linear exponential distribution

This is an open access article distributed under the terms of the Creative Commons CC BY license, which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited. ISSN (Online): 2319-7064. Distribution theory plays a vital role in modelling lifetime data not only in life insurance but also in various fields like reliability, queuing theory, and other related areas. This distribution is one of the best models to fit data with increasing failure rate (IFR). The bias, standard error, and coverage length (length of 95% confidence interval) for the MLE of each parameter are evaluated for each case. When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. where (x):=(x;1,2,3)=1x+22x23 and (x):=1+2x. how much do fdm pay. Therefore, accounting the flexibility of PDF and HRF, the NEGLED model can be utilized as an effective model for lifetime data applications. We had, however, found several mistakes upon reading this paper. Cited by lists all citing articles based on Crossref citations.Articles with the Crossref icon will open in a new tab. Table 1. It is not a good choice for modeling data that has decreasing, non linear increasing, or non-monotone failure rates [2]. Did you know that with a free Taylor & Francis Online account you can gain access to the following benefits? The function h ( x) must of course be non-negative. Let XNEGLED(). PDF 6: The Exponential Family and Generalized Linear Models 1 The Therefore, the distributions are modified or extended in the literature for further use. These values will help to test the goodness-of-fit of the NEGLED model compare to more familiar distribution models, namely GLFRD, GLED, RD, WD, and ED. 3099067 A note on the generalized linear exponential distribution Interpreting how much my linear model has improved after Box-Cox transformation, Data transformation prior to the logistic/linear regression model, Linear regression to predict both mean and SD of dependent variable, 'Undoing' a square root transformation for linear regression. E1X+22X23r=Er(X) =4(x)r+41(x)e4(x)1e4(x)1dxSubstituting u=4(x), on taking derivative we get du=4(x)41(x)dx. 0 The maximum likelihood estimations (MLE) of unknown parameters are also discussed. Exponentiated generalized linear exponential distribution (2003). Let we draw two independent random samples, i.e. We derive the expressions for regression function, product moments and the stress-strength parameter P ( X < Y) for the BGLE distribution. R. In the context of the reliability of a system, it is very important to study the system performance referred to as the stressstrength parameter. Comparing AIC values from Table 5, we mention that the NEGLED model has the smallest AIC value among all the considered distribution models. A set of real data is employed to illustrate the results given in this paper. 0000025551 00000 n (2014b), and Wu et al. Position where neither player can force an *exact* outcome. Exponential Distribution - Meaning, Formula, Calculation - WallStreetMojo are derived. It is a particular case of the gamma distribution. Table 3. You will get the summary of the exponential model setting the argument dispersion=1 in the summary function . All Answers (4) You can do this using the glmer function. The NEGLED model is best model in the sense that it has the largest p-value among all the used models here to fit the Leukemia dataset. To implement the LR test GLFRD, GLED, RD, WD, and ED have been considered as the null distributions, meanwhile, the NEGLED model has been taken as the alternative distribution. Density plots of NEGLED model at different values of , Figure 4. Robust Estimators for Marshal-Olkin Extended Linear Exponential on a single platform through NEGLED. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. Can lead-acid batteries be stored by removing the liquid from them? A problem about independent, i.i.d, random variables and exponential distribution. (2020), Bier (2019), C.