23 results on '"Katherine A Davies"'
Search Results
2. Progressively Type-II censored competing risks data from the linear exponential distribution
- Author
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Katherine F. Davies and William Volterman
- Subjects
Statistics and Probability ,010104 statistics & probability ,021103 operations research ,Exponential distribution ,Maximum likelihood ,0211 other engineering and technologies ,Econometrics ,02 engineering and technology ,0101 mathematics ,Competing risks ,01 natural sciences ,Mathematics - Abstract
Across different types of lifetime studies, whether it be in the medical or engineering sciences, the possibility of competing causes of failures needs to be addressed. Typically referred to as com...
- Published
- 2020
3. On the number of failed components in a k-out-of-n system upon system failure when the lifetimes are discretely distributed
- Author
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Katherine F. Davies and Anna Dembińska
- Subjects
021110 strategic, defence & security studies ,021103 operations research ,Component (thermodynamics) ,Order statistic ,0211 other engineering and technologies ,Conditional probability ,02 engineering and technology ,Industrial and Manufacturing Engineering ,Random variate ,Probability mass function ,Probability distribution ,Statistical physics ,Safety, Risk, Reliability and Quality ,Random variable ,Mathematics ,Variable (mathematics) - Abstract
In this paper, we examine properties of k-out-of-n systems when the component lifetimes are discretely distributed. The primary focus is the random variable which represents the number of failed components upon system failure. For this variate we derive the probability mass function under the most general setting of possibly dependent and heterogeneous components. We also present conditional probabilities for this variable given some information about the time of system failure. In addition to these results, we show that, in the special case of series systems, this variable exhibits some aging properties, and we establish several characterizations of probability distributions based on such properties. For illustration, we provide three numerical examples.
- Published
- 2019
4. Stochastic EM algorithm for generalized exponential cure rate model and an empirical study
- Author
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Joynob A. Siddiqua, Suvra Pal, and Katherine F. Davies
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Statistics and Probability ,021103 operations research ,media_common.quotation_subject ,0211 other engineering and technologies ,Articles ,02 engineering and technology ,Poisson distribution ,01 natural sciences ,Exponential function ,010104 statistics & probability ,Bernoulli's principle ,symbols.namesake ,Promotion (rank) ,Empirical research ,Goodness of fit ,Expectation–maximization algorithm ,symbols ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,media_common ,Parametric statistics - Abstract
In this paper, we consider two well-known parametric long-term survival models, namely, the Bernoulli cure rate model and the promotion time (or Poisson) cure rate model. Assuming the long-term survival probability to depend on a set of risk factors, the main contribution is in the development of the stochastic expectation maximization (SEM) algorithm to determine the maximum likelihood estimates of the model parameters. We carry out a detailed simulation study to demonstrate the performance of the proposed SEM algorithm. For this purpose, we assume the lifetimes due to each competing cause to follow a two-parameter generalized exponential distribution. We also compare the results obtained from the SEM algorithm with those obtained from the well-known expectation maximization (EM) algorithm. Furthermore, we investigate a simplified estimation procedure for both SEM and EM algorithms that allow the objective function to be maximized to split into simpler functions with lower dimensions with respect to model parameters. Moreover, we present examples where the EM algorithm fails to converge but the SEM algorithm still works. For illustrative purposes, we analyze a breast cancer survival data. Finally, we use a graphical method to assess the goodness-of-fit of the model with generalized exponential lifetimes.
