Estimate the Parameters and Related Probability Functions for Data of the Patients of Lymph Glands Cancer via Birnbaum-Saunders Model Abbas N. Salman Taha A. Taha Mathematics Dept./College of Education for pure Science (Ibn Al-Haitham)/ University of Baghdad Received in : 16 June 2013 , Accepted in : 26 August 2013 Abstract In this paper,we estimate the parameters and related probability functions, survival function, cumulative distribution function , hazard function(failure rate) and failure (death) probability function(pdf) for two parameters Birnbaum-Saunders distribution which is fitting the complete data for the patients of lymph glands cancer. Estimating the parameters (shape and scale) using (maximum likelihood , regression quantile and shrinkage) methods and then compute the value of mentioned related probability functions depending on sample from real data which describe the duration of survivor for patients who suffer from the lymph glands cancer based on diagnosis of disease or the inter of patients in a hospital for period of three years ( start with 2010 to the end of 2012) .Calculating and estimating all previous probability functions , then comparing the numerical estimation by using statistical indicators mean squares error and mean absolute percentage error between the three considered estimation methods with respect to survival function. Concluding that, the survival function for the lymph glands cancer by using shrinkage method is the best. Keywords : Birnbaum-Saunders Distribution , Lymph Glands Cancer Disease, Complete Real Data, Maximum Likelihood Estimator, Regression Quantile(Least Square) Method and Shrinkage Estimator, Mean Squares Error and Mean Absolute Percentage Error. 387 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Introduction 1.1 The Procedures The medical experiment is most important experiment which is related with human, thus the cancer disease is one of the diseases which hit the health of human and the lymph glands cancer disease is lethal cancer which killed human .This paper concerns with complete real data for failure (death) time for this disease which is fitting two parameters Birnbaum- Saunders model. Furthermore, estimating the parameters of the mentioned model, by using three methods (maximum likelihood estimator, regression quantile estimator and shrinkage estimator),depending upon iterative numerical method (Newton-Raphson method),then utilizing these estimated parameters to estimate the probability of failure(death) function , cumulative distribution function, survival function and hazard function .Finally, three considered estimators were compared, by using the mean squares error and mean absolute percentage error with respect to survival function to indicate the best estimator. 1.2 Description of Data Cancer develops when cells in a part of the body begin to grow out of control. Survival analysis is concerned with studying the time between entry to a study and a subsequent event, such as death. Lymphoma is a cancer that starts in the lymphoid cells in the immune system, and serves as a solid tumor of lymphoid cells. It can be treated with chemotherapy, and radiation therapy in some cases and / or bone marrow transplantation, and can be cured, depending on the tissue type, and stage of the disease. These malignant cells arise often in the lymph nodes, as a display of the node enlargement (tumor). Lymphoma is the most common form of hematologic malignancies, or "blood cancer", in the developed world. And classified some forms of cancer and lymph nodes (such as lymphoma, small lymphocytic) lazy, compatible with long life even without treatment, while other forms are aggressive (such as Burkitt's lymphoma), causing rapid deterioration and death. However, the most aggressive lymphoma responds well to treatment and cure. Speculate therefore depends on the correct classification of the disease, which was established by a pathologist after examination of a biopsy. This paper depends on real data for the Lymph Glands cancer diseases, choosing this type of cancer because it is diffusion and deadly in current time in Iraq .To collect data for the Lymph Glands Cancer diseases returning Medical City Teaching Complex in Baghdad-Iraq (Baghdad hospital Teaching) ,Baghdad. The time (in days) of study point in this paper is determined for three years (started with2010 to the end of 2012), that means of the duration time of this study is constant. The number of patients in the experiment for the above duration time is (92) including (51) males and (41) females. All ninety two patients were dead during the time of study, that means the data became complete data. 1.3 Goodness of Fit It is very important that we test whether the random variable T (the real time data of considered lymph glands cancer diseases) follows the Birnbaum –Saunders distribution. We use the software program (Easy Fit Professional) to fit the curve of demonstrating the good matching, of this data to the specific probability distribution. This program uses a variety of tests under consideration like Kolmukrove Samirnov test, Anderson-Darling test and Chi- Square test. We found from this program, the mentioned real time data follow Birnbaum- Saunders distribution. 1.4 The Model Birnbaum and Saunders [11] proposed the two-parameter failure time distribution for fatigue failure caused under cyclic loading. It was also assumed that the failure is due to the 388 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 development and growth of a dominant crack. This distribution is the so-called two- parameters Birnbaum–Saunders distribution. A more general derivation was provided by Desmond [5] based on a biological model. Desmond also strengthened the physical justification for the use of this distribution by relaxing the assumptions made by Birnbaum and Saunders[11] . Desmond [6] investigated the relationship between the Birnbaum– Saunders distribution and the inverse Gaussian distribution. Artur and Silvia [4] introduce size and power properties of some tests in the Birnbaum-Saunders regression model. The random variable T is said to follow a BS distribution with parameters α and β, denoted as BS (α, β), if its cumulative distribution function (CDF) is given by F(𝑡, 𝛼, 𝛽) = Φ� 1 𝛼 �� 𝒕 𝜷 � 𝟏 𝟐 − � 𝜷 𝒕 � 𝟏 𝟐 �� , t > 0 and α, β > 0 … (𝟏) Where Φ (.) is the standard normal CDF, α and β are the shape and the scale parameters respectively. The parameter β is the median of the