ESTIMATION AND OUTPUT ANALYSIS (L&K Chapters 9, 10) Review performance measures (means, probabilities and quantiles).
|
|
- Agatha Wood
- 6 years ago
- Views:
Transcription
1 ESTIMATION AND OUTPUT ANALYSIS (L&K Chapters 9, 10) Set up standard example and notation. Review performance measures (means, probabilities and quantiles). A framework for conducting simulation experiments and reporting results. Review point estimators. Measuring and controlling error for absolute and relative performance. Measuring and controlling error in steady-state simulation. 14
2 Public Sub TTF(Lambda As Double, Mu As Double, Sum As Double) sub to generate one replication of the ttf for the airline reservation system variables State = number of operational computers Lambda = computer failure rate Mu = computer repair rate Sum = generate ttf value that is returned from the Call STANDARD EXAMPLE Dim State As Integer Dim Fail As Double Dim Repair As Double State = 2 Sum = 0 While State > 0 If State = 2 Then Fail = Exponential(1 / Lambda) Sum = Sum + Fail State = 1 Else Fail = Exponential(1 / Lambda) Repair = Exponential(1 / Mu) If Repair < Fail Then Sum = Sum + Repair State = 2 Else Sum = Sum + Fail State = 0 End If End If Wend End Sub 15
3 NOTATION AND PERFORMANCE MEASURES Let Y 1, Y 2,..., Y n denote outputs across n replications. Thus, Y 1, Y 2,..., Y n are independent and indentically distributed (i.i.d.) with common, unknown cumulative distribution function (cdf) F. We want to estimate mean µ = E[Y ] = y df (y) probability p = Pr{a < Y b} = b a df (y) quantile θ, where for given 0 < γ < 1 Pr{Y θ} = θ df (y) = γ 16
4 Example: In the TTF example, µ could represent the expected time to failure; p the probability failure occurs in less than 700 hours; and θ the number of hours so that failure occurs on or before then 80% of the time. Comments: Means and probabilities are the same problem, since a probability is the mean of an indicator random variable Z = I(a < Y b). E[Z] = 0 Pr{Y a or Y > b} + 1 Pr{a < Y b} Higher moments (variances, skewness, kurtosis, etc.) viewed as means. can also be Quantiles cannot be (conveniently) expressed as means. 17
5 A FRAMEWORK FOR CONDUCTING EXPERIMENTS AND REPORTING RESULTS We should distinguish between absolute and relative performance measures. The expected TTF of a system is an absolute performance measure. The difference in expected TTF between two proposed systems is a relative performance measure. When we report simulation-based performance measures, the minimum we report is either a point estimate (our best guess) and a measure of error, or a point estimate with only significant digits. Without a measure of error, a point estimate may be meaningless. 18
6 REPORTING POINT ESTIMATES Suppose we estimate the expected TTF to be What should we report? point estimate and a measure of error ± or [ , ] Report 998 ± 61 or [937, 1059] a point estimate with only significant digits with standard error Rule of Thumb: Use the first nonzero digit of the standard error to indicate which digit to round. Report
7 CONTROLLING ERROR When we control point-estimator error, we can control absolute or relative error. Absolute error: Estimate expected TTF to within ±20 hours. Relative error: true value. Estimate expected TTF to within ±5% of its We should reduce error to the level required for the decision we have to make. The primary ways we control error are through the experiment design (such as number of replications) and variance-reduction techniques. 20
