Statistics and Probability Commons^™

Strong Large Deviation And Local Limit Theorms, Narasinga Rao Chaganty, Jayaram Sethuraman

Mathematics & Statistics Faculty Publications

Most large deviation results give asymptotic expressions for log P(Y_n ≥ y_n), where the event {Y_n ≥ y_n} is a large deviation event, that is, P(Y_n ≥ y_n) goes to 0 exponentially fast. We refer to such results as weak large deviation results. In this paper we obtain strong large deviation results for arbitrary random variables {Y_n}, that is, we obtain asymptotic expressions for P(Y_n ≥ y_n), where {Y_n ≥ y_n} is a large deviation event. These strong large deviation results …

Reliability Trend Analyses With Statistical Confidence Limits Using The Luke Reliability Trend Chart, Stephen R. Luke

Engineering Management & Systems Engineering Theses & Dissertations

In electronic systems, it is interesting to understand exactly how the reliability is changing with time. Dynamic performance changes when a system passes from infant mortality stage into useful life phase and when the system passes from useful life phase into wearout phase. Dynamic performance also changes when the system is redesigned or when the system is acted on by a number of other outside forces such as a change in maintenance policy, escalation of alignment problems, or a change in training program. It is important to know when a system is changing dynamically in order to assess design, policy …

Estimation In A Marked Poisson Error Recapture Model Of Software Reliability, Rajan Gupta

Mathematics & Statistics Theses & Dissertations

Nayak's (1988) model for the detection, removal, and recapture of the errors in a computer program is extended to a larger family of models in which the probabilities that the successive programs produce errors are described by the tail probabilities of discrete distribution on the positive integers. Confidence limits are derived for the probability that the final program produces errors. A comparison of the asymptotic variances of parameter estimates given by the error recapture and by the repetitive-run procedure of Nagel, Scholz, and Skrivan (1982) is made to determine which of these procedures efficiently uses the test time.

The Effect Of Age Misreporting In China On The Calculation Of Mortality Rates At Very High Ages, Ansley J. Coale, Shaomin Li

Management Faculty Publications

When mortality rates by age are calculated from recorded deaths and enumerated populations, rates at higher ages are typically in error because of misstated ages. Mortality rates for China in 1981 have been calculated from the number of deaths in 1981 in each household recorded in the 1982 census, and from the census population back-projected one year. Because age was determined from date of birth, and because persons of the Chinese culture have very precise knowledge of date of birth, the mortality rates even at high ages should be unusually accurate. This expectation is fulfilled for most of China, but …

The Truncated Cauchy Distribution: Estimation Of Parameters And Application To Stock Returns, Paul G. Staneski

Mathematics & Statistics Theses & Dissertations

The problem addressed in this dissertation is the existence and estimation of the parameters of a truncated Cauchy distribution. It is known that when a number of distributions with infinite support are truncated to a finite interval that the maximum likelihood estimator of the scale parameter fails to exist with positive probability. In particular, necessary and sufficient conditions which give rise to instances of non-existence have been found for the exponential (Deemer and Votaw (1955)), gamma (Broeder (1955), Hegde and Dahiya (1989)), Weibull (Mittal and Dahiya (1989)) and normal distribution (Barndorff-Nielsen (1978), Mittal and Dahiya (1987), Hegde and Dahiya (1989)). …

A Model Of Dc Glow Discharges With Abnormal Cathode Fall, Karl H. Schoenbach, Hao Chen, G. Schaefer

Bioelectrics Publications

A model for an abnormal glow discharge, including a self‐consistent analysis of the cathode fall, was developed. It combines microscopic particle simulation by means of Monte Carlo methods with a fluid model of the gas discharge. The model allows calculations of the steady‐state electrical field distribution, the charged‐particle densities, and the current densities along the axis of the discharge. The model was used to simulate a glow discharge in 80% He and 20% SF₆ at a pressure of 8 Torr with a current density of 1 A/cm². The computed discharge voltage agrees well with measured values. The …

Software Reliability Models, Syed Afzal Hossain

Mathematics & Statistics Theses & Dissertations

The problem considered here is the building of Non-homogeneous Poisson Process (NHPP) model. Currently existing popular NHPP process models like Goel-Okumoto (G-O) and Yamada et al models suffer from the drawback that the probability density function of the inter-failure times is an improper density function. This is because the event no failure in (0, oo] is allowed in these models. In real life situations we cannot draw sample(s) from such a population and also none of the moments of inter-failure times exist. Therefore, these models are unsuitable for modelling real software error data. On the other hand if the density …

Detection Of Outliers And Influential Observations In Regression Models, Anwar M. Hossain

Mathematics & Statistics Theses & Dissertations

Observations arising from a linear regression model, lead one to believe that a particular observation or a set of observations are aberrant from the rest of the data. These may arise in several ways: for example, from incorrect or faulty measurements or by gross errors in either response or explanatory variables. Sometimes the model may inadequately describe the systematic structure of the data, or the data may be better analyzed in another scale. When diagnostics indicate the presence of anomalous data, then either these data are indeed unusual and hence helpful, or contaminated and, therefore, in need of modifications or …

Mathematical Models Of Prevascular Tumor Growth By Diffusion, Sophia A. Maggelakis

Mathematics & Statistics Theses & Dissertations

A study of several complementary mathematical models that describe the early, prevascular stages of solid tumor growth by diffusion under various simplifying assumptions is presented. The advantage of these models is that their degree of complexity is relatively low, which ensures fairly straightforward comparisons with experimental or clinical data (as it becomes available), yet they are mathematically sophisticated enough to capture the main biological phenomena of interest.

