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Certificate
This is to certify that the dissertation entitled “**Study** on the **problem** of **estimation** of **parameters** of **Generalized** **Exponential** **distribution**” is a bonafide record of independent reaserch work done by Sonu Munda, Roll no. 413MA2064 under the guidance of Dr. Suchandan Kayal and submitted to National Institute of Technology, Rourkela in partial fulfilment of the award of the degree of Master of Science in Mathematics.

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We calculate the average estimates (AEs), mean square errors (MSEs) and biases. The outcomes of the Monte Carlo simulation **study** are presented in Table 1. The findings of simulated results indicate that as ’n’ increases the MSE decrease and approaching towards zero, as usually expected under the first-order asymptotic theory. The aver- age parameter estimates tend to be closer to the true **parameters** as the sample size ’n’ increases. An obvious fact can be seen during **estimation** of **parameters** is that the asymptotic normal **distribution** provides a satisfactory approximation to the finite sam- ple **distribution** of the estimates. This normal approximation can be upgraded by the adjustment of bias to the estimates. First-order bias correction plays an excellent role in bias reduction but MSE might increase. Correction of bias is beneficial in practice depends mainly on the shape of the bias function and the variance of the MLE.

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In recent years there has been a growing interest in the **study** of inference problems associated with stress-strength model and record values. The **estimation** **problem** of R = P(Y < X) based on record values was firstly considered by Baklizi (2008) who estimated the reliability function based on URV for one and two **parameters** **exponential** **distribution**. Subsequent papers extended this work assuming various lifetime distributions for stress and strength random variables, for instance Baklizi (2012) estimated R based on URV for the Weibull **distribution**. Wang and Zhang (2013) estimated R for a class of distributions. Latterly, Salehi and Ahmadi (2015) considered the **estimation** of R based on URRSS from one-parameter **exponential** **distribution** and studied its performance.

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In this paper, we discussed the **estimation** **problem** of the unknown **parameters** of the GPW **distribution** based on progressive type-II censoring scheme, type-II censoring data and complete censoring data. We used MLE and Bayesian **estimation** methods to estimate the unknown **parameters**. The performances of the maximum likelihood estimators are also quite satisfactory. We obtained the Bayes estimates based on binary loss function and square error loss function under the assumption of independent gamma priors. A real data set is used to show how the scheme works in practice. The performance of the different estimator’s optimal censoring schemes is compared based on simulation **study** to determine the optimal censoring schemes by using MSE and Bias. It is observed that Bayesian estimates with respect to the gamma priors behave quite similarly with the corresponding MLE in terms of mean squared errors. We note that the complete censoring is a special case of the progressive type II censoring scheme that can be obtained simply by taking , and note that the usual type II censoring scheme is a special case of the progressive type II censoring scheme that can be obtained simply by taking . We note that, the

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interval, bootstrap interval and interval using the **generalized** pivot variable) of the stress- strength reliability in two-parameter **exponential** **distribution** based on upper records has been obtained by Baklizi (2014 b). Baklizi (2008 b) and Wong and Wu (2009) have discussed the MLE, Bayesian **estimation** and interval **estimation** of P ( X Y ) respectively using lower record values from the **generalized** **exponential** **distribution**. Hassan et al. (2015) have described the **estimation** of stress-strength reliability for exponentiated inverted weibull **distribution** based on lower record values. Basirat et al. (2016) have derived the **estimation** of stress strength parameter for proportional hazard rate models for upper record values. Condino et al. (2018) have considered a similar **problem** for proportional reversed hazard model based on lower records. Khan and Arshad (2016) have studied the UMVU **estimation** of reliability function and stress- strength reliability from proportional reverse hazard family based on lower records. MLE, approximate Bayes estimator and the exact CIs of stress-strength reliability for the two-parameter bathtub-shaped lifetime **distribution** based on upper record values have been deduced by Tarvirdizade and Ahmadpour (2016). Mahmoud et al. (2016) have deduced the result for the Bayesian **estimation** of P ( X Y ) for the Lomax **distribution** based on upper record values. In this paper, Mahmoud et al. (2016) described the MLE of stress-strength reliability in two cases, when all the **parameters** are unknown and when scale parameter is common and known. Amin (2017) has discussed the **estimation** of stress-strength reliability based on upper record values for Kumaraswamy **Exponential** **distribution**. Recently, Dhanya and Jeevavand (2018) have considered the Bayesian

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We note the greater the sample size, the greater the efficiency of the estimator in terms of lower values of MSE, Bias and L.C.I. If the ratio of effective sample sizes (r) increases for censoring sample, then the value of the MSE decreases for the **parameters** of WGED. The focus is on the output of the small ratio of effective censoring sample sizes, where the rare cases occur, and the small problems of loss. The **study** confirms the compatibility of the result in large Sampling sizes. The previous results confirm the compatibility of the Bayesian **estimation** for Linex loss function, which has the lowest of MSE values compared to other estimators, followed by SE loss function and finally the maximum likelihood **estimation**. scheme III is the best censoring scheme where it has the lowest MSE and the narrower L.C.I.