- Published
- 2020
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5. Computing moments of discrete order statistics from non-identical distributions
- Author
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Anna Dembińska and Katherine F. Davies
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Independent and identically distributed random variables ,021103 operations research ,Applied Mathematics ,Order statistic ,0211 other engineering and technologies ,Nonparametric statistics ,02 engineering and technology ,Geometric distribution ,16. Peace & justice ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Computational Mathematics ,Beta-binomial distribution ,Probability distribution ,Statistical physics ,0101 mathematics ,Inverse distribution ,Mathematics ,L-moment - Abstract
In life-testing and reliability studies, it is commonly assumed that the lifetimes come from a continuous distribution. While this often makes derivations more tractable, under certain circumstances, it may be more realistic to assume a discrete distribution. Recent literature has heavily focused on the former and with this, results have been aimed at deriving properties in the case of independent and identically distributed random variables and associated order statistics. In this paper, we consider the case of discrete order statistics from non-identical distributions. We derive expression for single and product moments, and with this, covariances. We demonstrate the results for the geometric and binomial distributions.
- Published
- 2018
6. Exact Nonparametric Meta-Analysis of Lifetime Data From Systems With Known Signatures
- Author
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Hon Keung Tony Ng, Katherine F. Davies, Narayanaswamy Balakrishnan, and William Volterman
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021103 operations research ,0211 other engineering and technologies ,Coverage probability ,Nonparametric statistics ,Prediction interval ,02 engineering and technology ,Expected value ,01 natural sciences ,010104 statistics & probability ,Component (UML) ,Statistics ,0101 mathematics ,Electrical and Electronic Engineering ,Safety, Risk, Reliability and Quality ,Algorithm ,CDF-based nonparametric confidence interval ,Weibull distribution ,Quantile ,Mathematics - Abstract
In this paper, a mixture representation is derived for the pooled system lifetimes arising from a life-test on two or more independent samples. The components of each system are assumed to have the same common absolutely continuous distribution, but the system signature may vary between the samples. These mixtures are then used for developing exact nonparametric inference in the form of confidence intervals for quantiles of component or system lifetimes, as well as prediction intervals for future component or system lifetimes. Examples are finally provided to illustrate the developed methods. It is noted that testing with systems rather than components directly can reduce the expected number of failures while maintaining nominal coverage probability.
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- 2016
7. Minimal–maximal correlation-type goodness-of-fit tests
- Author
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Josie A. White and Katherine F. Davies
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Statistics and Probability ,Applied Mathematics ,Carry (arithmetic) ,05 social sciences ,Closeness ,Order statistic ,Type (model theory) ,01 natural sciences ,Test (assessment) ,Power (physics) ,Correlation ,010104 statistics & probability ,Goodness of fit ,Modeling and Simulation ,0502 economics and business ,Statistics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Algorithm ,050205 econometrics ,Mathematics - Abstract
In this paper we consider correlation-type tests based on plotting points which are modifications to the simultaneous closeness probability plotting points as recently introduced in the literature. In particular, we consider a maximal correlation test and a minimal correlation test. Furthermore, we provide two methods to carry out each test, where one method uses plotting points which are data dependent and the other test uses plotting points which are not. Some numerical properties on the associated correlation statistics are provided for various distributions, as well as a comprehensive power study to assess their performance in comparison to correlation-type tests based on more traditional plotting points. Two illustrative examples are also provided to demonstrate the tests. Finally, we make some observations and provide ideas for future work.
- Published
- 2015
8. Two-sample Pitman closeness comparison under progressive Type-II censoring
- Author
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William Volterman, Narayanaswamy Balakrishnan, and Katherine F. Davies
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Statistics and Probability ,Current sample ,Censoring (clinical trials) ,Closeness ,Order statistic ,Statistics ,Two sample ,Statistics, Probability and Uncertainty ,Standard normal table ,Mathematics ,Exponential function ,Quantile - Abstract
In this paper, we consider two problems concerning two independent progressively Type-II censored samples. We first consider the Pitman closeness (PC) of order statistics from two independent progressively censored samples to a specific population quantile. We then consider the point prediction of a future progressively censored order statistic and discuss the determination of the closest progressively censored order statistic from the current sample according to the simultaneous closeness probabilities. For both these problems, explicit expressions are derived for the pertinent PC probabilities, and then special cases are given as examples. For various censoring schemes, we also present numerical results for the standard uniform, standard exponential, and standard normal distributions. Finally, a distribution-free result for the median is obtained.
