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Characterization of Probability Distribution through Generalized Order Statistics
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A.H. Khan, M. Faizan, Ziaul Haque
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Volume 3, Issue 3, 2011
pp.
38 - 47
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Received
09 April 2011,
Accepted
17 November 2011
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Abstract.
A family of probability distribution has been characterized through the difference of two conditional expectations, conditioned on a non-adjacent generalized order statistic. Also a result based on the unconditional expectation and a conditional expectation is used to characterize a family of distribution. Further, some of its deductions are also discussed.
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Keywords.
Characterization of distributions; Conditional expectation; Probability distributions; Generalized order statistics; Order statistics; Record statistics.
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On the Central Limit Theorem for m Dependent Random Variables with Unbounded m
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Yilun Shang
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Volume 3, Issue 3, 2011
pp.
32 - 37
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Received
05 August 2011,
Accepted
28 October 2011
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Abstract.
For each $k=1,2,\cdots$, let $n=n(k),m=m(k)$ and suppose $\{X_{ki}\}_{i\ge 1}$ is an $m$-dependent sequence of random variables. Under some regularity conditions, we show that $N_{kn}^{-1/2}(X_{k1}+\cdots+X_{kN_{kn}})$ is asymptotically normal, where $\{N_{ki}\}_{i\ge1}$ is a sequence of positive integer-valued random variables tending to infinity.
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Keywords.
Central limit theorem; $m$-Dependent random variables.
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Statistical Inference and Characterizations from Independent and Identical Exponential-Bernoulli Mixture Distribution
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M.M. Mohie El-Din, M.M. Amein, E.H. Hafez
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Volume 3, Issue 3, 2011
pp.
15 - 31
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Received
31 May 2011,
Accepted
15 September 2011
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Abstract.
In this paper, the recurrence relations for moments of order statistics for exponential-Bernoulli mixture distribution and its characterizations are obtained. Estimation for parameters of exponential-Bernoulli mixture distribution are given in two ways. First, using maximum likelihood method based on order statistics. Second, using best linear unbiased estimators technique.
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Keywords.
Order statistics; Single moments; Product moments; Recurrence relations; Characterizations; Exponential-Bernoulli; Inference.
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Recurrence Relations for Quotient Moments of Generalized Order Statistics and Characterization
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M.M. Mohie El-Din, M.S. Kotb
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Volume 3, Issue 3, 2011
pp.
1 - 14
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Received
12 March 2011,
Accepted
14 September 2011
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Abstract.
In this paper, we established recurrence relations satisfied by the quotient moments of generalized order statistics for a general class of distribution. Also, we presented characterization by considering the conditional expectation for general class satisfied by the quotient moments of generalized order statistics.
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Keywords.
Generalized order statistics; Quotient moments; Recurrence relations; Generalized Pareto distribution; Characterizations.
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On the Mixed Sum of Doubly Infinite and Mid Truncated Independent Random Variables
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M. M. Mohie-El-Din, Ahmed M. T. Abd El-Bar, A. M. Abd El-Raheem
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Volume 3, Issue 2, 2011
pp.
58 - 69
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Received
07 June 2011,
Accepted
06 September 2011
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Abstract.
The aim of the present paper is to study the distribution of the mixed sum of two random variables. Here we derive a theorem which gives the probability density function $(pdf)$ of sum of doubly infinite and mid truncated independent random variables. The distribution of the infinite and mid truncated independent random variables is given as a corollary. As an application of these results we have obtained a distribution of sum of bilateral exponential distribution with the mid truncated uniform distribution , the bilateral exponential distribution with the mid truncated exponential distribution and the exponential distribution with the mid truncated uniform distribution.
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Keywords.
Doubly infinite distribution; Exponential distribution; Exponential bilateral distribution; Mid truncated distribution; Mid truncated exponential distribution; Mid truncated uniform distribution; Mixed sum.
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On Mid-Truncated Distributions and its Applications in Order Statistic
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Hassan Okasha, Moustafa Mohie El-Din, Mohamed Mahmoud
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Volume 3, Issue 2, 2011
pp.
47 - 57
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Received
23 June 2011,
Accepted
15 August 2011
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Abstract.
In this paper, we will define the Mid-truncated distribution and we obtain the recurrence relation for single and double moments of order statistics in random samples of size n from the Mid-truncated distribution. Applications for Weibull and exponential distributions are given to illustrate the main results.
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Keywords.
