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by: Orval Funk


Orval Funk
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This 37 page Class Notes was uploaded by Orval Funk on Monday September 28, 2015. The Class Notes belongs to STAT991 at University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 29 views. For similar materials see /class/215439/stat991-university-of-pennsylvania in Statistics at University of Pennsylvania.

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Date Created: 09/28/15
Nonparametric Regression and Con dence Bands When Errors are Lognormal with Application to Bank Call Center Data Haipeng Shen 3 b The Wharton School University of Pennsylvania August 13 2002 aJoint work with Lawrence D Brown bThanks are due to Avi Mandelbaum Anat Sakov and Sergey Zeltyn for their initial analysis Q the data Bank Call Center Data Lognormal Service Time Methodology Application References Call Center 0 A service network D agents provide telephone based services D the customers and the service agents are remote from each other D connected via telephone lines o Consists of D callers customers D servers agents D invisible queues The Vast Call Center Worldl o 70 of all customer business interactions occur in call centers 0 3 of the US working population is currently employed in call centers D 155 6 million agents D more than in agriculture A Call Center of Bank Anonymous of IsraelI 0 Small 15 seats at most 0 Types of service D information for current and prospective customers D transactions of bank accounts D stock trading D IT support for users of the bank s website 0 Working hours D Sundays Thursdays 7AM 12AM D Fridays 7AM 2PM D Saturdays 8PM 12AM 10012010m0 gt Event history of an incoming callI units of rates are calls per month 13K VRUIVR NZZK Queue Service ZPK View if i291lt End of SerVICe Abandon End Of Abandon N5 SerVICe N1 5 80 The Call Center Datal o The data D Collected by Avishai Mandelbaum in 2000 D Empirically documented in Brown et al 2001 D Statistically analyzed in Brown et al 2002 o Consists of the Whole history of every agentseeking call in 1999 0 450000 Observations Service Times 0 Naive de nition time to serve a customer over the phone 0 Complications D Do we include holding time in the middle D How about multiple Visits D Do we include after call work D And how about A N 0 Figure 1 Service Time Distribution from a Call Center Jan Oct Mean 185 SD 238 400 500 600 700 800 900 Time Service Times Cont I 0 Service time distribution is one of the key inputs for queueing theory The mean is especially important D system delay D workload D staf ng 0 Standard queueing theory assumes exponential distribution D simple D memoryless property 0 But Figure 2 I Lognormal Service Time I Histogram 0f L0gSemz39ce Time N011 Dec 04 w 03 roponion 02 x 01 0 2 4 6 8 00 LogService Time Figure 3 Lognormal QQ Plot of Service Time N0 Dec 3000 I 5 2000 I Sen ce nm 1000 I I I I 1000 2000 3000 c Lognormal o Lognormality has been found before in contexts other than service time D Queueing data gtllt telecommunication line usage Bolotin 1994 gtllt psychology conjectures about the lognormality of parallel information process time Ulrich and Miller 1993 Breukelen 1995 D Biomedical Economics and Finance stock return 0 Mean service time as a function of time of day Nonparametric Regression 0 Mean service time across different categories like service types day of week Anova We can do a better job by taking into account the lognormality Note If Z has a lognormal distribution with mean V 02 Y logZ N Nu 02 then we know 1 e VT Similarly we have How quot 02 1X e 2 1 The proposed method is based on K Estimation of A Lognormal Meanl Suppose Yr log i39kd39 N 0102 for 239 1 n Want to estimate the lognormal mean 1 e r De ne S2 72 11 0 Crow and Shimizu 1988 has a detailed literature review 0 Commonly used estimators D 7 most common Zhou et al 1997 D evz the usual plug in MLE D 67 heavily biased o Finney 1941 derives the UMVU estimator as the sum of an in nite series lEstimation of A Lognormal Mean cont I 0 Among the Class of estimators as exp 7 l 0322 C 2 0 Shen 2002 suggests using 7Si2 D e 2n4gt under the square error loss 872 D ey2quot1gt under the loss functlon L6 1 10g261 Con dence Interval for A Lognormal Meanl o Cox s con dence interval Land 1972 2 2 4 2n1 1S izl a2 V nS 12ns13 D Approximate yet accurate coverage When sample size is large and 02 is not large Shen 2002 o Uniformly most accurate unbiased UMAU interval Land 1971 D Exact D No closed form yet necessary for small sample size and large variance and implementable Shen 2002 0 Theorem Shen 2002 The UMAU con dence interval and Cox s interval are asymptotically equivalent The Problem The data Xt 2321 X 2 Where Z X E has a lognormal distribution For example in a regression setup X the time of day of a call and Z the corresponding service time We are interested in estimating V1c EZX IE with con dence band attached Transform the original problem into a problem With normal errors Make inferences based on the transformed data Back transform the inference results to the original scale The Procedure XquL gt XMG With Y7 10gZ7 