- Published
- 2013
9. Pitman closeness results for Type-I censored data from exponential distribution
- Author
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Narayanaswamy Balakrishnan and Katherine F. Davies
- Subjects
Statistics and Probability ,Exponential distribution ,Maximum likelihood ,Closeness ,Statistics ,Estimator ,Statistics, Probability and Uncertainty ,Type (model theory) ,Mathematics - Abstract
Recent work on Pitman closeness has compared estimators under Type-II censored samples from exponential distribution based on observed number of failures. In this paper, we carry out similar Pitman closeness comparisons for Type-I censored samples from exponential distribution based on time under test.
- Published
- 2013
10. Some Pitman closeness properties pertinent to symmetric populations
- Author
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Katherine F. Davies, Narayanaswamy Balakrishnan, and Mohammad Jafari Jozani
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Statistics and Probability ,Statistics::Theory ,Order statistic ,Closeness ,Estimator ,Sample Median ,Context (language use) ,Pitman closeness criterion ,Combinatorics ,Distribution (mathematics) ,Mathematics::Probability ,Statistics::Methodology ,Statistics, Probability and Uncertainty ,Focus (optics) ,Mathematics - Abstract
In this paper, we focus on Pitman closeness probabilities when the estimators are symmetrically distributed about the unknown parameter θ. We first consider two symmetric estimators θˆ1 and θˆ2 and obtain necessary and sufficient conditions for θˆ1 to be Pitman closer to the common median θ than θˆ2. We then establish some properties in the context of estimation under the Pitman closeness criterion. We define Pitman closeness probability which measures the frequency with which an individual order statistic is Pitman closer to θ than some symmetric estimator. We show that, for symmetric populations, the sample median is Pitman closer to the population median than any other independent and symmetrically distributed estimator of θ. Finally, we discuss the use of Pitman closeness probabilities in the determination of an optimal ranked set sampling scheme (denoted by RSS) for the estimation of the population median when the underlying distribution is symmetric. We show that the best RSS scheme from symmetric p...
- Published
- 2013
11. Simultaneous Pitman closeness of progressively type-II right-censored order statistics to population quantiles
- Author
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William Volterman, Narayanaswamy Balakrishnan, and Katherine F. Davies
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Statistics and Probability ,education.field_of_study ,Population ,Closeness ,Order statistic ,Exponential function ,Normal distribution ,Distribution (mathematics) ,Bounded function ,Statistics ,Statistics, Probability and Uncertainty ,education ,Mathematics ,Quantile - Abstract
In this work, we extend prior results concerning the simultaneous Pitman closeness of order statistics (OS) to population quantiles. By considering progressively type-II right-censored samples, we derive expressions for the simultaneous closeness probabilities of the progressively censored OS to population quantiles. Explicit expressions are deduced for the cases when the underlying distribution has bounded and unbounded supports. Illustrations are provided for the cases of exponential, uniform and normal distributions for various progressive type-II right-censoring schemes and different quantiles. Finally, an extension to the case of generalized OS is outlined.
- Published
- 2013
12. Pitman closeness results concerning ranked set sampling
- Author
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Narayanaswamy Balakrishnan, Mohammad Jafari Jozani, and Katherine F. Davies
- Subjects
Statistics and Probability ,RSS ,Closeness ,Order statistic ,Neighbourhood (graph theory) ,Estimator ,computer.file_format ,Pitman closeness criterion ,Simple random sample ,Measure (mathematics) ,Statistics ,Econometrics ,Statistics, Probability and Uncertainty ,computer ,Mathematics - Abstract
We first compare the ranked set sampling (RSS) and simple random sampling schemes for the estimation of the population median through Pitman’s measure of closeness. Then, by using Banks’ criterion, we determine the frequency with which an individual order statistic is closer to the median than some symmetric estimator in an ϵ -neighbourhood of the median. Finally, we consider the simultaneous Pitman closeness criterion and identify the median RSS as the optimal RSS scheme for estimating the population median.