Mid-truncated; Order statistics; Weibull distribution; Exponential distribution; Single moments; Double moments; Recurrence relations; Incomplete beta function.
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Effect of Estimation Method on the Empirical Characteristic Function Goodness-of-Fit Test
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Suzanne A. Allam
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Volume 3, Issue 2, 2011
pp.
28 - 46
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Received
23 March 2011,
Accepted
04 July 2011
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Abstract.
For studying the null hypothesis that a random sample belongs to a specific probability distribution, the unknown parameters of this distribution must be estimated first. Then, the test statistic; which depends on these estimated parameters; will be calculated from the random sample. The goodness-of-fit test of [8], which is based on the empirical characteristic function, is applied to the generalized exponential distribution. This is done after estimating the unknown parameters of the generalized exponential distribution by different methods of estimation. Then the effect of the estimation method on the power of the test is studied using extensive simulations.
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Keywords.
Goodness-of-fit tests; Empirical characteristic function; Generalized exponential distribution; Maximum likelihood estimation; Moment estimation.
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Estimation and Prediction for the Inverse Weibull Distribution Based on Records
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M.M. Mohie El-Din, Fathy H. Riad
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Volume 3, Issue 2, 2011
pp.
20 - 27
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Received
27 January 2011,
Accepted
26 June 2011
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Abstract.
In this paper, we obtained based on record value, the maximum likelihood, minimum variance unbiased and Bayes estimators of the two parameters of the Inverse Weibull distribution are computed and compared. A Bayesian prediction interval for the $sth$ future record is obtained in a closed form. Based on simulated record values, numerical computations and comparisons between the different estimators are given.
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Keywords.
Inverse Weibull distribution; Record values; Bayesian inference Prediction.
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Random Sum of Mid Truncated Exponentiated Exponential Distribution
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M. M. Mohie-El-Din, Ahmed M.T. Abd El-Bar and A. M. Abd El-Raheem
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Volume 3, Issue 2, 2011
pp.
1 - 19
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Received
15 February 2011,
Accepted
27 May 2011
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Abstract.
The concept of random sum of distributions in actuarial science is used to represent such variables as the total amount of claims or losses payable by an insurer. In this note we are concerned with the random sum $s_{N}$ and sums $s_{n}$, where every constituent follows the mid truncated exponentiated exponential distribution. We derive the $pdf$ \ of random sum of mid truncated exponentiated exponential distribution. Finally, the pdf of sum of exponentiated exponential distribution is obtained.
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Keywords.
Exponentiated exponential distribution; Mid truncated distribution; Unit step function; Random sum; Complete multinomial expansions.
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Reliability of Connected (1,1,2) or (1,2,1) or (2,1,1)-out-of-(n,2,2): F Lattice Systems
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M. Gharib, E.M. El-Sayedb, I.I.H. Nashwan
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Volume 3, Issue 1, 2011
pp.
47 - 56
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Received
25 October 2010,
Accepted
04 April 2011
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Abstract.
A three dimensional consecutive $k$-out-of-$n$: $F$ system is a cuboid grid of edges $n_{1}, n_{2}, n_{3}$ containing $n_{1}\times n_{2}\times n_{3}$ components and fails if and only if there is at least a cuboid of edges $r_{1}, r_{2}, r_{3}$, where $r_{1}\leq r_{1}, r_{2}\leq n_{2}, r_{3}\leq n_{3}$, containing $r_{1}. r_{2}. r_{3}$ of failed components. Few researchers set numerous algorithms to compute the reliability of special cases of such systems.\\ A connected $(1,1,2)$ or $(1,2,1)$ or $(2,1,1)$-out-of-$(n,2,2)$: $F$ lattice system, consists of $4n$ components, arranged in a cuboid of $n$ layers, each layer consists of $2\times2$ components. This system fails if and only if it contains at least 2 connected failed components.\\ In this paper, the reliability of the connected $(1,1,2)$ or $(1,2,1)$ or $(2,1,1)$-out-of-$(n,2,2)$: $F$ lattice system is obtained utilizing a general formula that is driven using a Markov chain technique and a recursive algorithm.
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Keywords.
Three dimensional consecutive $(r_{1},r_{2},r_{3})$-out-of-$(n_{1},n_{2},n_{3})$: $F$ system; Connected $(1,1,2)$ or $(1,2,1)$ or $(2,1,1)$-out-of-$(n,2,2)$: $F$ lattice system; Markov chain; Recursive algorithm.
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