8 CONDUCTING EXPERIMENTS Experiments are conducted in stages. 1. Exploratory experiments provide information needed to form the designed experiments (sometimes called production runs ). They do not control error, but they do measure it. L&K refer to these as fixed-sample-size experiments. 2. Designed experiments provide the performance estimates and control error. 3. We redesign and repeat experiments if goals are not achieved. L&K refer to these as sequential experiments. Save raw output data when possible, and use both graphical and numerical exploration. 21
9 Estimate a mean µ (or probability p) by POINT ESTIMATION FOR MEANS where X t may be Y t or Z t. X = 1 n n X t t=1 Properties: E[ X] = E[X] (either µ or p; unbiased) se = Var[ X] = σ/ n if a mean, then σ = Var[Y t ] if a probability, then σ = p(1 p) 22
10 ESTIMATING QUANTILES Example: if n = 100, γ = 0.9 Y (1) Y (2) Y (89) Y (90) Y (100) 1. Initialization: n and 0 < γ < 1 2. Data: y[j], the jth output 3. Sort y[j] from smallest to largest so that y[1] y[2] y[n] 4. i (n + 1) γ if i < 1 output y[1] if i n output y[n] otherwise f (n + 1) γ i; output (1 f) y[i] + f y[i + 1] 23
11 Properties: E[ˆθ] θ (the estimator is biased) ˆθ is consistent (converges to θ) For i.i.d. data and large n se γ(1 γ) df 1 (γ) n du Example: How hard is it to estimate a quantile? Suppose F (y) = 1 e λy (the exponential distribution). γ n for equivalent se
12 MEASURES OF ERROR FOR ABSOLUTE PERFORMANCE A point estimate without a measure of error (MOE) may be meaningless. The goal of simulation output analysis is to provide an appropriate MOE. We will focus on two MOEs: standard error roughly the average deviation of an estimator from what it estimates; use for rounding confidence interval a procedure that captures the unknown quantity with prespecified probability With i.i.d. data (i.e., replications) this is classical statistics, but we take advantage of opportunities available in simulation. 25
13 BATCHING TRANSFORMATION When we have a large number of replications, we can use a transformation that partitions n outputs into m batches of size b (or m macroreps of b microreps): Y 1 Y b Y b+1 Y 2b Y (m 1)b+1 Y mb We can compute (within) batch means Ȳ h = 1 b b Y (h 1)b+j j=1 batch probabilities ˆp h = 1 b b j=1 I(A < Y (h 1)b+j B) batch quantiles (sort within each batch) ˆθ h = Y ((h 1)b+ (b+1)γ ) 26
14 BATCHING TRANSFORMATION, CONT. We can also batch continuous-time output processes {Y (t); 0 t T }. The ith batch mean of size b = T/m time units is Ȳ i = 1 b ib (i 1)b Y (t) dt After the first batching, it is easy to double the batch size by just averaging pairs of batch means; for other batch sizes we have to recompute the integral. 27
15 Why batch? Batch means and probabilities tend to be more nearly normally distributed than the raw outputs (due to the central limit theorem). If batching gives us approximately i.i.d. normal data then we can use classical statistics. Batch statistics aggregate the data, but less so than summary statistics. Little is lost provided m is not too small. For instance, consider a confidence interval based on the t distribution. t 0.975, = 1.96 while t 0.975,30 = 2.04 Maintain from 10 to 40 batch means. Batch sizes can be used to adjust variability. If S 2 i (n) > S2 l (n), then set b i S 2 i (n)/s2 l (n) and b l = 1. 28
16 MOE FOR i.i.d. DATA Goal: Estimate E[X] (note that X may be a batch quantile) Data: X 1, X 2,..., X m i.i.d. (reps or batch statistics) Point Estimator: X Estimated Standard Error: ŝe = S/ m where S 2 = mi=1 (X i X) 2 m 1 = mi=1 X 2 i ( m j=1 X i ) 2 /m m 1 Confidence interval (normal data): X ± t 1 α/2,m 1 ŝe where t 1 α/2,m 1 is the 1 α/2 quantile of the t distribution with m 1 degrees of freedom. Note: S 2 /m = p(1 p)/(m 1) when X is an indicator random variable. 29