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Simulation **Study** of **Generalized** **Exponential** **Distribution** In the simulation **study**, sample sizes were chosen at n = 25, 50, and 100 to represent small, medium, and large data sets. The scale parameter is estimated for **Generalized** **Exponential** **distribution** with Maximum Likelihood and Bayesian using Jeffrey’s & extension of Jeffrey’s prior methods. For the scale parameter, α = 0.5, 1.0, and 1.5. The values of Jeffrey’s extension are chosen as c 1 = 1.0, 1.5, and 2. The value for the loss parameter c 2 = ±1.0 and ±2.0. This was iterated 5000 times and the scale parameter for each method was calculated. A simulation **study** was conducted in R-software to examine and compare the performance of the estimates for different sample sizes with different values for Jeffrey’s prior and the extension of Jeffrey’s prior under different loss functions. The results are presented in tables for different selections of the **parameters** and c extension of Jeffrey’s prior. In Table 2 , Bayes’ **estimation** with Al-Bayyati’s Loss function under Jeffrey’s

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, which gives the Bayes’
estimator under PLF using Jeffery’s prior.
Simulation **Study** of **Generalized** **Exponential** **Distribution** In the simulation **study**, sample sizes were chosen at n = 25, 50, and 100 to represent small, medium, and large data sets. The scale parameter is estimated for **Generalized** **Exponential** **distribution** with Maximum Likelihood and Bayesian using Jeffrey’s & extension of Jeffrey’s prior methods. For the scale parameter, α = 0.5, 1.0, and 1.5. The values of Jeffrey’s extension are chosen as c 1 = 1.0, 1.5, and 2. The value for the loss parameter c 2 = ±1.0 and ±2.0. This was iterated 5000 times and the scale parameter for each method was calculated. A simulation **study** was conducted in R-software to examine and compare the performance of the estimates for different sample sizes with different values for Jeffrey’s prior and the extension of Jeffrey’s prior under different loss functions. The results are presented in tables for different selections of the **parameters** and c extension of Jeffrey’s prior. In Table 2 , Bayes’ **estimation** with Al-Bayyati’s Loss function under Jeffrey’s

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In recent times, researchers have shown more interest in using Bayes approach over MLE to estimate **parameters**. Statistical software programs, such as WinBUGS and OpenBUGS, use the Bayes approach for parameter **estimation**. Full Bayes (FB) and EB are two different approaches that have been proposed in highway safety research. The FB approach is more flexible when compared to EB method which makes FB approach easier to use to model crash data (Miranda-Moreno, 2006). Researchers have shown interest in using hierarchical Bayes model to model crash data by Markov Chain Monte Carlo (MCMC) method (Miaou and Song, 2005; Miranda Moreno et al, 2007; Miaou and Lord, 2003). Lord and Park (2013) provided the sampling procedure for MCMC simulation by using slice sampling algorithm within Gibbs sampling. The formulation of the Poisson-Gamma model is given below:

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3 Department of Statistics and Operations Research, Modibbo Adama University, Yola, Adamawa State, Nigeria
Abstract We provide another generalization of the inverted **exponential** **distribution** which serves as a competitive model and an alternative to both the **generalized** inverse **exponential** **distribution** and the inverse **exponential** **distribution**. The model is positively skewed and its shape could be decreasing or unimodal (depending on its parameter values). The statistical properties of the proposed model are provided and the method of Maximum Likelihood **Estimation** (MLE) was proposed in estimating its **parameters**.

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As noted, θ ˆ and hence its MSE MG cannot be put in a convenient closed form. Therefore, MSE’s of the estimators are empirically evaluated based on a Monte-Carlo simulation **study** of 1,000 samples by MATLAB mainly for small sample sizes. The simulation **study** was carried out for θ = 1 with sample sizes n = 6, 9, 12, 15, 18 and 20. These samples were placed into five intervals ( k = 4 ) with δ = 1 . The loss and prior **parameters** are arbitrarily taken as c = −1.5, −1, −0.5, 0.5, 1 and 1.5, α = 2 and

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the proposed estimators with maximum likelihood estimators in terms of their risks. Raqab and Madi [13] discussed the classical and Bayesian inferential procedure for progressively type II censored data from the **generalized** Rayleigh **distribution**. The results showed that the maximum likelihood estimators of the scale and shape **parameters** can be obtained via EM algorithm based on progressive censoring. Krishna and Kumar [14] discussed the inference problems in Lindley **distribution** and the results shows that Lindley **distribution** provide good parametric fit under progressive censoring scheme for some real life situations. Also, some of the recent work on progressive censoring include but not limited to Kumar et al. [15], Pak et al. [16] and Rastogi and Tripathi [17]. As far as we know, no one has described the EM algorithm for determining the MLEs of the **parameters** of the Poisson-**Exponential** **distribution** based on progressive type- II censoring scheme.

