One-parameter Exponential Distribution Research Articles

Effect of the Seasonal Rainfall Distribution on Storm-Water Quality Capture Volume Estimation

Thewater quality capturevolume (WQCV) is a design parameter that is used for sizing urban storm-water treatment facilities. The storage, treatment, overflow, and runoff model (STORM) in NetSTORM was used to translate historical hourly rainfall records into storm event runoff data and to compute the WQCV (set at the 85th percentile capture of the annual runoff volume) of eight cities located in various climatological areas of South Korea. For each city, one-parameter and two-parameter exponential distributions were fitted to the set of storm event runoff volumes that were computed by STORM, and the 85th percentile runoff volume that was determined from the STORM was compared to the 85th percentile runoff volumes that were obtained from the two exponential distributions. The results demonstrate that the more easily applied distribution approach is adequate in South Korea for determining the WQCV. In Seoul, which is a city located in the northwestern part of South Korea, the two-parameter exponential distribution yielded the best estimate of the WQCV computed by STORM. For Chuncheon, Wonju, and Daejeon, which are cities located in the north-central and central parts of South Korea, the WQCVs were accurately estimated using both the one-parameter and two-parameter exponential distributions. Finally, in the cities of Gangneung, Gwangju, Daegu, and Busan, which are located in the northeastern and southern parts of South Korea, the WQCVs were best estimated using the one- parameter exponential distribution. It was found that seasonal variations in rainfall and rainfall event depth are important factors in assessing the WQCV. This study also found that the one-parameter exponential distribution approach, which is the most common approach used in the United States, may not be adequate for WQCVestimation if the standard deviation for average monthly rainfall is significant. DOI: 10.1061/ (ASCE)WR.1943-5452.0000204. © 2013 American Society of Civil Engineers. CE Database subject headings: Nonpoint pollution; Parameters; Rainfall; Seasonal variations; South Korea; Stormwater management; Water quality.

Exact inference for the two-parameter exponential distribution under Type-II hybrid censoring

Epstein (1954) introduced the Type-I hybrid censoring scheme as a mixture of Type-I and Type-II censoring schemes. Childs et al. (2003) introduced the Type-II hybrid censoring scheme as an alternative to Type-I hybrid censoring scheme, and provided the exact distribution of the maximum likelihood estimator of the mean of a one-parameter exponential distribution based on Type-II hybrid censored samples. The associated confidence interval also has been provided. The main aim of this paper is to consider a two-parameter exponential distribution, and to derive the exact distribution of the maximum likelihood estimators of the unknown parameters based on Type-II hybrid censored samples. The marginal distributions and the exact confidence intervals are also provided. The results can be used to derive the exact distribution of the maximum likelihood estimator of the percentile point, and to construct the associated confidence interval. Different methods are compared using extensive simulations and one data analysis has been performed for illustrative purposes.

Marginal Distributions of the Counting Process Associated with Generalized Order Statistics

By using results from the distribution theory of generalized order statistics, the finite dimensional marginal distributions of the associated counting process are derived in terms of Meijer's G-functions. Moreover, some properties of the corresponding transition probabilities are given. The results are obtained by using a new representation of the cumulative distribution function of a single generalized order statistic in terms of the G-function. As an application, a Bayes estimator based on observations of the counting process of generalized order statistics from a one-parameter exponential distribution is determined.

Ordered ranked set samples and applications to inference

Ranked set sampling (RSS) was first proposed by McIntyre [1952. A method for unbiased selective sampling, using ranked sets. Australian J. Agricultural Res. 3, 385–390] as an effective way to estimate the unknown population mean. Chuiv and Sinha [1998. On some aspects of ranked set sampling in parametric estimation. In: Balakrishnan, N., Rao, C.R. (Eds.), Handbook of Statistics, vol. 17. Elsevier, Amsterdam, pp. 337–377] and Chen et al. [2004. Ranked Set Sampling—Theory and Application. Lecture Notes in Statistics, vol. 176. Springer, New York] have provided excellent surveys of RSS and various inferential results based on RSS. In this paper, we use the idea of order statistics from independent and non-identically distributed (INID) random variables to propose ordered ranked set sampling (ORSS) and then develop optimal linear inference based on ORSS. We determine the best linear unbiased estimators based on ORSS (BLUE-ORSS) and show that they are more efficient than BLUE-RSS for the two-parameter exponential, normal and logistic distributions. Although this is not the case for the one-parameter exponential distribution, the relative efficiency of the BLUE-ORSS (to BLUE-RSS) is very close to 1. Furthermore, we compare both BLUE-ORSS and BLUE-RSS with the BLUE based on order statistics from a simple random sample (BLUE-OS). We show that BLUE-ORSS is uniformly better than BLUE-OS, while BLUE-RSS is not as efficient as BLUE-OS for small sample sizes ( n < 5 ).