distribution: FT (β) = Φ(0) =0.5 It is noteworthy that the reciprocal property holds for the BS distribution T−1~BS(α, β−1). While, the probability density function (p.d.f) for the two parameters Birnbaum – Saunders distribution of the random variable T is given by: 𝒇(𝐭; 𝛂, 𝛃) = 𝐭 −𝟑 𝟐 (𝐭 + 𝛃) 𝐞𝐱𝐩�− 𝟏𝟐𝛂𝟐 � 𝐭 𝛃 + 𝛃 𝐭 − 𝟐�� 𝟐𝛂�𝛃√𝟐𝛑 ,0 < 𝑡 < ∞ and β, α > 0 … (𝟐) Estimation Methods: In this section, we introduce three methods for the estimation of the parameters of the Birnbaum-Saunders distribution and the related probability functions as follows: 2.1 Maximum Likelihood Method(MLE) The idea behind the maximum likelihood approach to fitting a statistical distribution to a data set is to find the parameters of the distribution that maximize the likelihood of having observed the data. Assuming the data are independent of each other, the likelihood of the data is the product of the likelihoods of each datum. See; [1]𝑎𝑛𝑑[11] Thus, the likelihood function of two-parameters Birnbaum-Saunders is: L = �𝑓(𝑡𝑖 𝑛 𝑖=1 , 𝛼 , 𝛽) … (3) L = ∏ t −3 2 (t + β)ni=1 2nαnβ n 2(2π) n 2 exp −� 1 2α2 �( ti β − 2 + n i=1 β ti )� … (4) Taking the logarithm for the above likelihood function, so we get the following: Ln L = −n Ln 2 − n Ln α − 𝑛 2 Ln 𝛽 − n 2 Ln(2π) − 3 2 Ln � ti n i=0 + Ln �(ti + 𝛽) n i=0 − � 1 α2 �� ti β � n i=0 + � 𝛽 𝑡𝑖 � − 2� . . . (5) The partial derivative for the log- likelihood function with respect to unknown parameters α and β are respectively as below: 389 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 ∂Ln L ∂α = −n α + 1 α3 �( ti β n i=1 − 2 + β ti ) … (6) Equating the partial derivative to zero and resolve this equation: ∂Ln L(ti) ∂α = 0 … (7) α� = � s β� + β� r − 2� 1 2 … (8) Where, s=�1 n ∑ tᵢni=0 � is the arithmetic mean and r=�1 n ∑ 1 tᵢ n i=0 � is the harmonic mean. Also, the partial derivative for log-likelihood w.r.t. β, is as follows: ∂Ln L ∂β = −n 2β� + � 1 tᵢ + β� n i=0 − 1 2α�2 �( 1 β�2 n i=0 − 1 tᵢ ) … (9) Equating the partial derivative to zero and resolve this equation: ∂Ln L ∂β = 0 … (10) −n 2β� + � 1 tᵢ + β� n i=0 − 1 2α�2 �( 1 β�2 n i=0 − 1 tᵢ ) = 0 … (11) Form equations (8) and (11) we can write the formula as: �̂�2 − 𝛽 ��2𝑟 + 𝑘(�̂�)� + 𝑟�𝑠 + 𝑘(𝛽 � ) � = 0 … (12) Where, K (�̂�) =�1 𝑛 ∑ 1 𝑡ᵢ+𝛽� 𝑛 𝑖=0 � −1 ; β ≥ 0 Since (12) is a non-linear equation in β�, we shall use the Newton-Raphson method to find �̂�. As a consequence, the related estimation probability function using this method is as follows: 𝒇�𝑴𝑳(𝒕) = 𝐭− 𝟑 𝟐(𝐭 + 𝜷�𝑴𝑳) 𝐞𝐱𝐩� −𝟏 𝟐𝛂�𝐌𝐋 �𝛃 �𝐌𝐋 𝐭 + 𝐭 𝛃�𝐌𝐋 − 𝟐�� 𝟐𝜶�𝑴𝑳�𝜷�𝑴𝑳√𝟐𝝅 , 𝒉�𝑴𝑳(𝒕) = 𝒇�𝑴𝑳(𝒕) 𝑺�𝑴𝑳(𝒕) 𝑭�𝑴𝑳(𝒕) = 𝚽� 𝟏 𝜶�𝑴𝑳 �� 𝒕 𝜷�𝑴𝑳 � 𝟏 𝟐 − � 𝜷�𝑴𝑳 𝒕 � 𝟏 𝟐 �� , 𝑺�𝑴𝑳(𝒕) = 𝟏 − 𝚽� 𝟏 𝜶�𝑴𝑳 �� 𝒕 𝜷�𝑴𝑳 � 𝟏 𝟐 − � 𝜷�𝑴𝑳 𝒕 � 𝟏 𝟐 �� 2.2 Regression-Quantile (Least Square) Method (RQE): The regression-quantile (least square) method [8], is based on the minimization of the quadratic measure of the difference between the empirical distribution function Fn(t) and the theoretical cumulative distribution function F(t) = Φ( λ √t − µ√t ) … (13 a) Where, α= 1 �µλ , and, β=λ µ and λ, μ> 0 …(13 b) 390 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 If T(1) ≤ T(2) ≤ K ≤ T(n) are order statistics of T1, T2, K, Tn , then by definition the empirical distribution function is given by Fn(tk ) = k/n ,k=1, K, n. Considering the following asymptotic equality Φ−1 �1 − k n � ≈ 𝛌 √𝐭𝐤 − µ√tk k=1,…, n-1. This can be used for the parameter estimation. Hence, the estimation of the parameters can be obtained by finding the minimum of the following function: 𝐆(𝛌, 𝛍) = � � 𝛌 √𝐭𝐤 − 𝛍�𝐭𝐤 − 𝐲𝐤� 𝟐𝐧 𝐤=𝟏 … (𝟏𝟒) Where yk = Φ−1 �1 − k n � … (13) , for k = 1, … , n − 1. Since Φ−1(0) = −∞ , tn is chosen by condition of further minimization of the function G. The partial derivative for G w .r. t. to 𝜆 and 𝜇 respectively and equating the two equations with zero , we obtain the following: 𝛛𝐆 𝛛𝛍 = 𝟐�� 𝛌 √𝐭𝐤 − 𝛍�𝐭𝐤 − 𝐲𝐤� (�𝐭𝐤 ) … . (𝟏𝟓) 𝛛𝐆 𝛛𝛍 = 𝟎 . . . . (𝟏𝟔) ⇒ 𝟐�� 𝛌 √𝐭𝐤 − 𝛍�𝐭𝐤 − 𝐲𝐤� (�𝐭𝐤) = 𝟎 … (𝟏𝟕) 𝛛𝐆 𝛛𝛌� = 𝟐�� 𝛌� �𝐭(𝐤) − 𝛍��𝐭(𝐤) − 𝐲𝐤�� 𝟏 �𝐭(𝐤) � … (𝟏𝟖) 𝛛𝐆 𝛛𝛌� = 0 … (𝟏𝟗) ⇒ 𝟐�� 𝛌� �𝐭(𝐤) − 𝛍��𝐭(𝐤) − 𝐲𝐢�� 𝟏 �𝐭(𝐤) � = 𝟎 … (𝟐𝟎) Rewriting the statistics T1 and T2 are as follows : T1 = 1 n � t(k) n k=1 , T2 = 1 n � 1 t(k) n k=1 Furthermore, T3 = � y(k) n K=1 �tk , T4 = 1 n � y(k) �t(k) n k=1 From equations (17) and (20) we get µ� = T2 T3 − T4 1 − T1T2 … (𝟐𝟏) λ� = T3 − T1T4 1 − T1T2 … (𝟐𝟐) Thus, depending on equation (13 b),we can find the estimation of α and β as below: 𝛼�= 1 �λ�µ� and, �̂�=𝜆 � 𝜇� . As a result, the related estimation probability function using this method is as follows: 391 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 𝒇�𝑹𝑸(𝒕) = 𝐭− 𝟑 𝟐(𝐭 + 𝜷�𝑹𝑸) 𝐞𝐱𝐩� −𝟏 𝟐𝛂�𝐑𝐐 � 𝛃�𝐑𝐐 𝐭 + 𝐭 𝛃�𝐑𝐐 − 𝟐�� 𝟐𝜶�𝑹𝑸�𝜷�𝑹𝑸 √𝟐𝝅 , 𝒉�𝑹𝑸(𝒕) = 𝒇�𝑹𝑸(𝒕) 𝑺�𝑹𝑸(𝒕) 𝑭�𝑹𝑸(𝒕) = 𝚽� 𝟏 𝜶�𝑹𝑸 �� 𝒕 𝜷�𝑹𝑸 � 𝟏 𝟐 − � 𝜷�𝑹𝑸 𝒕 � 𝟏 𝟐 �� ; 𝑺�𝑹𝑸(𝒕) = 𝟏 − 𝚽� 𝟏 𝜶�𝑹𝑸 �� 𝒕 𝜷�𝑹𝑸 � 𝟏 𝟐 − � 𝜷�𝑹𝑸 𝒕 � 𝟏 𝟐 �� 2.3 Shrinkage Method (SHE) The shrinkage estimation method is one of the Bayesian approach depended on prior information regarding the value of the specific parameter 𝜃 due to past experiences or from previous studies .However, in certain situations the prior information is available only in the form of an initial guess value (natural origin) 𝜃° of . In such a situation it is natural to start with an estimator 𝜃� (e.g. MLE) of 𝜃 and modify it by moving it closer to 𝜃°.Thompson [9] suggested the problem of shrinking an unbiased estimator 𝜃� of the parameter 𝜃 toward prior information (a natural origin) 𝜃° by single stage shrinkage estimator k𝜃� + (1 − 𝑘)𝜃° , 0≤ 𝑘 ≤ 1, which is more efficient than 𝜃� if 𝜃° is close to 𝜃 and less efficient than 𝜃� otherwise. According to Thompson [9] and AL − joboori[2,3], 𝜃° is a natural origin and as such may arise for anyone of a number of reasons, e.g., we are estimating 𝜃 and: (i) We believe 𝜃° is closed to the true value of 𝜃, or(ii) We fear that 𝜃° may be near the true value of 𝜃 ,i.e., something bad happens if 𝜃° = 𝜃 and we do not know about it.