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1 Use of Asymmetric Loss Functions in Sequential Estimation Problem for the Multiple Linear Regression Raghu Nandan Sengupta Department of Industrial and Management Engineering Indian Institute of Technology Kanpur

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 2 What is this talk all about? Use of Asymmetric Loss Functions in Sequential Estimation Problem for the Multiple Linear Regression Raghu Nandan Sengupta JOURNAL OF APPLIED STATISTICS (2008), 35, 245-261 Abstract: While estimating in a practical situation, asymmetric loss functions are preferred over squared error loss functions, as the former is more appropriate than the latter in many estimation problems. We consider here the problem of fixed precision point estimation of a linear parametric function in beta for the multiple linear regression model using asymmetric loss functions. Due to the presence of nuissance parameters, the sample size for the estimation problem is not known before hand and hence we take the recourse of adaptive multistage sampling methodologies. We discuss here some multistage sampling techniques and compare the performances of these methodologies using simulation runs. The implementation of the codes for our proposed models is accomplished utilizing MATLAB 7.0.1 program run on a Pentium IV machine. Finally we highlight the significance of such asymmetric loss functions with few practical examples. Key words and phrases: loss function; risk; bounded risk; asymmetric loss function; LINEX loss function; relative LINEX loss function; stopping rule; multistage sampling procedure; purely sequential sampling procedure; batch sequential sampling procedure

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 3 Background Any population is characterized by X (random variable) which has a particular distribution given by its cumulative distribution function (cdf), where the cdf is given by P[X x] = F(x ; ) Note: 1)In general we select a sample {X 1, X 2,….., X n } of random observations to estimate 2)The statistics is given by T n = T(X 1, X 2,….., X n ) which is an estimator of 3)If we consider as the error in our estimation process, then = (T n - )

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 4 Loss functions, the error results in different types of LOSS FUNCTIONS, denoted by L( ) and few examples are 1)Absolute error loss function L( ) = | | 2)Squared error loss (SEL) function L( ) = 2 3)Linear exponential (LINEX) loss function, [Zellner (1986)], a type of asymmetric loss function L( ) = b{e a - a -1} where a ( 0) is the shape parameter and b (>0) is the scale parameter 4)Balanced loss function (BLF), [Zellner (1994)] L( ) = w{g( ) – g(T n )} / {g( ) – g(T n )} + (1-w)(T n - ) 2 where 0 w 1

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 5 LINEX loss function L( ) = b{e a - a -1} L( ) L( ) a > 0 a < 0

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 6 Few examples where asymmetric loss functions can be used Marketing strategy Exponential survival model, where X is the life time of a component with a pdf f(x;, ) = (1/ )exp{-(x - )/ } where = minimum guarantee/warranty time/period 1/ = failure rate Note: Use a LINEX loss function with an appropriate value for a

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 7 Few examples where asymmetric loss functions can be used Construction of a dam Underestimation of height of dam is more serious than overestimation Note: Use a LINEX loss function with a (< 0)

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 8 Few examples where asymmetric loss functions can be used Reliability of equipments Exponential life time of equipments, where X is the life of an equipment with a pdf f(x; ) = (1/ )exp{-x/ } such that the reliability function R(t) is given by R(t) = P[X > t] = exp{-t/ } Over estimation of the reliability function can have marked consequence than under estimation Note: Use a LINEX loss function with a (> 0)

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 9 Risk RISK = EXPECTED LOSS Let us consider a simple example 1)Consider we choose X 1, X 2,….., X n (i.i.d) from X ~ N(, 2 ), but with both and 2 unknown. 2)We are interested in estimating using 3)The loss function for the LINEX loss is of the form 4)The risk is given by

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 10 Concept of Bounded Risk RISK w (given or a known quantity) 5)The risk is given by 6)For bounded risk we must have 7)As both and 2 are unknown we take the recourse of Sequential Sampling Techniques