AN ALTERNATIVE METHOD FOR PREDICTION INTERVALS OF AN ORDER OBSERVATION FROM ONE-PARAMETER EXPONENTIAL DISTRIBUTION BASED ON CENSORED SAMPLE

ABSTRACT In the paper, we construct a more precise prediction interval of the jth order observation from some pivotal quantities for the one-parameter exponential distribution based on Type II censored samples. Our method is more general in the sense that it can be applied to any data scheme. From the illustrated examples and the results of simulation, the performances of our method are better than the widely-used methods of Mann and Grubbs [12] and Kaminsky and Nelson [5,6]. For further study, our method is easily applied to other location and scale family distributions.

AN ALTERNATIVE METHOD FOR PREDICTION INTERVALS OF AN ORDER OBSERVATION FROM ONE-PARAMETER EXPONENTIAL DISTRIBUTION BASED ON CENSORED SAMPLE

In the paper, we construct a more precise prediction interval of the jth order observation from some pivotal quantities for the one-parameter exponential distribution based on Type Ⅱ censored samples. Our method is more general in the sense that it can be applied to any data scheme. From the illustrated examples and the results of simulation, the performances of our method are better than the widely-used methods of Mann and Grubbs [12] and Kaminsky and Nelson [5,6]. For further study, our method is easily applied to other location and scale family distributions.

Prediction intervals of future observation from one-parameter exponential distribution based on multiply type II censored samples

Wu and Yang [Quality and Reliability Engineering International 18 (2002) 149] (Ref. [12]) propose the weighted moments estimators (WMEs) of scale parameter θ for one-parameter exponential distribution under multiply type II censored sample Y ( r+1) <⋯< Y ( r+ k) < Y ( r+ k+ l+1) <⋯< Y ( n− s) . Hence, we use these WMEs, approximate maximum likelihood estimator (AMLE) and best linear unbiased estimator (BLUE) of scale parameter to find pivotal quantities and obtain the prediction intervals of the jth future observation ( Y ( j) , n − s < j ⩽ n) based on the above censored sample. Finally, we give one example and the Monte Carlo simulation to assess the behaviors of these pivotal quantities for establishing prediction intervals of the jth future observation ( Y ( j) , n − s < j ⩽ n).

Prediction Intervals of an Ordered Observation from One-Parameter Exponential Distribution Based on Multiple Type II Censored Samples

This paper provides a suitable pivotal quantity to present the prediction interval of the j(superscript th) future ordered-observation in a sample of size n from one-parameter exponential distribution in case of a multiple type II censored sample. In addition, we also discuss the approximate prediction interval, the best linear unbiased estimate and approximate maximum likelihood estimate of X(subscript j) based on the censored sample. As in application, the total duration time in a life test and the failure time of a j-out-of-n system may be predicated. Finally, a simulated study and three illustrative examples are included.

Remote sensing of mesospheric electric fields using MF radars

Large mesospheric electric fields can play an essential role in middle atmospheric electrodynamics (see, e.g., Goldberg, R. A., Middle Atmospheric Electrodynamics during MAP, Adv. Space Res. 10 (10) (1990) 209). The V/m electric fields of atmospheric origin can be the possible cause of large variations in the electron collision frequency at mesospheric altitudes, and this provides a unique opportunity to take measurements of electric fields in the lower ionosphere by using remote sensing instruments employing radiowave techniques. A technique has been proposed for making estimates of large mesospheric electric field intensities on the lower edge of the ionosphere by using MF radar data and the inherent effective electron collision frequency. To do this, data collected in Canada and Ukraine were utilized. The developed technique permits the changes in mesospheric electric field intensities to be derived from MF radar data in real time. The statistical analysis of data consistent with large mesospheric electric field intensities in the 60– 67 km region resulted in the following inferences. There are at least two mechanisms for the generation of large mesospheric electric fields in the mesosphere. The most likely mechanism, with a probability of 60–70%, is the summation of random fields from a large number of elementary small-scale mesospheric generators, which results in a one-parameter Rayleigh distribution of the total large mesospheric electric field intensity E with a mean value of approximately 0.7– 0.9 V/m in the 60– 67 km altitude region, or in the corresponding one-parameter exponential distribution of the intensity squared E 2 of large mesospheric electric fields. The second mechanism of unknown nature, with 5–15% probability, gives rise to the sporadic appearance of large mesospheric electric field intensities E>2.5 V/m with a mean of 4 V/m . Statistically significant seasonal differences in the averaged large mesospheric electric field parameters have not been revealed. The probability of the absence of local large mesospheric electric fields amounts to approximately 25% for Ukraine and approximately 30% for Canada. A comparison of the Ukrainian and Canadian data indicates the possible existence of a latitudinal dependence in mean large mesospheric electric field features. Hence, the large electric fields are an additional source of electron heating that must be taken into account in studying a disturbed lower ionosphere and radio wave propagation within it.