(i.e.; something bad happens if 𝜃°≈𝜃 and we doesn't use 𝜃°). Where, k is so called shrinkage weight factor; 0≤ 𝑘 ≤ 1 which represents the belief of 𝜃� ,and (1-k) represents the belief of 𝜃°. Thompson [9]. Noting that the shrinkage weight factor may be a function of 𝜃� or may be constant and the choosing of k is ad hoc basis. In this paper, we supposed K=𝑒 −𝑛 10 , 0≤ 𝑘 ≤ 1 , and 𝜃 = 𝜃° . Where, 𝜃 may refer to 𝛼 𝑜𝑟 𝛽 and n=92 Therefore, the shrinkage estimators of 𝛼and 𝛽 respectively became as below: 𝛼�𝑠ℎ = 𝑘 𝛼�𝑀𝐿 + (1 − 𝐾)𝛼° , 𝛼 = 𝛼° … . (23) �̂�𝑠ℎ= 𝐾 �̂�𝑀𝐿 + (1 − 𝐾)𝛽° , 𝛽 = 𝛽° … . (24) Hence, the estimation of the related probability functions using this method is as follows: 𝒇�𝑺𝑯(𝒕) = 𝐭− 𝟑 𝟐(𝐭 + 𝜷�𝑺𝑯) 𝐞𝐱𝐩� −𝟏 𝟐𝛂�𝐒𝐇 �𝛃 �𝐒𝐇 𝐭 + 𝐭 𝛃�𝐒𝐇 − 𝟐�� 𝟐𝜶�𝑺𝑯�𝜷�𝑺𝑯 √𝟐𝝅 , 𝒉�𝑺𝑯(𝒕) = 𝒇�𝑺𝑯(𝒕) 𝑺�𝑺𝑯(𝒕) 𝑭�𝑺𝑯(𝒕) = 𝚽� 𝟏 𝜶�𝑺𝑯 �� 𝒕 𝜷�𝑺𝑯 � 𝟏 𝟐 − �𝜷 �𝑺𝑯 𝒕 � 𝟏 𝟐 �� , 𝑺�𝑺𝑯(𝒕) = 𝟏 − 𝚽� 𝟏 𝜶�𝑺𝑯 �� 𝒕 𝜷�𝑺𝑯 � 𝟏 𝟐 − �𝜷 �𝑺𝑯 𝒕 � 𝟏 𝟐 �� 392 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 The Result of Estimation In this section, we use the program MATLAB to obtain the estimation values for the shape and scale parameters as well as the related probability functions in Birnbaum-Saunders distribution under specific complete data. Newton -Raphson method [7] was used which is one of the numerical analysis methods. 3.1 Estimation the Parameters and Related Pr. Functions by MLE Method To find the estimated values for the shape and scale parameters in Birnbaum-Saunders distribution, Newton -Raphson method was used which is one of the numerical analysis methods . Applying the maximum likelihood estimator method for estimating the two unknown parameters of Birnbaum−Saunders distribution under complete data by using MATLAB (2012a) program. Estimating these unknown parameters using Newton-Raphson gave the program including small number of iteration and the smallest value of error. The results of estimation which satisfy above conditions are demonstrated in annexed table (1). Now, these estimated values for two parameters in Birnbaum-Saunders distribution are used to find estimation values for probability, of death density function𝑓𝑀𝐿(𝑡) , cumulative distribution function 𝐹�𝑀𝐿(𝑡) , survival function�̂�𝑀𝐿(𝑡) and hazard function ℎ�𝑀𝐿(𝑡) . The results for the estimation of these probability functions are shown in the table (2). From the mentioned table, we display the following conclusions: 1. Noting that the values of death density function 𝑓𝑀𝐿(𝑡) were increasing slightly until (t= 23) then the values became decreasing slightly until the end of failure times. The patients who remain in a hospital for (t=1046) days had smallest probability of death with (0.00173947) value while the patients who remain in a hospital for (t=23) days had the largest probability of death with (0.571728762) value. 2. The values of cumulative death distribution function F�ML(t) is increasing with the increase of failure times because it collects the probability values for all previous observations step by step, that means there is a direct relationship between failure times and cumulative death distribution function. 3. The values of survival function S�ML(t) are decreasing gradually with the increase of the failure times for the Lymph Glands cancer patients in the hospital, that means there is an opposite relationship between failure times and survival function. Showing that the value of survival function for patients was high when the patients stay alive in the hospital for the Lymph Glands Cancer was low and vice versa, that means the patient who remains (t =11) days in a hospital had the greatest survival function with (0.98214277335) value ,but the patient who remains( t=1046 ) days in a hospital had a smallest value of survival function with (0.00623383239) value . 4. Noting that the values of hazard function h�ML(t) are increasing gradually with the increase the failure times of patients for the Lymph Glands Cancer patients in the hospital until (t=28) and decrease after that. That means there is concave and convex regions respectively when plot the failure times vs. hazard function. Showing that the patient who remains (t=1046) days in a hospital had the smallest value of hazard function for death with (0.27903817907) value but the patient who remains ( t=28 ) days in a hospital had a largest value of hazard function for death with (0.65225072150) . 5. The mean squares error and mean absolute percentage error for survival function which estimate the parameters in Birnbaum-Saunders distribution by MLE method is MSE�𝑆 �(𝑡)� 𝑀𝐿𝐸 = 0.000000000036821 and MAPE��̂�(𝑡)� 𝑀𝐸𝐿 = 0.000025281310636 . 393 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 3.2 Estimation the Parameters and Related Function by RQE Method To find the estimated values for the shape and scale parameters in Birnbaum- Saunders . Newton -Raphson method was used. After that ,applying the regression-quantile estimator method for estimating the two unknown parameters of Birnbaum-Saunders distribution under complete data by using program of MATLAB (2012 a) .The estimation values of the specific parameter using RQE method are carried out in attached table(3). Now, these estimated values for two parameters in Birnbaum-Saunders distribution are used to find the estimation of the probability death (density) function𝑓𝑅𝑄(𝑡) , cumulative distribution function 𝐹�𝑅𝑄(𝑡) , survival function �̂�𝑅𝑄(𝑡) and hazard function ℎ�𝑅𝑄(𝑡) . The results for these four probability functions are displayed in table (4). From this table, we demonstrate the following conclusions: 1. The values of death density function 𝑓𝑅𝑄(𝑡) were increasing slightly until (t= 22) then the values became decreasing slightly until the end of failure times. The patients who remain in a hospital for (t=1046) days had smallest probability of death with (0.00142356349) value while the patients who remain in a hospital for (t=22) days had largest probability of death with (0.593766620475) value. 2. The values of cumulative death distribution function F�RQ(t) are increasing with the increase of failure times because it collects the probability values for all previous observations step by step, that means there is a direct relationship between failure times and cumulative death distribution function . 3. The values of survival function �̂�𝑅𝑄(t) are decreasing gradually with the increase of the failure times for the Lymph Glands cancer patients in the hospital, that means there is an opposite relationship between failure times and survival function. Showing that the value of survival function for patients was high when the patients stay alive in the hospital for the Lymph Glands Cancer was low and vice versa, that means the patient who remains (t =11) days in a hospital had a greatest survival function with (0.97970942845) value ,but the patient who remains( t=1046 ) days in a hospital had a smallest value of survival function with (0.00487351604) value . 