17 algorithm batch means 1. Initialization: length of the replication, n; number of batches m; batch size b n/m ; sum 0; and sumsq 0 2. Data: y[t], the tth observation from a single replication. 3. Batch the outputs: (a) for j 1 to m: y[j] y[(j 1) b + 1] for t 2 to b: y[j] y[j] + y[(j 1) b + t], loop loop (b) Compute the standard error: for j 1 to m: y[j] y[j]/b sum sum + y[j] sumsq sumsq + y[j] y[j] loop 30
18 (note: batch means are now stored in y[1],..., y[m]) 4. Output: sqrt{(sumsq sum sum/m)/(m(m 1))}
19 SPECIAL CI FOR QUANTILES If the number of replications is too small for batching, there is a nonparametric c.i. for quantiles. Goal: Estimate θ such that Pr{Y θ} = γ Data: Y 1, Y 2,..., Y n i.i.d. (reps) Point Estimator: Y ( (n+1)γ ) (or interpolated version) By definition, Pr{Y i θ} = γ. Therefore we can find integers l and u such that Pr{Y (l) θ Y (u) } = Pr{l #(Y i θ) < u} = u 1 j=l ( n j ) γ j (1 γ) n j 1 α The confidence interval is [Y (l), Y (u) ]. 31
20 For large n we can use the normal approximation to the binomial distribution to get l = nγ z 1 α/2 nγ(1 γ) + 1/2 u = nγ + z 1 α/2 nγ(1 γ) + 1/2 + 1 where z 1 α/2 is the 1 α/2 quantile of the standard normal distribution. Even with small n, the approximations for l and u provide a good place to start looking. Example: γ = 0.9, n = 200, α = 0.05 and z = 1.96 implies l = 172 and u = 189, meaning the confidence interval is [Y (172), Y (189) ]. Question: Why no batching allowed? 32
21 EXAMPLE Suppose we set λ = 1 and µ = 1000 in the TTF simulation and make n = 1000 replications. Our goal is to estimate µ = E[Y ], p = Pr{Y < 500} and the 0.9 quantile of Y, Pr{Y θ} = 0.9 For µ and p we will use both the raw data, and m = 100 batch means of size b = 10. For the quantile estimator, we use the nonparametric methods: i = (n + 1) γ = (1001)(0.9) = 900 f = (1001)(0.9) 900 = 0.9 θ = 0.1 Y (900) Y (901) l = (1000)(0.9) 1.96 (1000)(0.9)(0.1) + 1/2 = 881 u = (1000)(0.9) 1.96 (1000)(0.9)(0.1) + 1/2 + 1 =
22 CONTROLLING ABSOLUTE ERROR Our standard error estimator se = S/ n estimates the true standard error σ/ n. If S is not too far from σ, then we can predict what our absolute error will be if we increase n. Suppose we initially make n 0 replications or batches and compute S. To obtain an absolute error less than a we need n = ( t1 α/2,n0 1 S a ) Notice that we need to take 4n to cut se in half. This approach can be shown to be valid as a 0. 34
23 CONTROLLING RELATIVE ERROR Suppose we initially make n 0 replications or batches and compute X and S. To obtain a relative error less than r 100%, set n = ( t1 α/2,n0 1 S r X ) Comments: It is important to recheck the absolute or relative error after taking the additional samples. To obtain a stable estimate of σ, we would like n 0 10 reps or batches initially. Using n 0 1 degrees of freedom for t is conservative. When estimating a probability, planning can be done in advance since σ = p(1 p) which is largest at p = 1/2. 35
24 ERROR CONTROL EXERCISE Based on 1000 replications of TTF, a 95% confidence interval for the expected TTF is ± 60.96, based on a standard error of and t 0.975,999 = How many replications are needed to estimate expected TTF to within ±20 hours? How many replications are needed to estimate expected TTF to within ±1% of its true value? 36
25 STEADY-STATE SIMULATION When we attempt to estimate long-run, or steady-state, performance, we need to simulate each replication (perhaps only 1) long enough that output data are statistically stationary. All of the point estimators we have covered are valid for stationary, but dependent data. However, the MOEs are not valid unless they are applied to independent replications, or batches that are sufficiently large that the point estimators from different batches are (nearly) independent. Typically, the standard MOEs underestimate the true level of error when applied to positively dependent data. 37
