) n let A = {A(,1), A(,2), ..., A(,m)} be a partition on the set N. Define the process, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI), The new process y(t), called a function of Markov chain, need not be Markov. If y(t) is again Markov, whatever the initial...
Show moreConsider a Markov chain x(t), t = 0, 1, 2, ..., with a finite state space, N = {1, 2, ..., n}, transition probability matrix P = (p(,ij)) i, j (epsilon) N, and an initial probability vector V = (v(,i)) i (epsilon) N. For m (LESSTHEQ) n let A = {A(,1), A(,2), ..., A(,m)} be a partition on the set N. Define the process, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI), The new process y(t), called a function of Markov chain, need not be Markov. If y(t) is again Markov, whatever the initial probability vector of x(t), x(t) is said to be lumped to y(t) with respect to the partition A. If y(t) is again Markov for only certain initial probability vectors of x(t), x(t) is said to be weakly lumped to y(t) with respect to the partition A., Conditions under which x(t) can be lumped or weakly lumped to y(t) with respect to A, are introduced. Relationships between the two processes x(t) and y(t) and the properties of the new process y(t) are discussed., Criteria are developed to determine whether a given Markov chain can be weakly lumped with respect to a given partition in terms of an analysis of systems of linear equations. Necessary and sufficient conditions on the transition probability matrix of a Markov chain, a partition, A, on N and a subset S of probability vectors for weak lumpability to occur are given in terms of the solution classes to these systems of linear equations. Finally, given that weak lumping occurs, the class S of all initial probability vectors which allow weak lumping is determined as is the transition probability matrix of the lumped process, y(t)., Lumpability and weak lumpability are also studied for Markov chains which are not irreducible. This involves a study of the interplay between two partitions of the state space N, the partition C, induced by the closed sets of states of the Markov chain and the partition A, with respect to which lumpability is to be considered. Under the assumptions that lumpability occurs the relationships which must exist between sets of the two partitions A and C are obtained in detail. It is found, for example that if neither partition is a refinement of the other and (A,C) form an irreducible pair of partitions over N then for each A (epsilon) A and C (epsilon) C, A (INTERSECT) C (NOT=) (phi). Further conditions which the transition probability matrix P must satisfy if lumpability is to hold are obtained as are relationships which must exist between P and P*., Suppose a process y(t) is known to arise as a result of a weak lumping or lumping from some unknown Markov chain x(t). Let (chi)(t) be the class of all Markov chains x(t) with n states which yield this weak lumping or lumping. The problem of characterizing this class and a class S of initial probability vectors which allow this lumping is considered. A complete solution is given when n = 3 and m = 2., The importance of lumpability in application is discussed.
Show less  Date Issued
 1980, 1980
 Identifier
 AAI8109927, 3084860, FSDT3084860, fsu:74361
 Format
 Document (PDF)