-S. Lee and Tsai (2017), De Andrade and Zea (2018), Elbatal et al. Such as normal, binomial, Poisson and etc. Table 2. 127 21 0000015119 00000 n and corresponding log-likelihood functions l(;x) of above equation is l(;x)=mln4+mln+k=1mln(xk)+(41)k=1mln(xk)k=1m4(xk) +(1)k=1mln1e4(xk). Abstract This article introduces a new family of bivariate generalized linear exponential (BGLE) distributions, whose marginals are generalized linear exponential (GLE) distributions. Exponential random variable with parameter that is another exponential random variable. The linear part of the distribution is the hazard rate, which varies as a linear function of age or time [1]. Then we have (3.3) j=0ri=02jrj2ji(1)rj(12+223)rj1i2r2r2j+i2ji()=m=01(1)mr4+1(m+1)r4+1. Describe the form of predictor (independent) variables. The normal equations are given as l(;x)1=k=1m12xk+1+2(41)k=1mxk2xk2+21xk234241 (4.1) k=1mxk(2xk2+21xk23)411e22xk2+1xk341e22xk2+1xk34=0,(4.1) l(;x)2=k=1mxk2xk+1+(41)k=1mxk22xk2+21xk23424 (4.2) k=1mxk2(2xk2+21xk23)411e22xk2+1xk341e22xk2+1xk34=0,(4.2) l(;x)3=2(41)k=1m12xk2+21xk23+4241 (4.3) k=1m(2xk2+21xk23)411e22xk2+1xk341e22xk2+1xk34=0,(4.3) l(;x)4=m4+k=1mln22xk2+1xk3124k=1m(2xk2+21xk23)4 (4.4) ln22xk2+1xk31e22xk2+1xk341e22xk2+1xk34=0,(4.4) (4.5) l(;x)=m+k=1mln1e22xk2+1xk34=0.(4.5). Ue7.5 kdqPHKC V7``do1P[A@9qz An alternative to that would be to use a GLM with log link (in which case you'd be modelling $\log(E(y))$ as a linear function of parameters -- and expected values do come straight out of that model. Let X(1)X(2)X(m) denote the order statistics of the random sample X1,X2,,Xm from NEGLED() model. ;w~R!0``qs4/@ m We have r()=xrf(x;)dx =4xr(1+2x)1x+22x2341e1x+22x234 1e1x+22x2341dx. XNEGLED(), the rth raw moment of X is given by r()=i=0rj=0m=0riri2j1(1)i+m1i2j(12+223)ri2j2rj j4+1,(m+1)12+223224(m+1)j4+1+2ri2j(12+223)j2r+i2+j (3.2) ri24j4+1,(m+1)12+223224(m+1)ri24j4+1. But it does not provide a reasonable fit for modeling data with decreasing failure rate (DFR) and bathtub shaped failure rate (BTFR). So, the variance of NEGLE() distribution is derived as follow Var(X)=2()2 =22m=01(1)m14+1(m+1)14+11+3222. the exponential distribution against the linear exponential distribution is discussed. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. (3.3) Proof. For example, the Rayleigh distribution a submodel of the LE has PDF. There are many alternative ways than least squares to fit linear relationships, and some might be more suitable in the case of some non-normal conditional distributions. At the other end of the spectrum, the Tweedie distribution a member of the linear exponential family of distributions, has a PDF that is complex and cannot be expressed in closed form; its sometimes expressed as a series of functions. What is the rationale of climate activists pouring soup on Van Gogh paintings of sunflowers? It can be used to investigate a distributions most essential properties (e.g. The log-likelihood value of NEGLED model is largest compare to the considered models which indicates the best fit of NEGLED to the given dataset, see Table 5. The rth raw moment, with shape parameter 24 and scale parameter 22, of the Weibull distribution can be found by putting 1=3=0 and =1 in (3.2). Statistical Methods for Survival Data Analysis. Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine. MX(s), of X is given by MX(s)=r=0i=0rj=0m=0riri2j1(1)i+m1itrr!2j(12+223)ri2j2rj j4+1,(m+1)12+223224(m+1)j4+1+2ri2j(12+223)j2r+i2+j ri24j4+1,(m+1)12+223224(m+1)ri24j4+1. International Journal of Science and Research (IJSR). Remark 3.1. Therefore, the NEGLED model is chosen as the model with the best fit among all the distributions considered. Meanwhile, GLED does not provide reasonable fit to modelling phenomenon with bimodal density and constant hazard rate. It only takes a minute to sign up. The 1 and 2 are the scale parameters, 4 is the shape parameter, and is the exponentiation parameter. are 3.841, 5.991, 7.815, and 