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[8] N.R. Mann, Optimum estimators for linear functions of the location and scale **parameters**, Ann. Math. Statist. 40 (1969) 2149–2155.
[9] P. Pawlas, D. Szynal, Recurrence relations for single and product moments of **generalized** order statistics from Pareto, **generalized** Pareto and Burr distributions, Commun. Statist.-Theory Methods 30 (4) (2001) 739–746. [10] D. Pfeifer, Characterizations of **exponential** **distribution** of independent non-stationary record increments, J. Appl.

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The reversed **generalized** logistic RGL distributions are very useful classes of densities as they posses a wide range of indices of skewness and kurtosis. This paper considers the **estimation** **problem** for the **parameters** of the RGL **distribution** based on progressive Type II censoring. The maximum likelihood method for RGL **distribution** yields equations that have to be solved numerically, even when the complete sample is available. By approximating the likelihood equations, we obtain explicit estimators which are in approximation to the MLEs. Using these approximate estimators as starting values, we obtain the MLEs using iterative method. We examine numerically MLEs estimators and the approximate estimators and show that the approximation provides estimators that are almost as eﬃcient as MLEs. Also we show that the value of the MLEs decreases as the value of the shape parameter increases. An exact confidence interval and an exact joint confidence region for the **parameters** are constructed. Numerical example is presented in the methods proposed in this paper.

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The outline of the paper is as follows. In Section 2 we define the regression linear model with autoregressive errors, where the underlying **distribution** of the innovations is a **Generalized** **Exponential** **distribution**. In Section 3 we propose the MML estimators as a powerful methodology to deal with ML estimators which are intractable in the case of a **Generalized** **Exponential** **distribution**. In Section 4 we **study** the asymptotic properties of the proposed estimators. The main advantages of the proposed estimators are discussed via simulation studies in Section 5. Finally discussions and observations appear in Section 6 of the proposed model and the specific nu- merical results, attaching an Appendix which displays the details of asymptotic results.

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The simplicity of the logistic **distribution** and its importance as a growth curve have made it one of the most important statistical models. The shape of the logis- tic **distribution** (similar to that of the normal **distribution**) makes it simpler and also protable on suitable occasions to choose it as a model instead of the normal **distribution**. Pearl and Reed (1920, 1924), Schultz (1930) and Oliver (1982) ap- plied the logistic model as a growth model in human populations and in the **study** of the populations of some biological organisms. Some applications of logistic functions in bioassy problems were discussed by Berkson(1944) and Wilson and Worcester (1943). Other applications and signicant developments concerning the logistic **distribution** can be found in the book by Balakrishnan (1992). Balakrish- nan and Leung (1988) dened three types of **generalized** logistic distributions by compounding logistic **distribution** with some other well known models and named them as Type I, Type II and Type III **generalized** logistic distributions. Type III **generalized** logistic **distribution** was earlier derived by Gumbel (1944). In this ar- ticle, our main interest is to deal with **estimation** problems of Type III **generalized** logistic **distribution**.

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Let { X i , i ≥ 1 } be a sequence of independent and identically distributed (iid) random variables having an absolutely continuous cumulative **distribution** function (cdf) F (x) and probability density function (pdf) f (x). An observation X j is called a lower record if X j < X i for every i < j. An analogous definition deals with upper record values. In a number of situations, only observations that exceed or only those that fall below the current extreme value are recorded. Examples include meteorology, hydrology, athletic events and mining. Interest in records has increased steadily over the years since Chan- dler’s (1952) formulation. Useful surveys are given in Ahsanullah (1995) and Arnold et al. (1998). **Estimation** of **parameters** using record values and prediction of future record values have been studied by several authors, for details see Balakrishnan and Chan (1998), Raqab (2002), and Sultan et al. (2002). Bayesian **estimation** and prediction for some life distributions based on record values have been considered by Ahamadi and Doostparast (2006).

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In this section, we shall compare the estimators obtained under GELF with corresponding Bayes estimators under SELF and their mle's. The comparisons are based on the simulated risks(average loss over sample space) under GELF and SELF both. The exact expressions for the risks can not be obtained, therefore the risks of the estimators are estimated on the basis of Monte-Carlo simulation **study** of 5000 samples. It may be noted that the risks of the estimators will be the function of n , r , m , c , , , t and c 1 . In order to

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This article presents new method which is called partial **generalized** probability weighted moments for estimating the unknown **parameters** of distributions from censored samples. The PGPWMs will be used to estimate unknown **parameters** of EE **distribution** under doubly censored samples. Then, the PGPWMs estimators of the unknown **parameters** from left and right censored samples will be obtained as special cases. At the same time, the **generalized** probability weighted moments can be obtained as the special case from PGPWMs. To illustrate the properties of the new estimators, an extensive numerical **study** will be performed. Analysis of a real data set has been performed.

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This paper deals with the **estimation** of R=P(Y<X) when X and Y are two independent **generalized** **exponential** distributed random variables. We assume that the sample from each population contains one outlier. The MLE and Bayes estimator of R are obtained under the exchangeable and identifiable models. A simulation **study** is presented to