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 11 Different Sequential Sampling Methodologies 1)Two stage sampling procedure [Stein (1945)] 2)Purely sequential sampling procedure [Ray (1957)] 3)Three stage sampling procedure [Hall (1981), Mukhopadhyay (1980)] 4)Accelerated sequential sampling procedure [Hall (1983), Mukhopadhyay and Solanky (1991)] 5)Batch sequential sampling procedure [Liu (1997)]

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 12 Estimation problem for the multiple linear regression In the context of the multiple linear regression problem formulation we have just discussed for the first paper we now deal with the problem of estimation considering LINEX loss function

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 13 Estimation problem for the multiple linear regression Given n data points the usual least square error (LSE) estimator of and the forecasted value of are n = (X n X n ) –1 X n Y n

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 14 Estimation problem for the multiple linear regression However, Zellner(1986) has shown that when 2 is known, under LINEX loss, the estimator thus found for SEL is inadmissible, being dominated (in terms of risk) by the estimator

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 15 Estimation problem for the multiple linear regression Theorem: Under LINEX loss, the estimator is dominated by the estimator of the form even when 2 is unknown Here we replace 2 by it usual predictor 2 n

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 16 Estimation problem for the multiple linear regression Thus under asymmetric loss function the shrinkage estimators and dominates The corresponding risk of the shrinkage estimator is given by when 2 is known

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 17 Estimation problem for the multiple linear regression Now if the corresponding risk is bounded then we have But if 2 is unknown we have to solve the problem of finding the optimal sample size by taking the recourse of some adaptive sampling methodologies about which we have discussed before

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 18 Estimation problem for the multiple linear regression Purely sequential sampling procedure m m+1 N-1 N

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 19 Estimation problem for the multiple linear regression Purely sequential sampling procedure One sampling stops we consider the two forecasted values

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 20 Estimation problem for the multiple linear regression Purely sequential sampling procedure The corresponding risks are

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 21 Estimation problem for the multiple linear regression Batch sequential sampling procedure 1)Choose a positive integer k and consider 0 < 1 < 2 < …< k < 1, thus the objective is to estimate k fractions of the sample size using sequential type sampling, but taking batches of observations at each stage. 2)We specify decreasing batch sizes for these k sampling stages as r 1 > r 2 >…> r k > 1. 3)In the final stage, sampling is done purely sequentially 4)We start with an initial sample of size m and then, for t = 1,2,…, define m (m+r 1 *t 1 ) (N-1*t k ) N

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 22 Estimation problem for the multiple linear regression Batch sequential sampling procedure 1) 2). k-1) k)

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 23 Estimation problem for the multiple linear regression Batch sequential sampling procedure One sampling stops we consider the two forecasted values

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 24 Estimation problem for the multiple linear regression Batch sequential sampling procedure The corresponding risks are

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA25 Estimation problem for the multiple linear regression If the parameter value is very small then considering a relative LINEX loss function would be more practical and advisable than a LINEX loss function. The relative LINEX loss function and its corresponding risk is L(,T) = e a(T/ -1) - a(T/ - 1) -1 R(,T) = E[L(,T)] = E[e a(T/ - 1) - a(T/ - 1) -1]

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 26 Estimation problem for the multiple linear regression Similar bounded risk problem formulation for the estimated value was undertaken and corresponding sequential sampling methodologies were considered for the case of relative LINEX loss function

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 27 Simulation for the estimation problem for the multiple linear regression Data set used for simulation The manuscript describing the data can be found at www.spatial-statistics.com. One can refer Kelley and Barry (1997) for further details www.spatial-statistics.com The MLR is of the form

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 28 Simulation for the estimation problem for the multiple linear regression Data set used for simulation Where 1)MHV = Median house value 2)MI = Median income 3)MA = Housing median age 4)TR = Total rooms 5)B = Total bedrooms 6)P = Population 7)H = Households

