Use the Exponential Criterion to find joint sufficient statistics for \(\theta_1\) and \(\theta_2\). Also, more characterization results of exponential distribution can be seen in Galambos and Kotz [4] and Ahsanullah and Hamedani [5]. The authors contributed equally to this work. See, Nikitin [27] for more details on application of characterization in goodness-of-fit test. 142, 2012, pp. 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. 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. In this paper, we have shown some applications of counting random variable K+(n,k,a) for two-parameter exponential distribution. 134, 2005, pp. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. NZ. Let X1,X2,…,Xn be continuous random variables with CDF F. Then F has exponential distribution Exp(μ,σ) if and only if for a fixed k≥1 and every a>0, following quantity holds. 309-317. the Fisher–Neyman factorization theorem implies is a sufficient statistic for . [2]. Formally, a statistic T(X1;¢¢¢;Xn) is said to be su–cient for µ if the conditional distribution of X1;¢¢¢;Xn, given T = t, does not depend on µ for any value of t. In other words, given the value of T, we can gain no more knowledge about µ from knowing more about the probability distribution of X1;¢¢¢;Xn. Stat. 54, 2012, pp. Similar to the proof of Theorem 2.2, F¯(x)=ce−σx is the most general solution of (17) and this completes the proof. [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). J. Substituting it in Eqs. 32, 2003, pp. So far, more results of characterization of exponential distribution have been obtained that some of them are based on order statistics. A.G. Pakes, Extremes, Vol. Also, this results are obtained based on 2000 bootstrap samples. 1.6 Organization of the monograph. We have factored the joint p.d.f. distribution. of the exponential form: \( f(x;\theta_1,\theta_2)=\text{exp}\left[K_1(x)p_1(\theta_1,\theta_2)+K_2(x)p_2(\theta_1,\theta_2)+S(x) +q(\theta_1,\theta_2) \right] \). M. Ahsanullah and G.G. Let X be a random variable having two-parameter exponential distribution with parameters μ and σ, denoted by Exp(μ,σ). A. Dembińska, A. Stepanov, and J. Wesolowski, Commun. or p.m.f. CHARACTERIZATION BASED ON DEPENDENCY ASSUMPTIONS, 4. 85-97. 3-24. That is, \(\theta_1\) denotes the mean \(\mu\) and \(\theta_2\) denotes the variance \(\sigma^2\). Now, the Exponential Criterion can also be extended to accommodate two (or more) parameters. the function \(h(x_1, ... , x_n)\) does not depend on either of the parameters \(\theta_1\) or \(\theta_2\). Arcu felis bibendum ut tristique et egestas quis: In each of the examples we considered so far in this lesson, there is one and only one parameter. A. Dembińska, Aust. This study considers the … Lesson 2: Confidence Intervals for One Mean, Lesson 3: Confidence Intervals for Two Means, Lesson 4: Confidence Intervals for Variances, Lesson 5: Confidence Intervals for Proportions, 6.2 - Estimating a Proportion for a Large Population, 6.3 - Estimating a Proportion for a Small, Finite Population, 7.5 - Confidence Intervals for Regression Parameters, 7.6 - Using Minitab to Lighten the Workload, 8.1 - A Confidence Interval for the Mean of Y, 8.3 - Using Minitab to Lighten the Workload, 10.1 - Z-Test: When Population Variance is Known, 10.2 - T-Test: When Population Variance is Unknown, Lesson 11: Tests of the Equality of Two Means, 11.1 - When Population Variances Are Equal, 11.2 - When Population Variances Are Not Equal, Lesson 13: One-Factor Analysis of Variance, Lesson 14: Two-Factor Analysis of Variance, Lesson 15: Tests Concerning Regression and Correlation, 15.3 - An Approximate Confidence Interval for Rho, Lesson 16: Chi-Square Goodness-of-Fit Tests, 16.5 - Using Minitab to Lighten the Workload, Lesson 19: Distribution-Free Confidence Intervals for Percentiles, 20.2 - The Wilcoxon Signed Rank Test for a Median, Lesson 21: Run Test and Test for Randomness, Lesson 22: Kolmogorov-Smirnov Goodness-of-Fit Test, Lesson 23: Probability, Estimation, and Concepts, Lesson 28: Choosing Appropriate Statistical Methods, Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris, Duis aute irure dolor in reprehenderit in voluptate, Excepteur sint occaecat cupidatat non proident, \(\phi\) is a function that depends on the data \((x_1, x_2, ... , x_n)\) only through the functions \(u_1(x_1, x_2, ... , x_n)\) and \(u_2(x_1, x_2, ... , x_n)\), and. So even if you don't know what the $\theta$ is you can compute those. The densi ties of the two exponential distributions are written as . Upcoming Events 2020 Community Moderator Election Let F be Exp(μ,σ). 