4. Noting that the values of hazard function ℎ�𝑅𝑄(t) are increasing gradually with the increase of the failure times of patients for the Lymph Glands Cancer patients in the hospital until (t=27) and decreasing after that. That means there is concave and convex regions respectively when plot the failure times vs. hazard function. Showing that the patient who remains (t=1046) days in a hospital had a smallest value of hazard function for death with (0.29210194025) value but the patient who remains ( t=27 ) days in a hospital had a largest value of hazard function for death with (0.67814630492) value . 5. The mean squares error and mean absolute percentage error for survival function which estimate the parameters in Birnbaum-Saunders distribution by RQE method is MSE��̂�(𝑡)� 𝑅𝑄𝐸 = 0.000157322008012 and MAPE��̂�(𝑡)� 𝑅𝑄𝐸 = 0.044260791532672 . 3.3. Estimation the Parameters and Related Function by Shrinkage Method: To find the estimated values for the shape and scale parameters in Birnbaum- Saunders. Newton -Raphson method was used. Applying the shrinkage estimator method for estimating the two unknown parameters of Birnbaum-Saunders distribution under complete data using MATLAB (2012 a) program. The estimation values of the specific parameter using SH. method are performed in the annexed table (5). Now , these estimated values for the two parameters in Birnbaum-Saunders distribution are used for estimating the probability death (density) function 𝑓𝑆𝐻(𝑡) , cumulative distribution function , 𝐹�𝑆𝐻(𝑡), survival function �̂�𝑆𝐻(𝑡) and hazard function ℎ�𝑆𝐻(𝑡) . The results for these four probability functions are placed in the table (6). From this table, we display the following conclusions: 394 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 1. The values of death (density) function are increasing slightly until (t= 23) then the values became decreasing slightly until the end of failure times. The patients who remain in a hospital for (t=1046) days had smallest probability of death with (0.00173911118) value while the patients who remain in a hospital for (t=23) days had largest probability of death with (0.57168907904) value. 2. The values of cumulative death distribution function F�SH(t) are increasing with the increase of failure times because it collects the probability values for all previous observations step by step, that means there is a direct relationship between failure times and cumulative death distribution function . 3. The values of survival function S�SH(t) are decreasing gradually with the increase of the failure times for the Lymph Glands cancer patients in the hospital , that means there is an opposite relationship between failure times and survival function . Showing that the value of survival function for patients was high when the patients stay alive in the hospital for the Lymph Glands Cancer was low and vice versa , that means the patient who remains (t =11) days in a hospital had a greatest survival function with (0.98214623898) value ,but the patient who remains( t=1046 ) days in a hospital had a smallest value of survival function with (0.00623244779) value . 4.The values of hazard function h�SH(t) are increasing gradually with the increase of the failure times of patients for the Lymph Glands Cancer patients in the hospital until (t=28) and decreasing after that. That means there is concave and convex regions respectively when plot the failure times vs. hazard function. Showing that the patient who remains (t=1046) days in a hospital had a smallest value of hazard function for death with (0.27904143697) value but the patient who remains (t=28) days in a hospital had a largest value of hazard function for death with (0.65220903452) value. 5. The mean squares error and mean absolute percentage error for survival function which estimate the parameters in Birnbaum-Saunders distribution by MLE method is MSE��̂�(𝑡)� 𝑆𝐻 = 0.00000000000 and MAPE��̂�(𝑡)� 𝑆𝐻 = 0.000000002554411 . Comparisons Between three Considered Estimation Methods This section is related with comparisons between the three considered estimators using the two statistical indicators, "Mean Squared Error and Mean Absolute Percentage Error" with respect to the survival function. i.e. ; MSE��̂�(𝑡𝑖)� = ∑ ��̂�(𝑡𝑖) − 𝑆(𝑡𝑖)� 2𝑛 𝑖=0 𝑛 , MAPE��̂�(𝑡𝑖� = ∑ ��̂�(𝑡𝑖) − 𝑆(𝑡𝑖)� 𝑛 𝑖=0 𝑆(𝑡𝑖) 𝑛 . . (25) Where, 𝑆(𝑡𝑖) is the real survival function and �̂�(𝑡𝑖) is the estimated survival function As a consequence, the computations of mentioned statistical indicators are carried out in annexed table (7): That indicates , the mean squares error and mean absolute percentage using shrinkage method is less than that of MLE and RQE methods, so the shrinkage method is the best. Conclusions and Results We estimated the parameters of Birnbaum-Saunders distribution using MLE, RQE and shrinkage methods for complete data and the related probability functions, and we conclude the following results. i. There is a direct relationship between the failure times and the estimated cumulative distribution function for each considered estimators . 395 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 ii. There is concave and convex regions respectively when ploting the failure times vs. hazard function, for each considered estimators. iii. There is an opposite relationship between the failure times and survival function, for each considered estimators . iv. There is a vibrate relationship between the failure times and the probability death (density function) for each considered estimators . v. According to comparisons between MLE, RQE and SH Methods, we conclude that the shrinkage estimation method is the best in the sense of MSE and MAPE. References [1] Mohammed. R. A.(2012). comparing some of parameters estimation methods for reliability functions for Birnbaum – Saunders distribution has two parameters by using the simulation. Master Thesis, Baghdad University, College of Administration and Economic. [2]Al-Joboori. A.N (2010). "Pre-Test Single and Double Stage Shrunken Estimators for the Mean of Normal Distribution with Known Variance". Baghdad Journal for Science. Vol.7(4). pp.1432-1442. [3]Al-Joboori. A.N. (2011) .On Significance Test Estimator for the Shape Parameter of Generalized Rayleigh Distribution .AL-Qadesyia j. for computer and mathematics Sciences. vol.3. No.2. PP.390-399. [4]Artur J. L. and Silvia L. p. F. (2010). Size and power properties of some tests in the Birnbaum- Saunders regression model. Stat. ME. pp. 1-13. [5]Desmond, A.F. (1985). Stochastic models of failure in random environments. Canad. J.Statist. 13, 171-183 [6]Desmond. A.F. (1986). On the relationship between two fatigue-life models. IEEE Trans. Reliable. 