Modèle statistique des hauteurs de pluies journalières en zone sahélienne: exemple du bassin amont du fleuve Sénégal / Frequency analysis of daily rainfall in the Sahelian area: case of the upstream basin of the Senegal River

Climatic variability in West and Central Africa has been detected using statistical tests for non-homogeneity in annual series of hydrological variables. This climatic variability is generally related to a shortage in water resources. This paper aims at looking for a statistical model for daily rainfall in the Sahelian area. This model should have a very low number of parameters, and be as flexible as possible to be able to adapt to the change introduced by the climatic variability. Raingauge stations have been selected where series of daily rainfall are long enough to reflect the climatic variability; these are situated in the upper part of the Senegal River basin, mainly in the countries of Mali and Senegal. One parameter exponential distribution truncated in frequency has been selected a priori and fitted graphically to each of these daily rainfall series. From the resulting plots, it appears that a two-component exponential distribution with four parameters is better for the fit. An iterative approach based on the least-squares method has been used for parameter estimation. Furthermore, the initial samples were divided into two parts, before and after the date of beginning of the climate variability, this date being assessed using statistical tests for non-homogeneity. A change in the statistical model was noted: two-component four-parameter exponential distribution before the beginning of climatic variability, and one-parameter exponential distribution afterwards, this parameter being estimated by ordinary least squares. The study shows that climatic variability introduces a change in the statistical model of daily rainfall, and that exponential distribution is well suited for frequency modelling of daily rainfall to represent this climatic variability.

PREDICTION INTERVALS OF AN ORDERED OBSERVATION FROM ONE-PARAMETER EXPONENTIAL DISTRIBUTION BASED ON MULTIPLE TYPE II CENSORED SAMPLES

ABSTRACT This paper provides a suitable pivotal quantity to present the prediction interval of the j th future ordered-observation in a sample of size n from one-parameter exponential distribution in case of a multiple type II censored sample. In addition, we also discuss the approximate prediction interval, the best linear unbiased estimate and approximate maximum likelihood estimate of X j based on the censored sample. As in applications, the total duration time in a life test and the failure time of a j-out-of-n system may be predicted. Finally, a simulated study and three illustrative examples are included.

Prediction intervals for the future record values from exponential distribution: comparative study

!n this paper we consider the predicf an problem of the future nth record value based an the first m (m < n) observed record values from one-parameter exponential distribution. We introduce four procedures for obtaining prediction intervals for the nth record value. The performance of the so obtained intervals is assessed through numerical and simulation studies. In these studies, we provide the means and standard errors of lower limits. upper limits and lengths of prediction intervals. Further, we check the validation of these intervals based on some point predictors.

On the Effect of Overdispersion On Exact Conditional Tests

Summary Let Y 1, . . ., Yn denote independent random variables such that Yj has a one-parameter exponential family distribution with canonical parameter θj=λ+ψXj; here X 1, . . ., Xn are known constants. Consider a test of the null hypothesis ψ=0. Under the null hypothesis, A=ΣYj is sufficient for λ and, hence, a test of ψ=0 may be based on the conditional distribution of T=ΣXj Yj given A, which is independent of λ. In this paper, the effects of overdispersion due to a mixture model on the conditional distribution of T given A are considered.

The Statistical Distribution of Wrist Movements during Sleep

The purpose of this work was to describe the basic statistical properties of the process of production of movements measured with a wrist actimeter, along a complete sleep period in normal human subjects. Two distinct types of random magnitudes were considered to analyze the data, the times between successive groups of movements and the number of movements at each fixed time (1 min) measurement epoch. Suitable probabilistic models for the two variates were chosen, fitting theoretical distribution functions to the observed data. It is concluded that interval data fit a one-parameter exponential distribution, while the number of movements fit a two-parameter negative binomial distribution. The estimated values of these parameters, besides being necessary to perform further statistical analysis, are a measure of the intensity and frequency of the movements. Finally the relationship between polysomnography measures and the elicited parameters was studied.