The model is Y MX7 0Xz39 i Where eiqu i39kd39 N01 Estimate Mac using any good existing nonparametric regression method eg local polynomial method Loader 1999 Estimate 0ZE using some good local nonparametric regression method like difference based estimate plus local polynomial smoothing We can get seMr from 696 The Procedure cont I 5 We also need to estimate the variance of 6296 in order to get se02 6 Suppose 196 is approximately unbiased then 1X l Z1aZS HX Will be an approximate 1001 00 con dence interval for WC Similarly a 1001 00 con dence interval for 02ZE is approximately 62x i Z1a2se02 Note Normal approximation is 0k since sample size is large The Procedure cont I 7 Back transform to the original scale and obtain the following plug in estimate for IZE nltxgtamp2x2 The corresponding 1001 00 con dence interval for 195 is emltxgt 2x2izla2xseix2sea2XV4 Note a and amp2I are asymptotically independent and very nearly independent at any sample size which gives us 86 113 amp2III2 V S H2 8602 11324 b The use of Z1a2 is supported by the property of Cox s interval 0 Use difference based estimator to estimate 02ZE via a two step procedure 1 le7 Yi b gt X2 1Y2 1X2 Y2 7l2l consecutive non overlapping pairs 02X2z39 lt D27 Y2rz 1 16022 2 Smooth X2 D2 2l to obtain 6296 0 Levins 2002 discusses the theoretical properties of using adjacent overlapping pairs Also see Dette et al 1998 Application 0 Problem Model the changing pattern of the mean service time across a day 0 Data D The weekdays of November and December in 1999 D The normal business hours 7AM to midnight D Service types gtllt Regular Services PS gtllt Internet Consulting IN Application 0 Figure 6 D for REGULAR SERVICE PS customers D n 42613 D a prominent and bimodal pattern longest service times around 1000AM and 300PM Figure 4 Mean of L0gSerVice Time PS VS Time Of day 95 CI n 42613 48 50 I Mean of LogService Time 46 I I I I I I I I I I I I I I I 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time of Day Figure 5 Variance of L0gSerVice Time PS vs Time of day 95 CI n 42613 08 I Variance of LogService Time 07 I I I I I I I I I I I I I I I 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time of Day Figure 6 Mean Service Time PS vs Time Of day 95 CI n 42613 240 I Mean Service Time I I I I I I I I I I I I I I I I I I 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time of Day Application 0 Figure 7 D for INTERNET CONSULTING IN customers D n 5066 D some non signi cant uctuations along the day D IN calls need longer services than PS calls Figure 7 Mean Service Time IN vs Time Of day 95 CI n 5066 400 450 500 I I I Mean Service Time 350 I 300 I I I I I I I I I I I I I I I I I I I 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time of Day Model Diagnostics I 0 Look at the residuals from the regression of LogService Time on Time of day for the PS calls 0 Figures 8 and 9 give the histogram and normal quantile plot of the residuals from Which we can see that the residuals are pretty normal 0 Consequently provides additional validation of our assumption of the log normality of the service times Time on Time Of day PS Proportion 03 04 I I 02 I 01 I 00 I Figure 8 Histogram of The Residuals from Modeling Mean L0gSerVice Residuals Time on Time Of day PS Figure 9 QQ plot of The Residuals from Modeling Mean L0gSerVice Residuals Quantiles of Standard Normal o Bolotin VA 1994 Telephone Circuit Holding Time Distributions Proceedings of the 14th International Teletra lc Congress 1257134 Breukelen GJPV 1995 THEORETICAL NOTE Parallel Information Processing Models Compatible With Lognormally Distributed Response Times Journal of Mathematical Psychology 39 3967399 Brown L D Gans N Mandelbaum A Sakov A Shen H Zeltyn S and Zhao L 2001 Empirical Analysis of a Telephone Call Center Technical Report Brown L D Gans N Mandelbaum A Sakov A Shen Zeltyn S and Zhao L 2002 Statistical Analysis of a Telephone Call Center A Queueing Science Perspective Working Paper Brown LB and Shen H 2001 Analysis of Service Times of A Telephone Call center Technical Report Crow EL and Shimizu K 1988 Lognormal Distributions Theory and Applications Marcel Dekker New York Dette H Munk A and Wagner T 1998 Estimating the variance in nonparametric regression What is a reasonable choice J R Statist Soc B 60 7517764 Land CE 1971 Con dence intervals for linear functions of the normal mean and variance Annals of Mathematical Statistics 42 118771205 7 1972 An evaluation of approximate con dence interval estimation methods for lognormal means Technometrics 14 1457158 Levins M 2002 On the New Local Variance Estimator in the Nonparametric Context Working PhD Thesis Loader C 1999 Local Regression and Likelihood Springer Verlag New York o Shen H 2002 Statistical Analysis of A Call Center and Estimation of Lognormal Mean With Con dence Interval Working PhD Thesis 0 Ulrich R and Miller J 1993 Information Processing Models Generating Lognormally Distributed Reaction Times Journal of Mathematical Psychology 37 5137525 0 Zhou XH Mel CA and Hui SL 1997 Methods for comparison of cost data Annals of Internal Medicine 127 7 5277 56


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