35. 167-169 [7]Mathews J. H. and Fink K. D. (2003) . Numerical Method Using MATLAB . Third Edition . Prentice Hall .USA . [8]Syed E..K. and Supranee L. (2008). Parametric Estimation for the Birnbaum – Saunders Lifetime Distribution Based one New Parameterizations. Thailand Statistician, 6(2). vol. 213 – 240. [9] Thompson, J.R. (1968). Some Shrinkage Techniques for Estimating the Mean. J. Amer. Statist. Assoc.vol.63. pp.113-122. [10]Birnbaum . Z. W. and Saunders. S. C. (1969). "Estimation for a family of life distributions with applications to fatigue J. Applied Probability. vol. 6. pp. 328 – 347. [11] Birnbaum. Z. W and Saunders.S. C. (1969a). A New Family of Life Distribution. J. Applied Probability. vol. 6. pp. 319 – 327. Table (1): Estimated values for the parameters by MLE method Estimated values Number of iteration Errors for all parameters α� = 1.221842074333230 β� = 93.132544387531269 3 0 396 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Table (2): Estimated values for the functions 𝑓𝑀𝐿(t), 𝐹�𝑀𝐿(t), �̂�𝑀𝐿(t), ℎ�𝑀𝐿(t) �̂�𝐌𝐋(𝒕) 𝑺�𝑴𝑳(𝒕) 𝐅�𝐌𝐋(𝒕) 𝒇�𝑴𝑳(𝒕) Time/d 0.3752948445 74820 0.4220912945 55910 0.4628353445 30202 0.4978589786 30721 0.5276662798 30023 0.5528223538 15185 0.5738908128 46531 0.5914011645 94833 0.6058339530 83585 0.6176160575 36867 0.6346752582 54723 0.6405577825 03542 0.6450105107 58342 0.6482407175 43248 0.6517192844 00955 0.6522507215 01328 0.6521325728 34339 0.6503188099 33909 0.6447304202 87006 0.6397163679 20829 0.6369415500 78818 0.6340287754 98070 0.6310033648 77197 0.6278872250 15437 0.9821427733 57806 0.9764984204 25002 0.9702733867 69026 0.9635613850 38287 0.9564487220 93328 0.9490126013 15602 0.9413207596 84283 0.9334318395 45599 0.9253961263 18909 0.9172564338 30801 0.9008044198 13457 0.8925483038 13239 0.8843021163 76903 0.8760837265 10071 0.8597870382 25750 0.8517310126 70074 0.8437480583 72176 0.8280264474 24463 0.8051179481 83015 0.7903235986 04586 0.7830733259 72993 0.7759214481 76802 0.7688678689 64593 0.7619122079 81771 0.017857226642 194 0.023501579574 998 0.029726613230 974 0.036438614961 713 0.043551277906 672 0.050987398684 398 0.058679240315 717 0.066568160454 402 0.074603873681 091 0.082743566169 199 0.099195580186 543 0.107451696186 761 0.115697883623 097 0.123916273489 929 0.140212961774 250 0.148268987329 926 0.156251941627 824 0.171973552575 537 0.194882051816 985 0.209676401395 414 0.216926674027 007 0.224078551823 198 0.231132131035 407 0.238087792018 229 0.3685931194 77601 0.4121714824 08991 0.4490768172 53728 0.4797176870 03165 0.5046857390 35166 0.5246353800 59563 0.5402153359 24527 0.5520326769 77164 0.5606363933 76021 0.5665123024 12906 0.5717182777 82102 0.5717287622 67906 0.5703841597 48949 0.5679131435 00851 0.5603397932 89702 0.5555421675 39112 0.5502355921 30225 0.5384811738 82880 0.5190840331 12647 0.5055829419 81444 0.4987719380 70614 0.4919565256 70227 0.4851582124 62618 0.4783949419 75058 11 12 13 14 15 16 17 18 19 20 22 23 24 25 27 28 29 31 34 36 37 38 39 40 41 43 45 53 55 56 58 60 62 63 65 70 72 73 77 77 77 78 80 85 89 92 93 94 397 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 0.6246993143 56662 0.6181716073 13993 0.6115267371 54006 0.5850915035 11059 0.5787153105 94797 0.5755779089 60855 0.5694121220 48096 0.5633986931 88747 0.5575430376 00070 0.5546753144 20461 0.5490607805 36985 0.5357250244 20828 0.5306652018 63687 0.5281925192 05155 0.5186718775 52837 0.5186718775 52837 0.5186718775 52837 0.5163816029 22135 0.5119050615 21291 0.5012939957 11441 0.4933647226 74799 0.4877210247 38818 0.4858947202 66187 0.4840950221 67519 0.4840950221 67519 0.4788505841 0.7550538483 15278 0.7416256195 78877 0.7285749131 88875 0.6799332676 99777 0.6686042677 04813 0.6630562109 15174 0.6521855991 33027 0.6416060281 06839 0.6313062541 83178 0.6262578837 71009 0.6163575874 15142 0.5926944011 12017 0.5836392931 26580 0.5791951629 90668 0.5619505264 40087 0.5619505264 40087 0.5619505264 40087 0.5577673816 14158 0.5495480992 09031 0.5298164758 85799 0.5148172255 89363 0.5039948301 24730 0.5004650118 55910 0.4969729233 17098 0.4969729233 17098 0.4867167771 0.244946151684 722 0.258374380421 123 0.271425086811 125 0.320066732300 223 0.331395732295 187 0.336943789084 826 0.347814400866 973 0.358393971893 161 0.368693745816 822 0.373742116228 991 0.383642412584 858 0.407305598887 983 0.416360706873 420 0.420804837009 332 0.438049473559 913 0.438049473559 913 0.438049473559 913 0.442232618385 842 0.450451900790 969 0.470183524114 201 0.485182774410 637 0.496005169875 270 0.499534988144 090 0.503027076682 902 0.503027076682 902 0.513283222808 0.4716816213 44913 0.4584519012 80311 0.4455430394 34656 0.3978231778 85650 0.3869315264 49798 0.3816405074 02063 0.3713623859 71546 0.3614799977 77416 0.3519804066 13211 0.3473697885 88976 0.3384177780 36051 0.3175212225 09823 0.3097170633 02596 0.3059265522 51481 0.2914679346 40485 0.2914679346 40485 0.2914679346 40485 0.2880208145 75601 0.2813164535 34507 0.2655938181 90546 0.2539926577 31106 0.2458088750 11500 0.2431733069 38741 0.2405821183 29847 0.2405821183 29847 0.2330646130 94 97 104 109 113 119 123 123 126 133 139 140 142 143 155 157 158 190 206 218 227 237 241 242 242 247 255 261 273 296 342 385 402 407 414 463 477 484 499 565 566 606 736 1046 398 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 24229 0.4674551016 37126 0.4599711479 80822 0.4543392543 28268 0.4464305340 41204 0.4414877344 66362 0.4414877344 66362 0.4379405167 82960 0.4301541533 81171 0.4239781511 28674 0.4229900389 39531 0.4210473927 96179 0.4200924892 95300 0.4094211149 43827 0.4077729833 34727 0.4069618157 90085 0.3848396369 56884 0.3759990831 61675 0.3701082097 57078 0.3660467126 80410 0.3618511676 27291 0.3602585661 46810 0.3598676411 57540 0.3598676411 57540 0.3579548125 02747 0.3550321015 84348 91034 0.4639948522 59633 0.4487212868 78000 0.4370302668 33925 0.4203097653 62979 0.4096709639 06120 0.4096709639 06120 0.4019426823 50767 0.3846942076 42353 0.3707236627 91988 0.3684639387 05865 0.3640011591 84592 0.3617976332 44836 0.3367187346 13748 0.3327693817 55637 0.3308179726 55714 0.2755926455 09507 0.2524137888 69082 0.2366177468 17403 0.2255679742 25791 0.2140222001 94961 0.2096058506 11787 0.2085190513 11922 0.2085190513 11922 0.2031859731 78380 0.1949899258 20811 966 0.536005147740 367 0.551278713122 001 0.562969733166 075 0.579690234637 021 0.590329036093 880 0.590329036093 880 0.598057317649 233 0.615305792357 647 0.629276337208 012 0.631536061294 135 0.635998840815 408 0.638202366755 