26 REASONS TO USE STEADY-STATE SIMULATION Appropriate planning horizon is unknown, but long. Appropriate initial conditions are unknown. Effects of initial conditions are substantial, but short lived. System conditions vary over time, but we want to design the system to tolerate sustained exposure to worst-case conditions. 38
27 CAUTIONS If the simulated time required to overcome the initial conditions is much longer than the planning horizon, then it makes no sense to do a steady-state simulation. Instead, work on determining initial conditions. Pretending that peak load is the norm may be overly conservative, particularly if the peak is short lived. In reality we may be able to tolerate an over-capacity situation for a short time. The approach to steady state is typically evaluated via stability of the mean, but this is only a necessary condition. 39
28 EXAMPLE: GI/G/1 QUEUE Let Y t represent the delay in queue of the tth customer, S t the customer s service time, and G t the interarrival-time gap. Let Y 0 = 0, S 0 = 0. Then Y t = max{0, Y t 1 + S t 1 G t } Under certain conditions Y t Y as t, where denotes weak convergence (convergence in distribution). Problem: Estimate performance measures associated with the limiting random variable Y. This is a more difficult problem because we cannot observe Y directly. 40
29 STATIONARITY (STEADY STATE) If Y t F t and Y F, then weak convergence ( ) means that F t (a) F (a) for every a at which F is continuous. To obtain a MOE, we will also want Y t to be covariance stationary independent of t. Cov[Y t, Y t+h ] = γ(h) Steady state does not mean constant. Long run must not depend on initial conditions, no regular cycles and cannot become trapped. Y 1, Y 2,... i.i.d. satisfies these conditions. 41
30 INITIAL-CONDITION BIAS Even if Y t Y as t, we can only observe Y 1,..., Y m. For a finite sample, the initial conditions do matter; e.g., for GI/G/1 the E[Y 1 ] = 0 but E[Y 2 ] > 0 Ideas: make m large delete Y 1,..., Y d for d < m sample Y 1 from F pick Y 1 close to E[Y ] or median or mode (in general, pick system state near mean or modal state). 42
31 REPLICATION-DELETION APPROACH Goal: Estimate steady-state mean (or other quantity). Method: Make n replications of length m, deleting d observations from each, and use replications means as the basic observations. Y 11 Y 1d Y 1,d+1 Y 1m Ȳ 1 Y 21 Y 2d Y 2,d+1 Y 2m Ȳ Y n1 Y nd Y n,d+1 Y nm Ȳ n The key then becomes determining the appropriate value of d. (In practice, d is often a deletion time, rather than a count.) 43
32 MEAN PLOT Goal: Find point where mean is nearly stationary. Idea: Plot E[Y t ] vs. t and look for convergence. Method: Make k replications of length m and average across. Y 11 Y 12 Y 1m Y 21 Y 22 Y 2m.... Y k1 Y k2 Y km Ȳ (1) Ȳ (2) Ȳ (m) Plot Ȳ (t) vs. t. Smoothing can help. Note: probabilities and quantiles may converge more slowly than means, and covariance function is even slower. 44
33 algorithm mean plot 1. Initialization: number of reps, k; length of each rep, m; array a[t] of length m; array s[t] 0 for t 1, 2,..., m; window width, w 2. Data: y[t, j], the tth observation from replication j 3. Calculate averages: (a) for t 1 to m: for j 1 to k: s[t] s[t] + y[t, j], endfor endfor (b) for t 1 to m: a[t] s[t]/k, endfor 45
34 4. Smooth plot: (a) for t 1 to w: s[t] 0 for l t + 1 to t 1: s[t] s[t] + a[l + t], endfor s[t] s[t]/(2t 1) endfor (b) for t w + 1 to m w: s[t] 0 for l w to w: s[t] s[t] + a[l + t], endfor s[t] s[t]/(2w + 1) endfor 5. Output: plot a[t] versus t for t 1 to m w
35 ERROR IN THE MEAN PLOT We can put a standard error or confidence interval around the mean plot to show the variability of the plot. Without Smoothing Let and plot S 2 (t) = 1 k 1 k i=1 ( Yit Ȳ (t) ) 2 Ȳ (t) ± t 1 α/2,k 1 S(t) k Since we are forming many confidence intervals, it may be sufficient to form one around every bth point only. A conservative estimate of the remaining bias is the difference between the lowest lower bound and the highest upper bound on the retained data. 46