The tumor growth and cell proliferation rate are assumed to depend on the local concentrations of nutrients and inhibitory factors. The effects of geometry and spatially non-uniform inhibitor production and non-uniform nutrient consumption …

Optimal Row-Column Designs For Correlated Errors And Nested Row-Column Designs For Uncorrelated Errors, Nizam Uddin

Mathematics & Statistics Theses & Dissertations

In this dissertation the design problems are considered in the row-column setting for second order autonormal errors when the treatment effects are estimated by generalized least squares, and in the nested row-column setting for uncorrelated errors when the treatment effects are estimated by ordinary least squares. In the former case, universal optimality conditions are derived separately for designs in the plane and on the torus using more general linear models than those considered elsewhere in the literature. Examples of universally optimum planar designs are given, and a method is developed for the construction of optimum and near optimum designs, that …

Large Deviation Local Limit Theorems For Ratio Statistics, Sanjeev V. Sabnis

Mathematics & Statistics Theses & Dissertations

Let {T„, n > 1} be an arbitrary sequence of non-lattice random variables and {S_n, n > 1} be another sequence of positive non-lattice random variables. Let the two sequences be independent. Let Ø_1n and Ø_2n be the moment genereating functions of {Tn, n > 1} and { S_n,n > 1} respectively. Let {a_n} be a sequence of real numbers such that a_n —»• oo.

Limit Theorems In The Area Of Large Deviations For Some Dependent Random Variables, Narasinga Rao Chaganty, Jayaram Sethuraman

Mathematics & Statistics Faculty Publications

A magnetic body can be considered to consist of n sites, where n is large. The magnetic spins at these n sites, whose sum is the total magnetization present in the body, can be modelled by a triangular array of random variables (X(n) 1,..., X(n) n). Standard theory of physics would dictate that the joint distribution of the spins can be modelled by dQ_n(x) = z_n^-1 exp[ -H_n(x)]Π dP(x_j), where x = (x₁,..., x_n) ∈ Rⁿ, where H_n is the Hamiltonian, z_n is …

Estimation In Truncated Exponential Family Of Distributions, Laxman M. Hegde

Mathematics & Statistics Theses & Dissertations

Estimating the parameters of a truncated distribution is a well known problem in statistical inference. The non-existence of the maximum likelihood estimator (m.l.e.) with positive probability in certain truncated distributions is not well known. To mention a few results in the literature:

(i) Deemer and Votaw 1955 show that the maximum likelihood estimator does not exist in a truncated negative exponential distribution on 0,T , T > 0 known, whenever the sample mean x (GREATERTHEQ) T/2.

(ii) Broeder 1955 shows that the maximum likelihood estimator of the scale parameter of a truncated gamma distribution, with the shape parameter being known, becomes …

Statistical Calibration Theory, James John Mckeon

Mathematics & Statistics Theses & Dissertations

A calibration method substitutes for measurements, X(,i), that are accurate but impractical or costly, a set of measurements, Y(,i), that are less accurate but simpler or less costly. There are two general types of calibration methods. The classical approach in which once the calibration sample is drawn, the estimates of the X values for a given unit is found without any consideration of the distribution of X values for the other units to be measured. This corresponds best to the literal meaning of the word "calibration". Maximum likelihood estimation is the statistical formulation of the classical approach.

The second approach …

Large Deviation Local Limit Theorems For Arbitrary Sequences Of Random Varibles, Narasinga Rao Chaganty, J Sethuraman

Mathematics & Statistics Faculty Publications

The results of W. Richter (Theory Prob. Appl. (1957) 2 206-219) on sums of independent, identically distributed random variables are generalized to arbitrary sequences of random variables T_n. Under simple conditions on the cumulant generating function of T_n, which imply that T_n/n converges to zero, it is shown, for arbitrary sequences {m_n}, that k_n (m_n), the probability density function of T_n/n at m_n, is asymptotic to an expression involving the large deviation rate of T_n/n. Analogous results for lattice …

Estimating The Parameters Of Truncated Distributions, Mukul Mohan Mittal

Mathematics & Statistics Theses & Dissertations

The problem considered here is the estimation of the parameters of some special truncated distributions. If the sample observations are restricted to the interval {O, T} with T known, then it is well known in the literature that the method of maximum likelihood fails to provide a finite estimate, for the mean of an exponential distribution, whenever the sample mean is greater than T/2 (Deemer and Votaw, 1955, Ann. Math. Statist. 26, 498-504). Not so well known is the nonexistence of the maximum likelihood estimator (m.l.e.), under certain conditions, for the scale parameter of a gamma distribution from a truncated …

An Intraseason Forecasting System For Commercial Marine Fisheries, Erik J. Barth

OES Theses and Dissertations

The reliability of an intraseason yield estimation technique which is commonly used by Pacific salmon harvest managers is evaluated for applicability to a variety of commercial finfish and crustacean fisheries. The estimation technique is known as the average timing or the average performance model. The method is not easily related to standard statistical models, but does show some similarity to both a single parameter linear regression model and the ratio estimator of sampling theory. A comparison of these models, a two parameter linear model, and a regression estimator is made to determine if the precision of forecasts of performance can …

On The First Passage Time Distribution For A Class Of Markov Chains, Mark Brown, Narasinga Rao Chaganty

Mathematics & Statistics Faculty Publications

Consider a stochastically monotone chain with monotone paths on a partially ordered countable set S. Let C be an increasing subset of S with finite complement. Then the first passage-time from i ∈ S to C is shown to be IFRA (increasing failure rate on the,av;rage). Several applications are presented including coherent systems, shock models, and convolutions of IFRA distributions.