An approach to fuzzy hypothesis testing

In this paper an approach is presented how to test fuzzily formulated hypotheses with crisp data. The quantitiesα andβ, the probabilities of the errors of type I and of type II, are suitably generalized and the concept of a best test is introduced. Within the framework of a one-parameter exponential distribution family the search for a best test is considerably reduced. Furthermore, it is shown under very weak conditions thatα andβ can simultaneously be diminished by increasing the sample size even in the case of testingH0 against the omnibus alternativeH1: notH0, a result completely different from the case of crisp setsH0 andH1: notH0.

Sequential interval estimation procedures for the mean exponential survival time and reliability function

The problems of confidence interval estimation are considered (a) for estimating the mean of a one-parameter exponential distribution and (b) for estimating the reliability function associated with the one-parameter exponential distribution. For the estimation problem (a), the confidence interval of ‘preassigned width and coverage probability’ is considered. For the estimation problem (b), the confidence interval of ‘fixed-ratio width and preassigned coverage probability’ is proposed. The failure of the fixed sample size procedures to deal with these estimation problems is established and sequential procedures are proposed to deal with them. The proposed sequential procedures are proved to be ‘asymptotically efficient and consistent’ in the Chow-Robbins [Chow and Robbins, Ann. Math. Statist. 36,457–462 (1965)] sense. Asymptotic distributions of the stopping times are derived and second-order approximations are obtained for the average sample numbers associated with them.

On the Shapiro-Wilk Test and Darling's Test for Exponentiality

Tests for exponentiality are widely used in studying time-structured phenomena. Especially in the analysis of behavioural data, however, hazard rates are constant only after a dead time during which no events can occur. To take this into account, Shapiro and Wilk (1972, Teclhnoinetrics 14, 355-370) developed a test for the two-parameter exponential distribution with unknown origin. They did not, however, consider the asymptotic distribution of the test statistic or its power properties. Although it has as yet been unnoticed, it is an elementary exercise to show that a transformed version of the Shapiro-Wilk test statistic is equal to Darling's (1953, Annals of Mathematical Statistics 24, 239-253) test statistic for the one-parameter exponential distribution. For this test, no small-sample critical values were known, but the asymptotic null distribution of the statistic is known to be normal, and the right-sided version of the test is locally most powerful against mixtures of exponentials. The two test statistics have the same distribution under the null hypothesis of exponentiality with unknown origin as well as under the alternative of a mixture of two-parameter exponentials with the same unknown origin. Since simulation results indicate that the convergence toward normality is rather slow, it is advised to use small-sample results for both test statistics. To this end we extend the table given by Shapiro and Wilk (1972) to values of n up to 500.

Prediction of STH Ordered Observation in Doubly Type-II Censored Sample from One-Parameter Exponential Distribution

Prediction of STH Ordered Observation in Doubly Type-II Censored Sample from One-Parameter Exponential Distribution

Preliminary test estimator of average life in a life test experiment

The exponential distribution occupies an important position in life time distribution, especially in the fields of engineering, medical and industrial life testing. In this paper an estimation procedure using a preliminary test of significance for the estimation of the average life of a one-parameter exponential distribution for time censored life data is given. The bias and mean square error of the proposed estimator are discussed. The proposed estimator, though slightly biased, will for the most part be more efficient than the usual maximum likelihood estimators. Based on an empirical study, recommendations are made regarding the use of the proposed estimator. The procedure is illustrated by an example.

The use of geometric and gamma-related distributions for frequency analysis of water deficit

This paper presents an approach to perform statistical frequency analysis of water deficit duration and severity using respectively the geometric and exponential distributions. Monthly mean water discharges are compared to a given threshold and classified in two mutually exclusive ways. This leads to a two state random variable such that: a success represents the absence of a water deficit event (mean monthly discharge exceeds threshold), and a failure, a water deficit event (mean monthly discharge is below threshold). If we suppose that this random variable gives rise to a Markov process of order 1, then the duration of a water deficit event X (consecutive months in deficit) will have a geometric distribution. In turn, the summation of discharges in deficit will give the severity of a water deficit event which can be represented by a one-parameter exponential distribution. The threshold or base level is taken as a percentile of the observed mean discharges of a given month. This base level, which varies from month to month, can be viewed as the limit of an acceptable deficit (or energetic failure) associated to a given empirical probability of being in deficit. The second step of the approach is to estimate the value of the parameter for each distribution using the maximum likelihood method. Expressions for the estimator of a given percentile, $$\hat x_q $$ , as well as its variance are deduced. Finally, the presented models are applied to observed data.