36 With Smoothing Typically, Var[Ȳ (t, w)] < Var[Ȳ (t)] so using the previous intervals around the smoothed plot would be conservative (L&K, exercise 9.13). However, 1 Ȳ (t, w) = 2w + 1 = 1 k = 1 k k i=1 k i=1 w l= w 1 2w + 1 Ȳ it (w) 1 k Y i,l+t k i=1 w Y i,l+t l= w Since Ȳ 1t (w),..., Ȳ kt (w) are i.i.d., form a c.i. based on them. 47
37 SINGLE-RUN IDEAS Suppose we want to minimize bias by making only a single run, so that a mean plot may be too noisy. It will be useful to think of the output as represented by where E[X t ] = 0 and b t 0. Cumulative Mean Plot Y t = µ + X t + b t Let Ȳ j = 1 j j t=1 Y t and plot Ȳ j vs. j. This is not a good idea (L&K, exercise 9.14). Cusum Plot Let S 0 = 0 and S j = j t=1 (Ȳ Y t ) = j(ȳ Ȳ j ). Plot S j vs. j. 48
38 Under our model S j = j t=1 ( X X t + b b t ) Therefore, S 0 = S m = 0 (tied to 0 at the ends) E[S j ] = j t=1 ( b b t ) No bias If b t = 0 then E[S j ] = 0 and S j = j t=1 ( X X t ) should cross 0 many times. 49
39 Bias Suppose b t < 0 for t t and b t 0 for t > t. For j < t we have b b t > 0 so E[S j ] > 0. For j > t E[S j+1 S j ] = E j+1 (Ȳ Y t ) j t=1 t=1 = E[Ȳ Y j+1 ] E[Ȳ ] µ (Ȳ Y t ) = Bias[Ȳ ] = b < 0 So the slope of the curve is negative and equal to the bias. Statistical tests for bias have been based on this idea. 50
40 EFFECT OF DEPENDENCE When the output data are dependent, the usual se is not valid. When the output data are positively dependent, the usual se is too small. For Y 1, Y 2,..., Y n stationary and ρ h = Corr[Y t, Y t+h ] Var[Ȳ ] = σ2 Y n n 1 h=1 ( 1 h ) n ρ h while E [ ] S 2 n = σ2 Y n n 1 n 1 h=1 ( 1 h ) n ρ h giving a bias of E[S 2 /n] Var[Ȳ ] = 2σ2 n 1 ( ) n 1 h=1 1 h n ρh. 51
41 BATCH SIZE (CLASSICAL) How do we choose the batch size? 1. generate data 2. form batch means 3. evaluate independence and/or normality of batch means 4. if test fails, increase batch size (and perhaps generate more data). Classic References: Fishman (1978), Management Science 24, Law and Carson (1979), Operations Research 27,
42 LAG CORRELATIONS Batching algorithms typically focus on dependence rather than normality. Evaluate dependence via the lag correlations Corr[Y t, Y t+h ] for h = 1, 2,.... If independent, then Corr[Y t, Y t+h ] = 0. Sample lag-h correlation ˆρ h = n h i=1 (X i X)(X i+h X)/(n h) S 2 Focus on lag-1 correlation because other lags typically smaller. ˆρ h can be quite biased for small n. 53
43 FISHMAN (1978) Fishman assumes total sample size n is fixed. Starts with b = 1 (m = n batches). Tests Corr[Ȳ h, Ȳ h+1 ] = 0. If test rejected, set b 2b and repeat. Comments: Not clear what the meaning of repeated tests on same data is. Little is lost provided m > 30 since t 0.975,30 = 2.04 vs. t 0.975, = Waste to start with m = n. Actually used. 54
44 LAW & CARSON Law and Carson assume total sample size n can be sequentially increased to obtain c.i. with fixed relative width. Calculates lag-1 correlation for large number of small batches. Projects what lag-1 correlation would be for small number (40) of large batches. If correlation too large, can increase both batch size and sample size. Comments: A complex procedure, but well tested. 55
45 ASYMPTOTIC VARIANCE CONSTANT Suppose that the asymptotic variance constant σ 2 = lim n nvar[ȳ ] exists. Then for large n Var[Ȳ ] σ2 n The more modern view of single-rep output analysis is attempting to estimate σ 2. Note that σ 2 = σy 2 for i.i.d. data. 56