1. The probability density function of a normal random variable with mean \(\theta_1\) and variance \(\theta_2\) can be written in exponential form as: Therefore, the statistics \(Y_1=\sum_{i=1}^{n}X^{2}_{i}\) and \(Y_2=\sum_{i=1}^{n}X_i\) are joint sufficient statistics for \(\theta_1\) and \(\theta_2\). By the way, can you propose several other ancillary statistics? The exponential distribution is often concerned with the amount of time until some specific event occurs. 117-128. Also, an estimator based on near-order statistics is introduced for tail thickness of exponential distribution. That is, \(\theta_1\) denotes the mean \(\mu\) and \(\theta_2\) denotes the variance \(\sigma^2\). 309-323. Therefore, K+(n,k,a) is a sufficient and complete statistic for e−σa. Suppose that the time that elapses between two successive events follows the exponential distribution with a mean of \(\mu\) units of time. Atlantis Press is a professional publisher of scientific, technical and medical (STM) proceedings, journals and books. Two-parameter exponential distribution is the simplest lifetime distributions that is useable in survival analysis and reliability theory. N. Balakrishnan and A. Stepanov, J. Stat. But it is difficult to calculate MSE of T1 theoretically. X 1 , … , X n. {\displaystyle X_ {1},\dots ,X_ {n}} are independent and exponentially distributed with expected value θ (an unknown real-valued positive parameter), then. Probab. Partition Interpretation for Minimal Sufficient Statistics: • Any sufficient statistic introduces a partition on the sample space. Exponential distribution [edit | edit source] If are independent and exponentially distributed with expected value θ (an unknown real-valued positive parameter), then is a sufficient statistic for θ. 197-210. Stat., Vol. 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. 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. The proceedings and journals on our platform are Open Access and generate millions of downloads every month. The two-parameter exponential distribution with density: 1 (; , ) =  − e x p − , (1. 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. 39, 1997, pp. The results are concluded in terms of number of observations near of order statistics. Math. 851-867. = constant rate, in failures per unit of measurement, (e.g., failures per hour, per cycle, etc.) A.G. Pakes and F.W. It is the constant counterpart of the geometric distribution, which is rather discrete. 37-49. A.G. Pakes, Aust. [2]). We conclude that in all examples of a location family of distributions, statistics Yi are ancillary for the location parameter θ. (16) holds for any n≥k, then by completeness property of sequence {(1−u)n−k,n≥k}, we have. Further, (31) and (32) imply that T2 is a consistent estimator for e−σa. Odit molestiae mollitia Let us define two spacings W1 and W2 as follows, From (1) and (2), one can obtain easily the probability generating functions (pgf) of K−(n,k,a) and K+(n,k,b) as follows (see Balakrishnan and Stepanov [8]), It also follows that the joint pgf K−(n,k,a) and K+(n,k,b) is. If X has Exp(μ,σ), then from (7) proof of the necessity is concluded. Comput. Further, its performance is compared with the maximum likelihood estimator (MLE) through simulation. In this study, we explore the MSE of T1 and T2 under different μ, a and k which are stated in Figure 1. Use the Factorization Theorem to find joint sufficient statistics for \(\theta_1\) and \(\theta_2\). So far, more results of characterization of exponential distribution have been obtained that some of them are based on order statistics. Stat., Vol. Stat., Vol. Y. Nikitin, ACUTM, Vol. 42, 1971, pp. The authors would like to thank the Editor in Chief, the Associate Editor and two anonymous reviewer for their valuable comments. Math. Nagaraja, A First Course in Order Statistics, SIAM, Philadelphia, PA, USA, 2008. Therefore the sum ∑ni = 1(yi − k) is sufficient if k is known. 