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 29 Simulation for the estimation problem for the multiple linear regression For the LINEX loss function the following sampling methodologies were considered with the starting sample size m=10 PS L :Purely Sequential (m = 10) BS L (1): Batch sequential (m = 10, k+1 = 3; 1 = 0.80, 2 = 0.90, r 1 = 24, r 2 = 16, r 3 = 8) BS L (2):Batch sequential (m = 10, k+1 = 3; 1 = 0.75, 2 = 0.85, r 1 = 15, r 2 = 10, r 3 = 5) For the relative LINEX loss function the following sampling methodologies were considered with the starting sample size m=10 PS RL :Purely Sequential (m = 10) BS RL (1):Bath sequential (m = 10, k+1 = 3; 1 = 0.60, 2 = 0.90, r 1 = 17, r 2 = 13, r 3 = 9) BS RL (2):Batch sequential (m = 10, k+1 = 3; 1 = 0.70, 2 = 0.80, r 1 = 10, r 2 = 7, r 3 = 4)

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 30 Simulation for the estimation problem for the multiple linear regression Consider a = -0.6 and w=0.03 for the relative LINEX loss function %save For BS RL (1) 42 4 0.020655 0.020421 -0.009788 -0.010013 84.60 For BS RL (2) 39 6 0.022570 0.022298 -0.007696 -0.007964 76.90 For PS RL 35 26 0.024785 0.024460 -0.005298 -0.005626 ----

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 31 Acknowledgements for my visit IUSSTF Fellowship Foundation Princeton University, USA Prof. Jianqing Fan, ORFE Department, Princeton University, USA Prof. Lawrence M. Seiford, IOE Department, University of Michigan, Ann Arbor Prof. Katta G. Murty, IOE Department, University of Michigan, Ann Arbor Prof. Romesh Saigal, IOE Department, University of Michigan, Ann Arbor

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 32 Contact Detail Raghu Nandan Sengupta Assistant Professor Industrial & Management Engineering Department Indian Institute of Technology Kanpur Kanpur 208 016, UP, INDIA Ph: +91-512-259-6607; Fax: +91-512-259-7553 Email: raghus@iitkac.inraghus@iitkac.in

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 33 Research Areas Sequential Analysis Financial Optimization Statistical Reliability Use of different Meta Heuristics Techniques for Optimization

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34 List of Publications 1.LINEX Loss Function and its Statistical Application – A Review ; (co- authors Saiobal Chattopadhyay and Ajit. Chaturvedi), DECISION, Jan- Dec, 1999, 26, 1-4, 51-76. 2.Sequential Estimation of a Linear Function of Normal Means Under Asymmetric Loss Function ; (co-authors Saibal Chattopadhyay and Ajit Chaturvedi), METRIKA, 2000, 52, 3, 225-235. 3.Asymmetric Penalized Prediction Using Adaptive Sampling Procedures; (co-authors Saibal Chattopadhyay and Sujay Datta), SEQUENTIAL ANALYSIS, 2005, 24, 1, 23-43. 4.Three-Stage and Accelerated Sequential Point Estimation of the Normal Mean Using LINEX Loss Function; (co-author Saibal Chattopadhyay), STATISTICS, 2006, 40, 1, 39-49. R.N.Sengupta, IME Dept., IIT Kanpur, INDIA

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35 List of Publications 5.Use of Asymmetric Loss Functions in Sequential Estimation Problem for the Multiple Linear Regression, JOURNAL OF APPLIED STATISTICS, 2008, 35, 8, 245-261. 6."Impact of information sharing and lead time on bullwhip effect and on- hand inventory" ; (co-authors, Sunil Agrawal and Kripa Shanker), EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, (Accepted and forthcoming). 7.Bankruptcy Prediction using Artificial Immune Systems", (co-author Rohit Singh), LECTURE NOTES IN COMPUTER SCIENCE (LNCS), L.N.de Castro, F.J.Zuben and H.Knidel (Eds.), 2007, 4628, 131-141.

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R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 36 Thank you all

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