46 NONOVERLAPPING BATCH MEANS (NOBM) NOBM estimates σ 2 by σ 2 nobm = n m(m 1) m h=1 (Ȳh Ȳ ) 2 where b Ȳ h = 1 b Y (h 1)b+j j=1 To have a low-bias estimate of σ 2, it is sufficient that m be small enough (b large enough) that the batch means are approximately independent. 57
47 OVERLAPPING BATCH MEANS Notice that, for a stationary process E ( Ȳ h Ȳ ) 2 is the same for each batch mean. The NOBM estimator is just an average of these terms But this will be true for any batch mean of b consecutive observations, and the expected value of a sum is the sum of the expected values. Thus, if we have a large enough batch size to be good for NOBM, there is no reason the batches have to be nonoverlapping. 58
48 Let X h = 1 b h+b 1 for h = 1, 2,..., n b + 1, all the overlapping batch means of size b. t=h Y t Then we can estimate σ 2 by σ obm 2 n b+1 = constant h=1 ( X h Ȳ ) 2 The constant that is often used is nb/((n b + 1)(n b)) which is correct for i.i.d. data. The asymptotic variance of the OBM estimator is 2/3 that of NOBM estimator. 59
49 OTHER METHODS There are many other methods, including ARMA modeling: fit a parametric model to the data for which σ 2 is known. Standardized Time Series: σ 2 appears in a functional central limit theorem for a standardized version of the output process. Regenerative Method: σ 2 appears as a function of the reward and length of a regenerative cycle. 60
50 ESTIMATING THE BATCH SIZE For many variance estimators, their asymptotic bias and variance can be derived; thus, we can figure out the batch size that minimizes the asymptotic MSE. Under certain conditions MSE[ σ 2 ] = Var[ σ 2 ] + Bias 2 [ σ 2 ] lim b,n/b nbbias[ σ 2 ] = c b γ 1 R 0 lim b,n/b n 3 b Var[ σ 2 ] = c v (γ 0 R 0 ) 2 γ 0 = lim n nh= n ρ h γ 1 = lim n nh= n h ρ h R 0 = Var[Y t ] and (c b, c v ) are constants that depend on the estimator [(1,2) for NOBM, (1, 4/3) for OBM] 61
51 Thus for n and b large MSE[ σ 2 ] ( bcv γ 2 0 n 3 + c2 b γ2 1 n 2 b 2 ) R 2 0 Taking the derivative with respect to b and setting it to 0 gives an approximation to the minimum MSE batch size: b = 1 + 2n c2 ( ) 2 1/3 b γ1 c v γ 0 We can estimate b by estimating γ 0 and γ 1. They are not easy to estimate, but in finite samples the MSE has been shown to be rather flat around the optimal. 62
52 CONSISTENT ESTIMATION vs. CANCELLATION We have been thinking about NOBM with the number of batches m fixed as the sample size n (and therefore the batch size b) increases. While this gives an asymptotically valid confidence interval, it does not give a consistent estimator of σ 2. Consistent estimation is required by some sequential procedures, and it also tends to give narrower confidence intervals. As the total run length n increases, a consistent estimator is obtained by letting both m and b grow as O( n). See, for instance, Fishman and Yarberry (1997), INFORMS Journal on Computing, 9,
53 Cancellation: With the number of batches fixed, as n so that m(ȳ µ) σ 2 (m) σ 2 (m) σ 2 (m) χ2 df df σ(m)n(0, 1) σ 2 (m)χ 2 df /df t df Consistent estimation: With both the batch size and number of batches increasing as n σ 2 σ 2 so that n(ȳ µ) σ σ N(0, 1) σ N(0, 1) 64
Slides 12: Output Analysis for a Single Model
Slides 12: Output Analysis for a Single Model Objective: Estimate system performance via simulation. If θ is the system performance, the precision of the estimator ˆθ can be measured by: The standard error
More informationChapter 11. Output Analysis for a Single Model Prof. Dr. Mesut Güneş Ch. 11 Output Analysis for a Single Model
Chapter Output Analysis for a Single Model. Contents Types of Simulation Stochastic Nature of Output Data Measures of Performance Output Analysis for Terminating Simulations Output Analysis for Steady-state
More informationComputer Science, Informatik 4 Communication and Distributed Systems. Simulation. Discrete-Event System Simulation. Dr.