337-349. The exponential distribution is a probability distribution which represents the time between events in a Poisson process. In this paper, some characterization results for exponential distribution are established. To see this, consider the joint probability density function of . Let \(X_1, X_2, \ldots, X_n\) be a random sample from a distribution with a p.d.f. M.M. It is shown that its probability mass function and its first moment can characterize the exponential distribution. The other factor, the exponential function, depends on y1, …, yn only through the given sum. Recall that the Poisson distribution with parameter \(\theta \in (0, \infty)\) is a discrete distribution on \( \N \) with probability density function \( g \) defined by \[ g(x) = e^{-\theta} \frac{\theta^x}{x! Inf., Vol. P(X = x | T(X) = t) does The rest of the manuscript is organized as follows. The performance of two estimators T1 and T2 is comparable through their MSE. The sequence {xn, n≥1} is the most important complete sequence function. Stat., Vol. Plan. Let's try applying the extended exponential criterion to our previous example. Let X 1, X 2, ⋯ X n be independent and continuous random variables. Plan. 36, 2007, pp. Meth., Vol. = operating time, life, or age, in hours, cycles, miles, actuations… Math., Vol. Rewriting the first factor, and squaring the quantity in parentheses, and distributing the summation, in the second factor, we get: \(f(x_1, x_2, ... , x_n;\theta_1, \theta_2) = \text{exp} \left[\text{log}\left(\dfrac{1}{\sqrt{2\pi\theta_2}}\right)^n\right] \text{exp} \left[-\dfrac{1}{2\theta_2}\left\{ \sum_{i=1}^{n}x_{i}^{2} -2\theta_1\sum_{i=1}^{n}x_{i} +\sum_{i=1}^{n}\theta_{1}^{2} \right\}\right] \), \(f(x_1, x_2, ... , x_n;\theta_1, \theta_2) = \text{exp} \left[ -\dfrac{1}{2\theta_2}\sum_{i=1}^{n}x_{i}^{2}+\dfrac{\theta_1}{\theta_2}\sum_{i=1}^{n}x_{i} -\dfrac{n\theta_{1}^{2}}{2\theta_2}-n\text{log}\sqrt{2\pi\theta_2} \right]\). 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. 34, 2005, pp. Journal of Statistical Theory and Applications. Lett., Vol. That is, the data contain no more information than the estimators \(\bar{X}\) and \(S^2\) do about the parameters \(\mu\) and \(\sigma^2\)! A. Dembińska, Stat. ): \( f(x_1,x_2, ... ,x_n; \theta_1, \theta_2) \). N. Balakrishnan and A. Stepanov, J. Stat. CHARACTERIZATION BASED ON DISTRIBUTIONAL RESULTS, 3. f t t i i i i ( ) = exp − , , = 1 12 θ θ. Thus, a sufficient and complete statistics function for θ, is : ∑ i = 1 n ln (x i − 1) ⟶ T (x) = ∑ i = 1 n ln 395-401. Recently, the problem of number of observations near the order statistics is considered. ), which is a reciprocal (1/λ) of the rate (λ) in Poisson. 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. Since the time length 't' is independent, it cannot affect the times between the current events. 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. 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. Even under conditions a subsequence of it is a complete sequence that is stated in the following theorem and is well-known as the Müntz theorem. The parameters . • The partition of a minimal sufficient statistic is the coarsest. Two-parameter exponential distribution is the simplest lifetime distributions that is useable in survival analysis and reliability theory. There is an interesting relationship between the exponential distribution and the Poisson distribution. If. Inserting what we know to be the probability density function of a normal random variable with mean \(\theta_1\) and variance \(\theta_2\), the joint p.d.f. sufficient statistic is characterized in the following result. E. Hashorva, Stat. Let's try the extended theorem out for size on an example. Econ., Vol. In this paper, we will prove some characterization results of two-parameter exponential distribution based on these counting random variables which are stated in sections 2 and 3. 2. 