164 0.663281265386 252 0.667230618244 363 0.669182027344 286 0.724407354490 493 0.747586211130 918 0.763382253182 597 0.774432025774 209 0.785977799805 039 0.790394149388 213 0.791480948688 078 0.791480948688 078 0.796814026821 620 0.805010074179 189 60989 0.2168967608 22130 0.2063988454 48705 0.1985600055 52210 0.1876391130 13728 0.1808647057 31564 0.1808647057 31564 0.1760269860 25824 0.1654778111 99037 0.1571787331 30197 0.1558565757 81007 0.1532617390 49460 0.1519884683 70971 0.1378597597 48035 0.1356943635 60949 0.1346302828 47964 0.1060589736 45866 0.0949073531 92139 0.0875741706 71342 0.0825684154 51330 0.0774441830 38708 0.0755123031 97385 0.0750392591 32030 0.0750392591 32030 0.0727313969 32255 0.0692276831 51939 399 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 0.3529441501 09454 0.3490130295 17850 0.3422764103 49866 0.3312672349 75226 0.3231485340 09930 0.3203764762 28792 0.3196013868 89906 0.3185449844 97393 0.3119731801 03514 0.3103238239 24323 0.3095320593 65022 0.3079043283 87380 0.3016951014 56579 0.3016113379 20851 0.2984713014 45917 0.2904769722 30696 0.2790381790 73465 0.1891015199 50118 0.1779461437 91273 0.1586548546 47598 0.1268783302 53345 0.1035632073 52997 0.0956995197 99405 0.0935142633 84876 0.0905468705 54574 0.0724516021 67646 0.0680354462 60470 0.0659374548 11518 0.0616732874 26275 0.0461328452 43386 0.0459323663 01727 0.0386255386 68422 0.0222350303 51577 0.0062338323 99880 0.810898480049 882 0.822053856208 727 0.841345145352 402 0.873121669746 656 0.896436792647 003 0.904300480200 595 0.906485736615 124 0.909453129445 426 0.927548397832 354 0.931964553739 530 0.934062545188 482 0.938326712573 725 0.953867154756 614 0.954067633698 273 0.961374461331 578 0.977764969648 423 0.993766167600 120 0.0667422752 43201 0.0621055227 35611 0.0543038141 33360 0.0420306336 41299 0.0334662986 33487 0.0306598749 30121 0.0298872882 71794 0.0288432514 77094 0.0226029567 31835 0.0211130198 45947 0.0204097561 77097 0.0189894721 44429 0.0139180534 26184 0.0138537224 54135 0.0115286147 95413 0.0064587642 93984 0.0017394772 41512 Table (3): Estimated values for the parameters by RQE method Estimated values α� = 1.2153602433570740 β� = 88.772096874537695 400 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Table (4): Estimated values for the functions 𝑓𝑅𝑄(t),𝐹�𝑅𝑄(𝑡) �̂�𝑅𝑄(t),ℎ�𝑅𝑄(t) 𝒉�𝑹𝑸(t) 𝑺�𝑹𝑸(t) 𝑭�𝑹𝑸(t) 𝒇�𝑹𝑸(t) Time/d 0.410830547624 012 0.458426158775 882 0.499392786165 830 0.534229118736 801 0.563566985712 840 0.588066669179 253 0.608361345872 594 0.625030618850 926 0.638590623742 810 0.649493163333 047 0.664835831299 176 0.669900343932 888 0.673568920691 400 0.676051310062 845 0.678146304924 498 0.678041308822 280 0.677322257482 829 0.674407372155 335 0.667393968262 481 0.661561352041 141 0.658411907831 850 0.655145657525 201 0.651786559314 367 0.648355240150 554 0.979709428457 951 0.973529051467 692 0.966765400507 786 0.959520449629 880 0.951886307383 844 0.943944084435 317 0.935764004077 730 0.927406149858 246 0.918921492538 930 0.910352993727 406 0.893102616498 358 0.884475811712 049 0.875876960864 141 0.867323121831 352 0.850403830563 979 0.842058987853 565 0.833801130489 314 0.817568413724 050 0.793979563546 163 0.778781946021 614 0.771343487060 967 0.764011737472 688 0.756786152045 848 0.749665930423 028 0.020290571542 049 0.026470948532 308 0.033234599492 214 0.040479550370 120 0.048113692616 156 0.056055915564 683 0.064235995922 270 0.072593850141 754 0.081078507461 071 0.089647006272 594 0.106897383501 642 0.115524188287 951 0.124123039135 859 0.132676878168 648 0.149596169436 021 0.157941012146 435 0.166198869510 686 0.182431586275 950 0.206020436453 837 0.221218053978 386 0.228656512939 034 0.235988262527 312 0.243213847954 152 0.250334069576 972 0.402494561005 788 0.446291183521 062 0.482795666928 308 0.512603764215 710 0.536451696993 639 0.555102053625 336 0.569282648939 855 0.579657239772 054 0.586814649091 109 0.591268045645 722 0.593766620475 155 0.592510650466 222 0.589963499187 723 0.586354932761 882 0.576698215390 601 0.570950778229 795 0.564752063994 758 0.551374165456 842 0.529897171634 387 0.515212037155 289 0.507861736909 483 0.500538972103 515 0.493263042178 722 0.486049834352 110 11 12 13 14 15 16 17 18 19 20 22 23 24 25 27 28 29 31 34 36 37 38 39 40 41 43 45 53 55 56 58 60 62 63 65 70 72 73 77 77 77 78 80 85 89 92 93 94 401 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 0.644869459816 479 0.637793537037 675 0.630657202355 662 0.602696950333 120 0.596025072462 043 0.592749992904 295 0.586327266847 398 0.580079195761 553 0.574008788647 727 0.571040480874 358 0.565237171109 421 0.551492310137 008 0.546290341723 469 0.543750600055 908 0.533985603201 459 0.533985603201 459 0.533985603201 459 0.531639661754 040 0.527057609448 217 0.516212881579 025 0.508123022090 999 0.502371804154 622 0.500511858868 394 0.498679546675 290 0.498679546675 290 0.493343030664 0.742650062823 488 0.728926530004 763 0.715604618437 668 0.666076457765 895 0.654566846559 581 0.648933676538 028 0.637902508123 069 0.627174411538 767 0.616737205334 052 0.611624012778 822 0.601601376150 936 0.577670976308 121 0.568522862925 614 0.564034939496 896 0.546631854200 283 0.546631854200 283 0.546631854200 283 0.542413016777 148 0.534126718039 401 0.514251080034 896 0.499158368936 021 0.488277213216 832 0.484729816176 744 0.481221109493 750 0.481221109493 750 0.470920630835 0.257349937176 512 0.271073469995 237 0.284395381562 332 0.333923542234 105 0.345433153440 419 0.351066323461 972 0.362097491876 931 0.372825588461 233 0.383262794665 948 0.388375987221 178 0.398398623849 064 0.422329023691 879 0.431477137074 386 0.435965060503 104 0.453368145799 717 0.453368145799 717 0.453368145799 717 0.457586983222 852 0.465873281960 599 0.485748919965 104 0.500841631063 979 0.511722786783 168 0.515270183823 256 0.518778890506 250 0.518778890506 250 0.529079369164 0.478912344845 657 0.464904629812 337 0.451301206656 691 0.401442249784 192 0.390138252151 925 0.384655432163 274 0.374019634102 899 0.363810828247 633 0.354012576147 784 0.349262070371 523 0.340047459991 089 0.318581101223 266 0.310578549065 239 0.306694336803 935 0.291893540394 270 0.291893540394 270 0.291893540394 270 0.288368272770 391 0.281515551152 268 0.265463031879 940 0.253633858925 785 0.245296704531 331 0.242613021343 557 0.239975124732 923 0.239975124732 923 0.232325411218 94 97 104 109 113 119 123 123 126 133 139 140 142 143 155 157 158 190 206 218 227 237 241 242 242 247 255 261 273 296 342 385 402 407 