Simulation Discrete-Event System Simulation Chapter 0 Output Analysis for a Single Model Purpose Objective: Estimate system performance via simulation If θ is the system performance, the precision of the
More informationFall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A.
1. Let P be a probability measure on a collection of sets A. (a) For each n N, let H n be a set in A such that H n H n+1. Show that P (H n ) monotonically converges to P ( k=1 H k) as n. (b) For each n
More informationOverall Plan of Simulation and Modeling I. Chapters
Overall Plan of Simulation and Modeling I Chapters Introduction to Simulation Discrete Simulation Analytical Modeling Modeling Paradigms Input Modeling Random Number Generation Output Analysis Continuous
More informationISyE 6644 Fall 2014 Test 3 Solutions
1 NAME ISyE 6644 Fall 14 Test 3 Solutions revised 8/4/18 You have 1 minutes for this test. You are allowed three cheat sheets. Circle all final answers. Good luck! 1. [4 points] Suppose that the joint
More informationSIMULATION OUTPUT ANALYSIS
1 / 71 SIMULATION OUTPUT ANALYSIS Dave Goldsman School of ISyE Georgia Tech Atlanta, Georgia, USA sman@gatech.edu www.isye.gatech.edu/ sman 10/19/17 2 / 71 Outline 1 Introduction 2 A Mathematical Interlude
More information2WB05 Simulation Lecture 7: Output analysis
2WB05 Simulation Lecture 7: Output analysis Marko Boon http://www.win.tue.nl/courses/2wb05 December 17, 2012 Outline 2/33 Output analysis of a simulation Confidence intervals Warm-up interval Common random
More informationSTAT 512 sp 2018 Summary Sheet
STAT 5 sp 08 Summary Sheet Karl B. Gregory Spring 08. Transformations of a random variable Let X be a rv with support X and let g be a function mapping X to Y with inverse mapping g (A = {x X : g(x A}
More informationPractice Problems Section Problems
Practice Problems Section 4-4-3 4-4 4-5 4-6 4-7 4-8 4-10 Supplemental Problems 4-1 to 4-9 4-13, 14, 15, 17, 19, 0 4-3, 34, 36, 38 4-47, 49, 5, 54, 55 4-59, 60, 63 4-66, 68, 69, 70, 74 4-79, 81, 84 4-85,
More informationSIMULATION OUTPUT ANALYSIS
1 / 64 SIMULATION OUTPUT ANALYSIS Dave Goldsman School of ISyE Georgia Tech Atlanta, Georgia, USA sman@gatech.edu www.isye.gatech.edu/ sman April 15, 2016 Outline 1 Introduction 2 Finite-Horizon Simulation
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation
Chapter 11 Output Analysis for a Single Model Banks, Carson, Nelson & Nicol Discrete-Event System Simulation Purpose Objective: Estimate system performance via simulation If q is the system performance,
More informationOutput Data Analysis for a Single System
Output Data Analysis for a Single System Chapter 9 Based on the slides provided with the textbook 2 9.1 Introduction Output data analysis is often not conducted appropriately Treating output of a single
More informationAsymptotic Statistics-III. Changliang Zou
Asymptotic Statistics-III Changliang Zou The multivariate central limit theorem Theorem (Multivariate CLT for iid case) Let X i be iid random p-vectors with mean µ and and covariance matrix Σ. Then n (
More informationConfidence Intervals, Testing and ANOVA Summary
Confidence Intervals, Testing and ANOVA Summary 1 One Sample Tests 1.1 One Sample z test: Mean (σ known) Let X 1,, X n a r.s. from N(µ, σ) or n > 30. Let The test statistic is H 0 : µ = µ 0. z = x µ 0
More informationSummary of Chapters 7-9
Summary of Chapters 7-9 Chapter 7. Interval Estimation 7.2. Confidence Intervals for Difference of Two Means Let X 1,, X n and Y 1, Y 2,, Y m be two independent random samples of sizes n and m from two
More informationSimulation of stationary processes. Timo Tiihonen