9.488, respectively. A comparison study between the maximum likelihood method, the unbiased estimates which are linear functions of. x=(x1,,xm) and y=(y1,,yn), from NEGLED(1,2,3,4,1) and NEGLED(1,2,3,4,2) of sizes m and n respectively. Can an adult sue someone who violated them as a child? (2020). were studied. Where to find hikes accessible in November and reachable by public transport from Denver? Pearson r has values between -1 and 1, where 1 is a perfect positive linear correlation, 0 is no linear . Let XNEGLED(1,2,3,4,1) and YNEGLED(1,2,3,4,2) be two independent random variables. [2] El-Damcese, M. & Marei, Gh. A New Generalization of Linear Exponential Distribution: Theory and =r=0i=0rj=0m=0riri2j1(1)i+m1itrr!2j(12+223)ri2j2rj j4+1,(m+1)12+223224(m+1)j4+1+2ri2j(12+223)j2r+i2+j ri24j4+1,(m+1)12+223224(m+1)ri24j4+1. Exponential distribution - Wikipedia The generalized linear exponential distribution - ScienceDirect The approximate 95% confidence intervals for 1,2,3,4, and are (0.2594, 0.2877), (0.2703, 0.2768), (0.1638, 0.3833), (1.0103, 1.5575), and (0.1156, 0.6628), respectively. 0. The failure rate function and the . If you begin with an exponential distribution with a constant failure rate, the result is the linear exponential distribution with a distribution function of [3], There are, however, a wide range of members in the linear exponential family, so youll come across a wide variety of different PDFs, which range from the basic to the complex. It is therefore clearly evident that at 5% level of significance we reject ED and none of the five models GLFRD, GLED, RD, WD, and NEGLED are rejected at the considerable level of significance. It was found that, compared to the other distributions (ED, RD, WD, GLED, and GLFRD), the NEGLED model offers a better fit to the Leukemia dataset. The PDF defined in (2.1) can also be written in simplified manner as f(x;)=4(x)41(x)e4(x)1e4(x)1,x>. In Table 4, KS test statistic values accompanying their p-values are shown for various modelling distributions. To overcome this drawback, we propose a new record-based transmuted . this is why I transform all x_s and y so that the histogram of them looks more or less normal. (2014), Mahmoud and Alam (2010), Okasha and Kayid (2016), Rajchakit et al. But it does not provide a reasonable fit for modeling data with decreasing failure rate (DFR) and bathtub shaped failure rate (BTFR). A New Exponentiated Generalized Linear E . 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Lemma 3.1. It has right-skewed, unimodal and bimodal density function. <<01DF24E6CF1E1B4CA9719B12A648DC6D>]>> It is easy to discuss various statistical properties of the GED, GLED, GRD, etc. MathJax reference. S. Lee and Kim (2019) introduced exponentiated generalized Pareto distribution. In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. (5.6) The function p ( x) depends only on x ( not on ), and the function q () is a normalizing constant. Then, for 1kOrder statistics from the linear-exponential distribution, part I question2: suppose I have a dataset with dependent variable y, and 1000 features (or independent variables) x_s, after removing some feature which are highly correlated with other features, I have, saying 300 x_s left, for whatever reasons, I decide to use linear regression (ols or glm, with or without regularization), I don't assume y is linearly dependent on x_s, but I assume the linear relation holds between f(y) and [g1(x1),g2(x2),], where f and g1,g2 are just some transformation function, such as log. If you expect the relationship between y and the x's to be linear, then a nonlinear transformation of y will make the relationship between it and the x's nonlinear. (2019), Louzada et al. Also, it is easy to get 2() on putting r=1 in (3.4) as follow E1X+22X23=m=01(1)m14+1(m+1)14+1 1+222()3=m=01(1)m14+1(m+1)14+1 2()=22m=01(1)m14+1(m+1)14+11+3.So, the variance of NEGLE() distribution is derived as follow Var(X)=2()2 =22m=01(1)m14+1(m+1)14+11+3222. Theorem 3.3. The maximum likelihood estimators, least squares and weighted least squares estimators, Anderson-Darling estimators, Cramr-von Mises estimators of the unknown parameters are constructed and their biases and mean squared errors are reported via Monte Carlo simulation study. =r=0srr()r! It is often used in actuarial science and survival analysis, where it is sometimes called the linear failure rate distribution. This work was supported by the No direct funding. startxref Thanks for contributing an answer to Cross Validated! An Introduction to the Exponential Distribution - Statology In addition to being used for the analysis of Poisson point processes it is found in var So P(YRecord-based transmuted generalized linear exponential distribution Relevant reliability measures of the linear exponential dis-tribution are also evaluated. For the random variable X with NEGLED(), i.e. In this paper, we study some statistical properties of this distribution such as negative moments, moments of order statistics, mean . Assume the distributions of the sample. Considering all the above results, we may conclude that the NEGLED model is superior competitor for lifetime datasets than the ED, RD, WD, GLFRD, and GLED models. Finally, the real data sets illustrate the goodness of fit and applicability of the proposed distribution; hence, suitable recommendations are forwarded. In recent time, many distributions are extended to the class of the exponentiated distributions. Then, for NEGLED, the joint density of X(1) and X(m) becomes of the form fX(1)X(m)(x1,xm;)=m(m1)422(x1)(xm)((x1)(xm))41e4(x1)4(xm) 1e4(x1)1e4(xm)11e4(xm)1e4(x1)m2. rev2022.11.7.43014. Finally, a dataset of 40 Leukemia patients was analysed for illustrative purpose proving that the NEGLED outperforms compared distributions. You should consider the spread about the relationship; if you know something about that it already it may help inform your choice of model (but beware your inferences if you're using the same data to identify the model as to make inferences about it). Mahmoud and Alam (2010) generalized LED to make it compatible with decreasing, increasing and bathtub shaped hazard rate and denoted by GLED(,,,). Now, we discuss the estimation of the model parameters by using the method of maximum likelihood estimation. No assumption about their distribution is made. Small values have relatively high probabilities, which consistently decline as data values increase. Then the log-likelihood function l(;x,y) of =(1,2,3,4,1,2) is l(;x,y)=(m+n)ln(4)+k=1mln(xk)+(41)k=1mln(xk)k=1m4(xk) +(11)k=1mln1e4(xk)+k=1nln(yk)+(41)k=1nln(yk) k=1n4(yk)+(21)k=1nln1e4(yk). Such as i = 0 + 1xi1 + + ipxip: Link function. Therefore, NEGLED can be used to model the phenomena with constant or non-monotone hazard rates. where is the mean of X, which can be derived by substituting r=1 in (3.2). This indicates that the MLE method provides good estimates of the parameters for the NEGLED model. The proposed distribution includes several known distributions as special cases, some of the most widely used distribution are shown in Table 1. Teleportation without loss of consciousness. Topp-Leone Linear Exponential Distribution | Request PDF A dataset of lifetimes (in days) for 40 patients suffering from leukemia type blood cancer, Table 4. Thus, the PDF of X(1) (the smallest order statistic) is fX(1)(x;)=m4(x)41(x)e4(x)1e4(x)111e4(x)m1,x. Figure 8. Making statements based on opinion; back them up with references or personal experience. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 0000002552 00000 n By closing this message, you are consenting to our use of cookies. R is R=11+2. (mk)!4(x)41(x)e4(x)1e4(x)k1 11e4(x)mk;x>. We introduce in this paper a new four-parameter generalized version of the linear exponential distribution wh ich is called Kumaraswamy linear exponential distribution.

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