- Published
- 2012
13. Pitman closeness as a criterion for the determination of the optimal progressive censoring scheme
- Author
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Katherine F. Davies, William Volterman, and Narayanaswamy Balakrishnan
- Subjects
Statistics and Probability ,Statistics::Theory ,Exponential distribution ,Statistics::Applications ,Computation ,Closeness ,Small sample ,Statistics::Other Statistics ,Pitman closeness criterion ,Censoring (statistics) ,Sample size determination ,Statistics ,Statistics::Methodology ,General algorithm ,Mathematics - Abstract
Selecting the optimal progressive censoring scheme for the exponential distribution according to Pitman closeness criterion is discussed. For small sample sizes the Pitman closeness probabilities are calculated explicitly, and it is shown that the optimal progressive censoring scheme is the usual Type-II right censoring case. It is conjectured that this to be the case for all sample sizes. A general algorithm is also presented for the numerical computation of the Pitman closeness probabilities between any two progressive censoring schemes of the same size.
- Published
- 2012
14. Nonparametric prediction of future order statistics
- Author
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Narayanaswamy Balakrishnan, Jafar Ahmadi, William Volterman, and Katherine F. Davies
- Subjects
Statistics and Probability ,Applied Mathematics ,Order statistic ,Coverage probability ,Nonparametric statistics ,Limiting case (mathematics) ,Sample (statistics) ,Absolute continuity ,Arbitrarily large ,Distribution (mathematics) ,Modeling and Simulation ,Statistics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Prediction of censored order statistics from a Type-II censored sample can be done with trivial bounds having perfect confidence. However, given independent samples from the same absolutely continuous distribution, improved bounds can be attained. In this regard, we develop here point prediction based on L-statistics for predicting order statistics (OS) from a future sample as well as for predicting censored OS from a Type-II censored sample. An example is taken to illustrate these ideas, and the limiting case wherein a single independent sample is arbitrarily large is also discussed.
- Published
- 2012
15. Computation of optimal plotting points based on Pitman closeness with an application to goodness-of-fit for location-scale families
- Author
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Jerome P. Keating, Katherine F. Davies, Robert L. Mason, and Narayanaswamy Balakrishnan
- Subjects
Statistics and Probability ,education.field_of_study ,Percentile ,Applied Mathematics ,Population ,Order statistic ,Nonparametric statistics ,Pitman closeness criterion ,Computational Mathematics ,Normality test ,Computational Theory and Mathematics ,Goodness of fit ,Statistics ,education ,Quantile ,Mathematics - Abstract
Plotting points of order statistics are often used in the determination of goodness-of-fit of observed data to theoretical percentiles. Plotting points are usually determined by using nonparametric methods which produce, for example, the mean- and median-ranks. Here, we use a distribution-based approach which selects plotting points (quantiles) based on the simultaneous-closeness of order statistics to population quantiles. We show that the plotting points so determined are robust over a multitude of symmetric distributions and then demonstrate their usefulness by examining the power properties of a correlation goodness-of-fit test for normality.
- Published
- 2012
16. Pitman Closeness Comparison of Best Linear Unbiased and Invariant Predictors for Exponential Distribution in One- and Two-Sample Situations
- Author
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Robert L. Mason, Katherine F. Davies, Narayanaswamy Balakrishnan, and Jerome P. Keating
- Subjects
Statistics and Probability ,Statistics::Theory ,Minimum-variance unbiased estimator ,Efficiency ,Exponential distribution ,Statistics ,Order statistic ,Statistics::Methodology ,Pitman closeness criterion ,Best linear unbiased prediction ,Invariant (mathematics) ,Lehmann–Scheffé theorem ,Mathematics - Abstract
Best linear unbiased, best linear invariant, and maximum likelihood predictors are commonly used in reliability studies for predicting either censored failure times or lifetimes from a future life-test. In this article, by assuming a Type-II right-censored sample from an exponential distribution, we compare best linear unbiased (BLUP) and best linear invariant (BLIP) predictors of the censored order statistics in the one-sample case and order statistics from a future sample in the two-sample case, in terms of Pitman closeness criterion. Some specific conclusions are drawn and supporting numerical results are presented.