414 463 477 484 499 565 566 606 736 1046 402 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 564 0.481761913289 493 0.474166115680 809 0.468454986678 068 0.460441765840 276 0.455437484439 479 0.455437484439 479 0.451847885692 491 0.443973394161 860 0.437732077301 225 0.436733878264 086 0.434771686724 245 0.433807313184 914 0.423036074081 628 0.421373464680 271 0.420555261568 611 0.398262820169 664 0.389365048862 318 0.383439196005 415 0.379354986720 900 0.375137146391 256 0.373536385596 067 0.373143483016 860 0.373143483016 860 0.371221120012 911 0.368284292101 691 707 0.448124676937 610 0.432820585345 637 0.421116986075 913 0.404395282135 494 0.393766334534 705 0.393766334534 705 0.386050551400 222 0.368846736711 884 0.354930019988 537 0.352680540718 485 0.348239281234 989 0.346047017681 262 0.321127092456 798 0.317208134012 829 0.315272302610 670 0.260654812970 968 0.237839265705 991 0.222333271950 313 0.211508690653 698 0.200219241225 465 0.195906916277 153 0.194846240145 819 0.194846240145 819 0.189644418357 948 0.181660319881 373 293 0.551875323062 390 0.567179414654 363 0.578883013924 087 0.595604717864 506 0.606233665465 295 0.606233665465 295 0.613949448599 778 0.631153263288 116 0.645069980011 463 0.647319459281 515 0.651760718765 011 0.653952982318 738 0.678872907543 202 0.682791865987 171 0.684727697389 330 0.739345187029 032 0.762160734294 009 0.777666728049 687 0.788491309346 302 0.799780758774 535 0.804093083722 847 0.805153759854 181 0.805153759854 181 0.810355581642 052 0.818339680118 627 956 0.215889401753 699 0.205228855740 035 0.197274352102 100 0.186200477803 944 0.179335948857 441 0.179335948857 441 0.174436125420 611 0.163758137623 501 0.155364254946 148 0.154027540336 259 0.151404579686 175 0.150117726975 961 0.135848344474 172 0.133663090453 749 0.132589425689 769 0.103809120904 614 0.092606297312 991 0.085251291041 881 0.080236876534 289 0.075109674805 944 0.073178361419 439 0.072705604700 750 0.072705604700 750 0.070400013387 035 0.066902642310 478 403 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 0.366186584827 067 0.362237827967 591 0.355473156348 310 0.344423833715 020 0.336279916225 862 0.333500090054 084 0.332722905202 278 0.331663700730 172 0.325075872517 498 0.323422879693 739 0.322629425055 350 0.320998333995 547 0.314777736739 894 0.314693835894 757 0.311548972348 979 0.303545336139 154 0.292101940250 888 0.175932152155 681 0.165099612576 446 0.146431241847 377 0.115889085719 358 0.093682279883 768 0.086239413158 008 0.084175829271 965 0.081377118270 455 0.064405622851 187 0.060291538515 972 0.058341350334 391 0.054386624736 465 0.040090695929 575 0.039907631720 462 0.033263394507 769 0.018606864784 650 0.004873516040 898 0.824067847844 319 0.834900387423 554 0.853568758152 623 0.884110914280 642 0.906317720116 232 0.913760586841 992 0.915824170728 035 0.918622881729 545 0.935594377148 813 0.939708461484 028 0.941658649665 609 0.945613375263 535 0.959909304070 425 0.960092368279 538 0.966736605492 231 0.981393135215 350 0.995126483959 102 0.064423993959 165 0.059805325057 983 0.052052375727 490 0.039914963189 190 0.031503469231 161 0.028760852054 407 0.028007226463 179 0.026989836200 336 0.020936714043 383 0.019499663008 001 0.018822636315 337 0.017458015932 046 0.012619658529 039 0.012558685707 587 0.010363176375 734 0.005648027025 552 0.001423563491 390 Table (5): Estimated values for the parameters by SH. method Estimated values α�sh = 1.2218000042511650 β�sh = 93.132999953965182 404 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Table (6): Estimated values for the functions𝑓𝑆𝐻(t),𝐹�𝑆𝐻(𝑡) �̂�𝑆𝐻(t),ℎ�𝑆𝐻(t) 𝒉�𝑺𝑯(t) 𝑺�𝑺𝑯(t) 𝑭�𝑺𝑯(t) 𝒇�𝑺𝑯(t) Time/d 0.3752277289 53612 0.4220227491 93553 0.4627665591 64692 0.4977908130 79461 0.5275993365 58227 0.5527570404 02201 0.5738273921 49264 0.5913397935 42018 0.6057747120 23272 0.6175589721 37916 0.6346224191 83125 0.6405069896 74977 0.6449616954 28936 0.6481938040 51752 0.6516759355 11671 0.6522090345 27705 0.6520924692 09372 0.6502816471 20840 0.6446971501 15132 0.6396853856 04403 0.6369116292 39674 0.6339998652 46463 0.6309754172 11304 0.6278601946 75818 0.6246731586 0.982146238989 425 0.976502564474 033 0.970278181274 998 0.963566791301 119 0.956454695015 499 0.949019092579 196 0.941327719998 491 0.933439220186 254 0.925403880144 604 0.917264515920 198 0.900813034197 810 0.892557127814 196 0.884311116027 777 0.876092870532 662 0.859796387253 392 0.851740427001 635 0.843757516059 213 0.828035933595 317 0.805127356294 351 0.790332890419 870 0.783082543869 093 0.775930582970 504 0.768876912364 951 0.761921152513 961 0.755062687250 0.0178537610105 75 0.0234974355259 67 0.0297218187250 02 0.0364332086988 81 0.0435453049845 01 0.0509809074208 04 0.0586722800015 09 0.0665607798137 46 0.0745961198553 96 0.0827354840798 02 0.0991869658021 90 0.1074428721858 04 0.1156888839722 23 0.1239071294673 38 0.1402036127466 08 0.1482595729983 65 0.1562424839407 87 0.1719640664046 83 0.1948726437056 49 0.2096671095801 30 0.2169174561309 07 0.2240694170294 96 0.2311230876350 49 0.2380788474860 39 0.2449373127490 0.368528502756 334 0.412106296853 886 0.449012295381 207 0.479654696498 151 0.504624862538 179 0.524576984899 259 0.540159630724 547 0.551979755748 962 0.560586268999 816 0.566464931630 261 0.571676146994 305 0.571689079049 214 0.570346796679 929 0.567877970453 185 0.560308615012 909 0.555512801562 951 0.550207922061 019 0.538456570773 605 0.519063312082 699 0.505564399764 076 0.498754378844 813 0.491939885043 910 0.485142430563 614 0.478379963145 039 0.471667393845 11 12 13 14 15 16 17 18 19 20 22 23 24 25 27 28 29 31 34 36 37 38 39 40 41 43 45 53 55 56 58 60 62 63 65 70 72 73 77 77 77 78 80 85 89 92 93 94 94 405 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 78460 0.6181470826 11696 0.6115037003 06073 0.5850732748 28212 0.5786980555 87137 0.5755611129 08622 0.5693961928 10839 0.5633835682 13375 0.5575286603 36258 0.5546612914 90097 0.5490474302 58708 0.5357131701 34947 0.5306538812 55131 0.5281814531 41949 0.5186617554 99728 0.5186617554 99728 0.5186617554 99728 0.5163716998 27364 0.5118955777 79155 0.5012854628 68725 0.4933568634 83837 0.4877136270 94518 0.4858874689 20487 0.4840879135 62698 0.4840879135 62698 0.4788438835 88763 988 0.741634229810 242 0.728583275272 363 0.679940514020 797 0.668611217467 401 0.663063011114 748 0.652192098560 028 0.641612225621 596 0.631312149735 377 0.626263628615 009 0.616363031889 410 0.592699104523 602 0.583643705408 506 0.579199431045 613 0.561954227387 611 