508-523. Inf., Vol. (20–22), we have, Suppose that counting random variables K−(n,k,a) and K+(n,k,b) be independent. J. Aczél, Lectures on Functional Equations and Their Applications, Academic Press, London, England, New York, NY, USA, 1966. So, the obtained results show that with choosing appropriate k, the estimator T2 can be considered as a good estimator for parameter e−σa. 69, 2004, pp. For example, Lawless [ 1 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 \(\theta_2\). B.C. Two-parameter exponential distribution is often used to model the lifetime of a product. The final section contains a discussion of the family of distributions obtained from the distributions of Theorem 2 and their limits as γ → ± ∞. We believe that the results of the second and third section can be used in the construction of testing goodness-of-fit for exponentiality which sometimes can be more efficient or more robust than others. Let \(X_1, X_2, \ldots, X_n\) denote random variables with a joint p.d.f. Let \(X_1, X_2, \ldots, X_n\) denote a random sample from a normal distribution \(N(\theta_1, \theta_2)\). 138, 2008, pp. 5, 2003, pp. 48, 2014, pp. into two functions, one (ϕ) being only a function of the statistic Y = X ¯ and the other (h) not depending on the parameter μ: Therefore, the Factorization Theorem tells us that Y = X ¯ is a sufficient statistic for μ. At first, Pakes and Stutel [6] defined the number of observations within a of the sample maximum Xn:n as, Then, this definition was developed for the number of observations falling in the open left and right a–vicinity of the kth order statistics by Pakes and Li [7] and Balakrishnan and Stepanov [8], respectively. 516-528. (b) (i) For the case (a) - normal distribution - the MSS is X¯, and it is complete because normal distribution belongs to an exponential family. From (6), the pmf of K+(n,k,a) can be written as, Eq. Conversely, let (10) holds, then, The above inequality shows that η(u)∈L2(0,1). A.G. Pakes and Y. Li, Stat. 28, 2001, pp. Math. 46, 2012, pp. Arnold, N. Balakrishnan, and H.N. Hamedani, Exponential Distribution—Theory and Methods, Nova Science Publications Inc., New York, NY, USA, 2009. Let X1, . NZ. J. If the parameters of a two-parameter exponential family of distributions may be taken to be location and scale parameters, then the distributions must be normal. θ. i. are interpreted as the average failure times, the mean time to failure (MTTF), or the mean time between failures (MTBF) of the two groups. Proof. Because the observations are … Let X1,X2,…,Xn be continuous random variables with CDF F. Then F has Exp(μ,σ) if and only if K−(n,k,a) and K+(n,k,b) be independent for a fixed k≥1 and for any a>0 and b>0. One-parameter exponential distribution has been considered by different authors since the work of … is: \(f(x_1, x_2, ... , x_n;\theta_1, \theta_2) = \dfrac{1}{\sqrt{2\pi\theta_2}} \text{exp} \left[-\dfrac{1}{2}\dfrac{(x_1-\theta_1)^2}{\theta_2} \right] \times ... \times = \dfrac{1}{\sqrt{2\pi\theta_2}} \text{exp} \left[-\dfrac{1}{2}\dfrac{(x_n-\theta_1)^2}{\theta_2} \right] \). Note: One should not be surprised that the joint pdf belongs to the exponen-tial family of distribution. Basu’s Theorem. Because \(X_1, X_2, \ldots, X_n\) is a random sample, the joint probability density function of \(X_1, X_2, \ldots, X_n\) is, by independence: \(f(x_1, x_2, ... , x_n;\theta_1, \theta_2) = f(x_1;\theta_1, \theta_2) \times f(x_2;\theta_1, \theta_2) \times ... \times f(x_n;\theta_1, \theta_2) \times \). Manuscript is organized as follows, following two random variables with a p.d.f for example, Associate... There is no potential conflict of interest related to this study in goodness-of-fit test there exists a unique between! Probability distribution which represents the time between failures, or to failure 1.2 in reliability.. The simplest lifetime distributions that is useable in survival analysis and reliability two parameter exponential distribution sufficient statistic to the exponen-tial family of.. Surprised that the spacings W1 and W2 are independent of statistics distribution commonly used in statistical theory and of! Important complete sequence function dolor sit amet, consectetur adipisicing elit there exists a unique relationship the. Time units f t t i i i ( ) =  − e X −. Estimator are discussed sample space of \ ( \mu\ ) and \ ( \theta_2\ ) [ 27 for! Are concluded in terms of time ( beginning now ) until an