- Published
- 2012
17. Pitman closeness, monotonicity and consistency of best linear unbiased and invariant estimators for exponential distribution under Type II censoring
- Author
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Jerome P. Keating, Katherine F. Davies, Narayanaswamy Balakrishnan, and Robert L. Mason
- Subjects
Statistics and Probability ,Exponential distribution ,Applied Mathematics ,Order statistic ,Estimator ,Best linear unbiased prediction ,Censoring (statistics) ,Efficiency ,Modeling and Simulation ,Statistics ,Statistics, Probability and Uncertainty ,Invariant (mathematics) ,Mathematics ,Parametric statistics - Abstract
Comparisons of best linear unbiased estimators with some other prominent estimators have been carried out over the last 50 years since the ground breaking work of Lloyd [E.H. Lloyd, Least squares estimation of location and scale parameters using order statistics, Biometrika 39 (1952), pp. 88–95]. These comparisons have been made under many different criteria across different parametric families of distributions. A noteworthy one is by Nagaraja [H.N. Nagaraja, Comparison of estimators and predictors from two-parameter exponential distribution, Sankhyā Ser. B 48 (1986), pp. 10–18], who made a comparison of best linear unbiased (BLUE) and best linear invariant (BLIE) estimators in the case of exponential distribution. In this paper, continuing along the same lines by assuming a Type II right censored sample from a scaled-exponential distribution, we first compare BLUE and BLIE of the exponential mean parameter in terms of Pitman closeness (nearness) criterion. We show that the BLUE is always Pitman closer th...
- Published
- 2011
18. Correlation-Type Goodness of Fit Test for Extreme Value Distribution Based on Simultaneous Closeness
- Author
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Katherine F. Davies, Jerome P. Keating, Narayanaswamy Balakrishnan, and Robert L. Mason
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Statistics and Probability ,Location parameter ,Goodness of fit ,Modeling and Simulation ,Statistics ,Generalized extreme value distribution ,Probability distribution ,Pitman closeness criterion ,Extreme value theory ,Scale parameter ,Weibull distribution ,Mathematics - Abstract
In reliability studies, one typically would assume a lifetime distribution for the units under study and then carry out the required analysis. One popular choice for the lifetime distribution is the family of two-parameter Weibull distributions (with scale and shape parameters) which, through a logarithmic transformation, can be transformed to the family of two-parameter extreme value distributions (with location and scale parameters). In carrying out a parametric analysis of this type, it is highly desirable to be able to test the validity of such a model assumption. A basic tool that is useful for this purpose is a quantile–quantile (QQ) plot, but in its use, the issue of the choice of plotting position arises. Here, by adopting the optimal plotting points based on Pitman closeness criterion proposed recently by Balakrishnan et al. (2010b), and referred to as simultaneous closeness probability (SCP) plotting points, we propose a correlation-type goodness of fit test for the extreme value distribution. W...
- Published
- 2011
19. Simultaneous closeness among order statistics to population quantiles
- Author
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Narayanaswamy Balakrishnan, Katherine F. Davies, Robert L. Mason, and Jerome P. Keating
- Subjects
Statistics and Probability ,education.field_of_study ,Percentile ,Applied Mathematics ,Order statistic ,Population ,Sampling distribution ,Sample size determination ,Bounded function ,Statistics ,Statistics, Probability and Uncertainty ,education ,Random variable ,Mathematics ,Quantile - Abstract
We derive expressions for the probability that an individual order statistic is closest to the target parameter among the order statistics from a complete random sample. Results are given for random variables with bounded and complete support. We then apply these general results to location-scale parameter families of distributions with specific applications to estimation of percentiles. In this case, simultaneous-closeness probabilities depend upon the parameters through the value of p in the percentile and the sample size, n. Results are finally illustrated with the estimation of percentiles for normal and exponential distributions.
