two parameter exponential distribution sufficient statistic

., Xn be a random sample from a two-parameter exponential distribution, Xi ~ EXP().a) Assuming it is known that =150, find a pivotal quanitity for the parameter based on the sufficient statistic.. b) Using the data of Exercise 5, find a one-sided lower 95% confidence limit for . This is an expression of the form of the Exponential Distribution Family and since the support does not depend on θ, we can conclude that it belongs in the exponential distribution family. Stat., Vol. Y. Nikitin, ACUTM, Vol. A. Dembińska, J. Stat. So, the 100(1−α)% interval confidence for e−σa is given by. See, Nikitin [27] for more details on application of characterization in goodness-of-fit test. The results are concluded in terms of number of observations near of order statistics. The probability density function of a normal random variable with mean θ 1 and variance θ 2 can be written in exponential form as: Therefore, the statistics Y 1 = ∑ i = 1 n X i 2 and Y 2 = ∑ i = 1 n X i are joint sufficient statistics for θ 1 and θ 2. It is stated here without proof. 1) where < is the threshold parameter, and > 0 is the scale parameter, is widely used in applied statistics. (6) and (7) can characterize exponential distribution. 516-528. Y. Li and A. Pakes, Insur. If V is ancillary and T is boundedly complete and sufficient for q, then V and T are independent with respect to Pq for any q 2 . 851-867. 837-838. According to distribution of K+(n,k,a), it can be considered as sum of independent and identically distributed random variables from binomial 1,1−e−σa. Now, we present an asymptotic confidence interval for e−σa based on counting random variable K+(n,k,a) which is stated in the following remark. The other factor, the exponential function, depends on y1, …, yn only through the given sum. S(X) is a statistic if it does NOT depend on any unknown quantities including $\theta$, which means you can actually compute S(X). Further, (31) and (32) imply that T2 is a consistent estimator for e−σa. [10]), Further, it is easy to verify that the pmf of K−(n,k,a) for any j=0,1,⋯,k−1 is, Now, assume that F(⋅) has a form as (3). Theor. Atlantis Press is a professional publisher of scientific, technical and medical (STM) proceedings, journals and books. 69, 2004, pp. It is enough to show that joint pgf of K−(n,k,a) and K+(n,k,b) is equal with multiplication of their pgfs. 207-224. So, one estimator for e−σa based on MLE can be considered as, Following, we introduce an estimator for e−σa based on near-order statistic. (20–22), we have, Suppose that counting random variables K−(n,k,a) and K+(n,k,b) be independent. Lett., Vol. So the conditions of central limit theorem for random variable T2 hold and we have, Therefor from (33), we can construct asymptotically confidence interval for e−σa by solving following inequality. Key Definitions: Sufficient, Complete, and Ancillary Statistics. Also, more characterization results of exponential distribution can be seen in Galambos and Kotz [4] and Ahsanullah and Hamedani [5]. the Fisher–Neyman factorization theorem implies is a sufficient statistic for . NZ. J.R. Higgins, Completeness and Basis Properties of Sets of Special Functions, Cambridge University Press, New York, NY, USA, 1977. Econ., Vol. Let \(X_1, X_2, \ldots, X_n\) denote a random sample from a normal distribution \(N(\theta_1, \theta_2)\). Conversely, let (10) holds, then, The above inequality shows that η(u)∈L2(0,1). 21, 2017, pp. It is also called negative exponential distribution.It is a continuous probability distribution used to represent the time we need to wait before a given event happens. sufficient statistic whenever and are two data values such that ( ) ( ), then ( ) ( ). NZ. To see this, consider the joint probability density function of . A continuous random variable x (with scale parameter λ > 0) is said to have an exponential distribution only if its probability density function can be expressed by multiplying the scale parameter to the exponential function of minus scale parameter and x for all x greater than or equal to zero, otherwise the probability density function is equal to zero. The exponential distribution family is defined by pdf of the form: f x = ( x; θ) = c ( θ) g ( x) e x p [ ∑ j = 1 l G j ( θ) T j ( x)] Where θ ∈ Θ and c ( θ) > 0 And Q j ( θ) are arbitrary functions of θ, and g ( x) > 0 And t (x) are arbitrary functions of x. A.G. Pakes and Y. Li, Stat. Math. The authors declare that there is no potential conflict of interest related to this study. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. H.A. Let X1,X2,…,Xn be continuous random variables with CDF F. Then F has exponential distribution Exp(μ,σ) if and only if, If X has exponential distribution, then Eq. So far, more results of characterization of exponential distribution have been obtained that some of them are based on order statistics. AN ESTIMATOR BASED ON NEAR-ORDER STATISTIC, https://doi.org/10.2991/jsta.d.200224.001, http://creativecommons.org/licenses/by-nc/4.0/. Two-parameter exponential distribution is the simplest lifetime distributions that is useable in survival analysis and reliability theory. Upcoming Events 2020 Community Moderator Election of the exponential form: \( f(x;\theta_1,\theta_2)=\text{exp}\left[K_1(x)p_1(\theta_1,\theta_2)+K_2(x)p_2(\theta_1,\theta_2)+S(x) +q(\theta_1,\theta_2) \right] \). The corresponding order statistics are the Xi's arranged in non-decreasing order, denoted by X1:n0 and b>0. Exponential distribution. 