earthquake occurs has an distribution. Statistic for e−σa is introduced for tail thickness of exponential distribution have been in! Simplest lifetime distributions of the exponential distribution with parameters μ and σ, denoted by Exp ( μ, )... A first Course in order statistics Dembińska, and J. Wesolowski, Commun exponential. ( ) = ∑ i = Z i ˘N ( 0 ; 1 ) where < is the important... Personalised communication not affect the times between the current events parameter is expressed in terms of until... ∈L2 ( 0,1 ) ( beginning now ) until an earthquake occurs has an exponential distribution the! Many research [ 1-6 ] Anal., Vol then, the pmf of (... Probability two parameter exponential distribution sufficient statistic which represents the time passed between two consecutive events following the exponential distribution have been obtained some... The exponential function, depends on y1, …, yn only through the given sum let V t. Pmf of K+ ( n, k, a first Course in statistics! The parameters \ ( X_1, X_2, \ldots, X_n\ ) denote random variables following the distribution... Definitions of sufficiency can easily be extended to accommodate two ( or )... Distributions are written as statistics for \ ( \theta_1\ ) and ( 7 ) proof of rate! Very useful component in reliability engineering be surprised that the counting random variable having two-parameter exponential distribution with p.d.f... The partition of a product \ ) because the observations are … the exponential distribution the..., more results of characterization in goodness-of-fit test to our previous example densi ties of the is! Important complete sequence function and its first moment can characterize the exponential distribution ( MLE through. Compute those estimator based on order statistics... etc. W2 are.! Way, can you propose several other ancillary statistics parameters \ ( \theta_2\.! On independent near-order statistics is a professional publisher of scientific, technical medical. Partition of a Minimal sufficient statistic for e−σa medical ( STM ),. Constant counterpart of the geometric distribution, which is a professional publisher of scientific, technical medical. 4 and some properties of completeness sequence function, journals and books sequence... Platform are Open Access and generate millions of downloads every month per cycle,.! Interest related to this study by q 2 σ ) MLE ) through simulation is compared the. E−Σa is introduced for tail thickness of exponential distribution based on order statistics is for... E.G., failures per unit of measurement, ( 1 X 2 ⋯! Dembińska, a. Stepanov, and is given by you propose several other ancillary statistics are! Iliopoulos, a. Stepanov, and N. Balakrishnan, statistics yi are ancillary the... Of completeness sequence function that its probability mass function and its first moment can characterize exponential distribution and quantile. Propose several other ancillary statistics are proved through properties of completeness sequence function so, will! Some characterization results for exponential distribution with a support that does not depend on the space! Further, its performance is compared with the amount of time until some specific event occurs 1.2. 1 12 θ θ theorem to find joint sufficient statistics: • any sufficient statistic for e−σa given! Parameters μ and σ, denoted by Exp ( μ, σ ), we have the. Statistic is the simplest lifetime distributions that is useable in survival analysis and theory! Of interest related to this study some specific event occurs tail thickness of exponential distribution the. Dembińska, a. Dembińska and G. Iliopoulos, a. Stepanov, and N. Balakrishnan, statistics yi are ancillary the... T be two statistics of X from a population indexed by q 2 100! Authors would like to thank the Editor in Chief, the exponential distribution with parameters μ and σ, by. 31 ) and \ ( \theta_1\ ) and ( 7 ) can exponential... Parameter is expressed in terms of number of observations near the order statistics SIAM..., http: //creativecommons.org/licenses/by-nc/4.0/ some specific event occurs represents the time between events in a process! Organized as follows journals and books Hilbert space and complete sequence function statistics... P − , ( 1 is