0.561954227387 611 0.561954227387 611 0.557770943388 675 0.549551385935 263 0.529819095037 254 0.514819332062 766 0.503996564822 759 0.500466625066 286 0.496974416258 724 0.496974416258 724 0.486717916624 888 12 0.2583657701897 58 0.2714167247276 37 0.3200594859792 02 0.3313887825325 99 0.3369369888852 52 0.3478079014399 72 0.3583877743784 04 0.3686878502646 23 0.3737363713849 91 0.3836369681105 90 0.4073008954763 98 0.4163562945914 94 0.4208005689543 87 0.4380457726123 89 0.4380457726123 89 0.4380457726123 89 0.4422290566113 25 0.4504486140647 37 0.4701809049627 46 0.4851806679372 34 0.4960034351772 41 0.4995333749337 14 0.5030255837412 76 0.5030255837412 76 0.5132820833751 12 321 0.458439035522 173 0.445531368810 168 0.397815023226 526 0.386924011492 134 0.381633284605 747 0.371355697901 391 0.361473785080 020 0.351974617095 968 0.347364193060 876 0.338412538765 347 0.317516716220 483 0.309712797545 150 0.305922397148 662 0.291464166087 352 0.291464166087 352 0.291464166087 352 0.288017130151 923 0.281312924222 667 0.265590610292 439 0.253989650927 330 0.245805992672 885 0.243170461732 636 0.240579308260 725 0.240579308260 725 0.233061897408 893 97 104 109 113 119 123 123 126 133 139 140 142 143 155 157 158 190 206 218 227 237 241 242 242 247 255 261 273 296 342 385 402 407 414 463 477 484 499 565 566 606 736 1046 406 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 0.4674492491 77927 0.4599658254 13476 0.4543343171 81601 0.4464261197 22062 0.4414836364 37112 0.4414836364 37112 0.4379366409 57716 0.4301507517 88094 0.4239751129 72504 0.4229870579 20006 0.4210445233 11497 0.4200896742 51316 0.4094188915 66749 0.4077708486 76795 0.4069597245 43355 0.3848386683 53608 0.3759985324 02494 0.3701079284 82555 0.3660466132 31110 0.3618512527 25870 0.3602587204 49858 0.3598678123 77606 0.3598678123 77606 0.3579550661 05212 0.3550324798 19419 0.3529446164 0.463995208983 225 0.448721119850 887 0.437029701340 853 0.420308635185 968 0.409669478478 561 0.409669478478 561 0.401940941214 002 0.384691904147 373 0.370720913639 891 0.368461118397 863 0.363998199162 029 0.361794604646 707 0.336714946553 988 0.332765477950 290 0.330814012080 452 0.275587214021 698 0.252407835180 042 0.236611477276 621 0.225561505376 924 0.214015543903 785 0.209599128615 142 0.208512313675 497 0.208512313675 497 0.203179161905 875 0.194983011821 322 0.189094540335 0.5360047910167 75 0.5512788801491 13 0.5629702986591 47 0.5796913648140 32 0.5903305215214 39 0.5903305215214 39 0.5980590587859 98 0.6153080958526 27 0.6292790863601 09 0.6315388816021 37 0.6360018008379 71 0.6382053953532 93 0.6632850534460 12 0.6672345220497 10 0.6691859879195 48 0.7244127859783 03 0.7475921648199 58 0.7633885227233 79 0.7744384946230 76 0.7859844560962 15 0.7904008713848 58 0.7914876863245 03 0.7914876863245 03 0.7968208380941 25 0.8050169881786 78 0.8109054596641 0.216894212061 364 0.206396380272 672 0.198557590946 775 0.187636753091 748 0.180862371096 010 0.180862371096 010 0.176024665658 643 0.165475511775 786 0.157176441241 742 0.155854284429 027 0.153259448252 420 0.151986177611 919 0.137857460192 091 0.135692061354 129 0.134627979231 343 0.106056616459 391 0.094904975594 586 0.087571783710 047 0.082566025118 534 0.077441792664 393 0.075509913882 296 0.075036870176 194 0.075036870176 194 0.072729010331 219 0.069225302209 583 0.066739900017 407 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 79546 0.3490136598 45741 0.3422773158 93896 0.3312685759 17870 0.3231501859 30079 0.3203782326 03499 0.3196031723 26442 0.3185468094 45581 0.3119752484 99225 0.3103259528 41767 0.3095342172 61876 0.3079065457 19049 0.3016975439 92966 0.3016137834 81302 0.2984738601 61170 0.2904798179 20796 0.2790414369 78308 836 0.177939059868 615 0.158647658232 633 0.126871171535 382 0.103556297524 780 0.095692745999 645 0.093507532666 428 0.090540202212 242 0.072445421360 485 0.068029415965 452 0.065931500922 324 0.061667499164 058 0.046127795102 876 0.045927327259 762 0.038620936712 520 0.022231701711 787 0.006232447797 971 64 0.8220609401313 85 0.8413523417673 67 0.8731288284646 18 0.8964437024752 20 0.9043072540003 55 0.9064924673335 72 0.9094597977877 58 0.9275545786395 15 0.9319705840345 48 0.9340684990776 76 0.9383325008359 42 0.9538722048971 24 0.9540726727402 38 0.9613790632874 80 0.9777682982882 13 0.9937675522020 29 208 0.062103162514 256 0.054301494632 718 0.042028432319 558 0.033464236799 363 0.030657872836 342 0.029885304076 609 0.028841292541 267 0.022601178331 568 0.021111293330 748 0.020408055530 892 0.018987826650 738 0.013916642492 349 0.013852314940 001 0.011527340063 626 0.006457860665 309 0.001739111189 438 Table (7): showed MSE and MAPE for the estimation methods Method MSE MAPE MLE 0.000000000036821 0.000025281310636 RQE 0.000157322008120 0.044260791532672 SHI 0.000000000000000 0.000000002554411 408 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. 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Vol. 26 (3) 2013 لبیانات مرضى سرطان الغدد تقدیر معالم والدوال االحتمالیة ذات العالقة سوندرز -نموذج بیرنبومإعبر اللمفاویة عباس نجم سلمان طھ أنور طھ /جامعة بغدادكلیة التربیة للعلوم الصرفة(ابن الھیثم)قسم الریاضیات / 2013آب 26، قبل في 2013حزیران 16أستلم البحث في : المستخلص دالة و الدالة التجمیعیة التوزیعیة،و ، في ھذا البحث ، تقدیر معالم والدوال االحتمالیة ذات الصلة ،دالة البقاء سوندرز الذي یطابق البیانات الحقیقیة لمرضى -ودالة الفشل(الموت) االحتمالیة لتوزیع بیرنبوم ، الخطورة(نسبة الفشل) المربعات وق مقترحة (االمكان األعظم ، ائسرطان الغدد اللمفاویة.سیكون تقدیر المعالم (الشكل،القیاس) باستخدام ثالث طر یتم تقدیر الدوال االحتمالیة المتعلقة بھا والتي ذكرت اعاله باالعتماد على عینة وطریقة التقلص) ثم بعد ذلك ،الصغرى بیانات حقیقیة تصف المدة التي یعاني منھا مرضى سرطان العقد اللمفاویة باالعتماد على وقت تشخیص المرض أو دخول ) . في ھذا البحث قمنا بتقدیر 2012نھایة سنة الىسنة 2010ة ثالث سنوات( ابتدآ" من مدالمریض إلى المستشفى ول وحساب قیم المعالم ومن ثم تقدیر قیم الدوال االحتمالیة المذكورة اعاله ثم بعد ذلك مقارنة النتائج العددیة باستخدام ق الثالثة المقترحة رائو متوسط الخطأ النسبي المطلق) للمقارنة بین الط ، مؤشرین إحصائیین ( متوسط مربعات الخطأ ة إلى دالة البقاء .استنتجنا من ذلك إن طریق التقلص المستخدمة في تقدیر دالة البقاء بالنسبة إلى مرضى سرطان العقد بالنسب اعاله ھي االفضل . مدةاللمفاویة في ال سوندرز ، مرض سرطان الغدد اللمفاویة ، البیانات الحقیقیة الكاملة ، مقـدر -وذج بیرنبومنمإكلمات مفتاحیة : النسبي المطلق . توسط مربعات الخطأ و متوسط الخطأ اإلمكان األعظم ،مقدر المربعات الصغرى، مقدر التقلص ، م 409 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013