- Published
- 2010
20. Some results on order statistics generated by two simulation methods
- Author
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William Volterman, Erhard Cramer, Katherine F. Davies, and Narayanaswamy Balakrishnan
- Subjects
Statistics and Probability ,Distribution function ,Probability theory ,Joint probability distribution ,Order statistic ,Statistics ,Probability distribution ,Asymptotic distribution ,Statistical physics ,Statistics, Probability and Uncertainty ,Fixed point ,Simulation methods ,Mathematics - Abstract
In this paper, we consider joint distributions of order statistics generated by two simulation methods. By using these distributions, we study the nature of dependence and exceedence probabilities between them.
- Published
- 2009
21. Pitman Closeness of Order Statistics to Population Quantiles
- Author
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Jerome P. Keating, Narayanaswamy Balakrishnan, and Katherine F. Davies
- Subjects
Statistics and Probability ,education.field_of_study ,Location parameter ,Population ,Closeness ,Order statistic ,Sample (statistics) ,Exponential family ,Sample size determination ,Modeling and Simulation ,Statistics ,Econometrics ,education ,Mathematics ,Quantile - Abstract
In this article, Pitman closeness of sample order statistics to population quantiles of a location-scale family of distributions is discussed. Explicit expressions are derived for some specific families such as uniform, exponential, and power function. Numerical results are then presented for these families for sample sizes n = 10,15, and for the choices of p = 0.10, 0.25, 0.75, 0.90. The Pitman-closest order statistic is also determined in these cases and presented.
- Published
- 2009
22. Pitman Closest Estimators Based on Convex Linear Combinations of Two Contiguous Order Statistics
- Author
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Robert L. Mason, Jerome P. Keating, Narayanaswamy Balakrishnan, and Katherine F. Davies
- Subjects
education.field_of_study ,Sample size determination ,Statistics ,Order statistic ,Population ,Pareto principle ,Applied mathematics ,Estimator ,Linear combination ,education ,Location-scale family ,Mathematics ,Quantile - Abstract
Comparisons of best linear unbiased estimators with some other prominent estimators have been carried out over the last six decades since the ground breaking work of Lloyd [13]; see Arnold et al. [1] and David and Nagaraja [9] for elaborate details in this regard. Recently, Pitman closeness comparison of order statistics as estimators for population parameters, such as medians and quantiles, and their applications have been carried out by Balakrishnan et al. [3, 4, 5, 7]. In this paper, we discuss the Pitman closest estimators based on convex linear combinations of two contiguous order statistics, which sheds additional insight with regard to the estimation of the population median in the case of even sample sizes. We finally demonstrate the proposed method for the uniform, exponential, power function and Pareto distributions.
- Published
- 2015
23. Nonparametric estimation for a two-dimensional renewal process
- Author
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Katherine F. Davies and B. Gail Ivanoff
- Subjects
avoidance probability ,Statistics and Probability ,Mathematical optimization ,renewal process ,Nonparametric statistics ,Estimator ,Point process ,60K05 ,Markov renewal process ,multiparameter martingale ,Local martingale ,62G05 ,Martingale difference sequence ,62M30 ,60G55 ,Renewal theory ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,product limit estimator ,Mathematics - Abstract
This paper introduces nonparametric estimation methods for spatial point processes satisfying a particular renewal property. Martingale methods yield a unified approach for renewal processes in both one and two dimensions, and can be used for both synchronous and asynchronous data. In each case, we obtain martingale estimators of the avoidance probabilities that characterize the renewal process. Asymptotic properties of the estimators are studied analytically and empirically.
- Published
- 2012
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