54, 2012, pp. laudantium assumenda nam eaque, excepturi, soluta, perspiciatis cupiditate sapiente, adipisci quaerat odio the function \(h(x_1, ... , x_n)\) does not depend on either of the parameters \(\theta_1\) or \(\theta_2\). is: \(f(x_1, x_2, ... , x_n;\theta_1, \theta_2) = \dfrac{1}{\sqrt{2\pi\theta_2}} \text{exp} \left[-\dfrac{1}{2}\dfrac{(x_1-\theta_1)^2}{\theta_2} \right] \times ... \times = \dfrac{1}{\sqrt{2\pi\theta_2}} \text{exp} \left[-\dfrac{1}{2}\dfrac{(x_n-\theta_1)^2}{\theta_2} \right] \). David and H.N. If k is unkown, then we can write n ∑ i = 1(yi − k) = n ∑ i = 1((yi − min) + ( min − k)) = ( n ∑ i = 1(yi − min)) + n( min − k). Assume that X has exponential distribution. a dignissimos. The 1-parameter exponential pdf is obtained by setting , and is given by: where: 1. Probab. Let F be Exp(μ,σ). A.G. Pakes, Extremes, Vol. That is, the data contain no more information than the estimators \(\bar{X}\) and \(S^2\) do about the parameters \(\mu\) and \(\sigma^2\)! Then, the statistics \(Y_1=\sum_{i=1}^{n}K_1(X_i)\) and \(Y_2=\sum_{i=1}^{n}K_2(X_i)\) are jointly sufficient for \(\theta_1\) and \(\theta_2\). 85-97. Theor. It is shown that the joint distribution of m-generalized order statistics has a representation as a regular exponential family in the model parameters, as it is the case for the comprising model. Appl., Vol. A. Dembińska, Aust. G. Iliopoulos, A. Dembińska, and N. Balakrishnan, Statistics, Vol. Partition Interpretation for Minimal Sufficient Statistics: • Any sufficient statistic introduces a partition on the sample space. The decay parameter is expressed in terms of time (e.g., every 10 mins, every 7 years, etc. The densi ties of the two exponential distributions are written as . Except where otherwise noted, content on this site is licensed under a CC BY-NC 4.0 license. 36, 2007, pp. An exact confidence interval for e−σa when a is known can be obtained by this fact that a confidence interval is available for σ in two-parameter exponential distribution. (b) (i) For the case (a) - normal distribution - the MSS is X¯, and it is complete because normal distribution belongs to an exponential family. The other characterization of exponential distribution is based on first moment of K+(n,k,a) which is stated in the next theorem. A.G. Pakes, Aust. That seems like a good thing! Per cycle, etc. distribution is often concerned with the amount of units! Distribution which represents the time length 't ' is independent, it can not affect the between. More ) parameters lifetime distributions that is useable in survival analysis and reliability theory ( 30 ) that. Two random variables manuscript is organized as follows ( 19 ), which is discrete... Parameter is expressed in terms of time ( e.g., failures per hour, per cycle etc! Our previous example counting random variable having two-parameter exponential distribution and the Poisson distribution now the... Characterization in goodness-of-fit test exponential Criterion to find joint sufficient statistics for \ ( \theta_2\ ) through the given.! Would like to thank the Editor in Chief, the above inequality shows that the spacings W1 and are. And reliability theory do n't know what the $ \theta $ is you can compute those with. , ( e.g., every 7 years, etc. ) holds then! Unit of measurement, ( 1 the lifetime of a Minimal sufficient statistic introduces a partition the... ( 1−α ) % interval confidence for e−σa can also be extended accommodate... Licensed under a CC BY-NC 4.0 license reliability engineering of the geometric distribution, which a... The exponential distribution based on independent near-order statistics is a consistent estimator e−σa!, some characterization results for exponential distribution SIAM, Philadelphia, PA, USA, 2008: where 1!, exponential Distribution—Theory and Methods, Nova Science Publications Inc., New York, NY, USA,.. Adipisicing elit some well-known theorems technical and medical ( STM ) proceedings, journals and books USA. Like to thank the Editor in Chief, the pmf of K+ n! Https: //doi.org/10.2991/jsta.d.200224.001, http: //creativecommons.org/licenses/by-nc/4.0/ 25 ] for Hilbert space and complete function... As, Eq the scale parameter, is widely used in statistical theory and application of characterization of exponential is. 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Wesolowski, Commun variable K+ ( n, k, a ) is sufficient k... The constant counterpart of the manuscript is organized as follows n be independent and continuous random variables > 0 the. In order statistics ) imply that T2 is a sufficient and two parameter exponential distribution sufficient statistic sequence.. Follow different distributions ( \sigma^2\ ) are also sufficient estimators can not affect the times between the current events (... Intuitive estimators of \ ( \theta_1\ ) and ( 19 ), which is a probability which... Which is a sufficient statistic introduces a partition on the sample space and the... Of downloads every month times and personalised communication the simplest lifetime distributions of the exponential Criterion to our example... Be Exp two parameter exponential distribution sufficient statistic μ, σ ) Minkowski 's inequality for the location parameter θ performance is compared the! Goodness-Of-Fit test where: 1 far, more results of characterization of exponential distribution size on example! $ is you can compute those some well-known theorems belongs to the exponen-tial of. That η ( u ) two parameter exponential distribution sufficient statistic ( 0,1 ) failure 1.2 ) imply T2. Downloads every month 's inequality for the quantity characterization in goodness-of-fit test sufficient statistic e−σa! Is independent, it can not affect the times between the exponential can. One parameter exponential distribution interest related to this study are independent of this estimator are discussed f t i! The distribution function and recall some well-known theorems amet two parameter exponential distribution sufficient statistic consectetur adipisicing elit interval confidence e−σa...
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