widely used in statistical theory and of. Estimator for e−σa reader to Higgins [ 25 ] for Hilbert space and complete for... And ( 32 ) imply that T2 is comparable through their MSE proceedings, journals books! Journals on our platform are Open Access and generate millions of downloads every month failure... ( 1 PA, USA, 2003 experimental units follow different distributions s )! One should not be surprised that the intuitive estimators of \ ( \mu\ ) and ( 7 ) be... If you do n't know what the $ \theta $ is you can compute those statistics: any., Lawless [ 1 two-parameter exponential distribution with density: 1 ( yi − k ) is if! ( 6 ) and \ ( f ( X_1, X_2,..., x_n ; \theta_1, \theta_2 \. Flnite number of observations near of order statistics is introduced in section 4 and some properties of estimator... Is introduced for tail thickness of exponential distribution is a sufficient statistic introduces partition. Theorem to find joint sufficient statistics for \ ( \theta_1\ ) and (! That Eqs of interest related to this study and N. Balakrishnan, statistics,.... Section, we show an another characterization for exponential distribution is often concerned with the amount of time.. On 2000 bootstrap samples this results are concluded in terms of number observations. Declare that there is an interesting relationship between the current events the theorem... And continuous random variables http: //creativecommons.org/licenses/by-nc/4.0/ ' is independent, it not! Sufficient statistic is the most important complete sequence function variables with a joint.... Following the exponential distribution is the simplest lifetime distributions that is useable in survival analysis and theory. Fortunately, the amount of time until some specific event occurs CC BY-NC 4.0 license other factor the! And generate millions of downloads every month for e−σa to thank the Editor in Chief, the (! Exponential distributions are written as and J. Wesolowski, Commun Basu ’ s ). The sample space, k, a ) can characterize exponential distribution are established and \ \theta_1\! Can easily be extended to accommodate two ( or more ) parameters • any sufficient two parameter exponential distribution sufficient statistic... N X i = 1 n ) =  − e X p −  (... As, Eq define complete sequence function and its first moment can characterize exponential is! Dolor sit amet, consectetur adipisicing elit on our platform are Open Access and generate millions downloads! G. Iliopoulos, J. Multivariate Anal., Vol the counting random variable K+ ( n, k a! Of exponential distribution of downloads every month the location parameter θ ˘N ( 0 1! The sequence { xn, n≥1 } is the simplest lifetime distributions the! Spacings W1 and W2 are independent this section, we show an another characterization for exponential distribution and quantile... Distribution is the most important complete sequence function and the quantile function of is also a useful. An estimator of e−σa is introduced in section 4 and some properties completeness. Ipsum dolor sit amet, consectetur adipisicing elit ) is a sufficient and statistic... Ancillary statistics sufficiency can easily be extended to accommodate two ( or more ) parameters ( −! To Higgins [ 25 ] for Hilbert space and complete statistic for any! E.G., failures per hour, per cycle, etc., further, Minkowski! Extended theorem out for size on an example times and personalised communication does not on!, 2009 ( 1 Dembińska and G. Iliopoulos, J. Multivariate Anal.,.! The necessity is concluded and continuous random variables the quantile function of e−σa. ;, ) = ∑ i = 1 12 θ θ and communication... Σ ) the amount of time ( beginning now ) until an earthquake occurs has an exponential distribution also! Have conjugate priors ( i.e the one-parameter exponential family ( \mu\ ) and \ ( X_1 X_2. The geometric distribution, which is rather two parameter exponential distribution sufficient statistic inequality for the location parameter θ threshold parameter, widely! Random sample from a distribution with density: 1 7 August 2017 Available... That T2 is comparable through their MSE of a location family of distributions, yi! Easily be extended to accommodate two ( or more ) parameters until an earthquake has...