Simulation of stationary processes Timo Tiihonen 2014 Tactical aspects of simulation Simulation has always a goal. How to organize simulation to reach the goal sufficiently well and without unneccessary
More informationEXAMINERS REPORT & SOLUTIONS STATISTICS 1 (MATH 11400) May-June 2009
EAMINERS REPORT & SOLUTIONS STATISTICS (MATH 400) May-June 2009 Examiners Report A. Most plots were well done. Some candidates muddled hinges and quartiles and gave the wrong one. Generally candidates
More informationCentral Limit Theorem ( 5.3)
Central Limit Theorem ( 5.3) Let X 1, X 2,... be a sequence of independent random variables, each having n mean µ and variance σ 2. Then the distribution of the partial sum S n = X i i=1 becomes approximately
More informationStat 710: Mathematical Statistics Lecture 31
Stat 710: Mathematical Statistics Lecture 31 Jun Shao Department of Statistics University of Wisconsin Madison, WI 53706, USA Jun Shao (UW-Madison) Stat 710, Lecture 31 April 13, 2009 1 / 13 Lecture 31:
More informationCPSC 531: System Modeling and Simulation. Carey Williamson Department of Computer Science University of Calgary Fall 2017
CPSC 531: System Modeling and Simulation Carey Williamson Department of Computer Science University of Calgary Fall 2017 Quote of the Day A person with one watch knows what time it is. A person with two
More informationReview. December 4 th, Review
December 4 th, 2017 Att. Final exam: Course evaluation Friday, 12/14/2018, 10:30am 12:30pm Gore Hall 115 Overview Week 2 Week 4 Week 7 Week 10 Week 12 Chapter 6: Statistics and Sampling Distributions Chapter
More informationMonte Carlo Studies. The response in a Monte Carlo study is a random variable.
Monte Carlo Studies The response in a Monte Carlo study is a random variable. The response in a Monte Carlo study has a variance that comes from the variance of the stochastic elements in the data-generating
More informationChapter 5. Statistical Models in Simulations 5.1. Prof. Dr. Mesut Güneş Ch. 5 Statistical Models in Simulations
Chapter 5 Statistical Models in Simulations 5.1 Contents Basic Probability Theory Concepts Discrete Distributions Continuous Distributions Poisson Process Empirical Distributions Useful Statistical Models
More informationBias Variance Trade-off
Bias Variance Trade-off The mean squared error of an estimator MSE(ˆθ) = E([ˆθ θ] 2 ) Can be re-expressed MSE(ˆθ) = Var(ˆθ) + (B(ˆθ) 2 ) MSE = VAR + BIAS 2 Proof MSE(ˆθ) = E((ˆθ θ) 2 ) = E(([ˆθ E(ˆθ)]
More informationB. Maddah INDE 504 Discrete-Event Simulation. Output Analysis (1)
B. Maddah INDE 504 Discrete-Event Simulation Output Analysis (1) Introduction The basic, most serious disadvantage of simulation is that we don t get exact answers. Two different runs of the same model
More informationCh 2: Simple Linear Regression
Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component
More informationOutput Data Analysis for a Single System
CHAPTER 9 Output Data Analysis for a Single System 9.1 Introduction... 9. Transient and Steady-State Behavior of a Stochastic Process...10 9.3 Types of Simulations with Regard to Output Analysis...1 9.4
More informationResampling and the Bootstrap
Resampling and the Bootstrap Axel Benner Biostatistics, German Cancer Research Center INF 280, D-69120 Heidelberg benner@dkfz.de Resampling and the Bootstrap 2 Topics Estimation and Statistical Testing
More informationRegression Estimation - Least Squares and Maximum Likelihood. Dr. Frank Wood
Regression Estimation - Least Squares and Maximum Likelihood Dr. Frank Wood Least Squares Max(min)imization Function to minimize w.r.t. β 0, β 1 Q = n (Y i (β 0 + β 1 X i )) 2 i=1 Minimize this by maximizing
More informationEE/PEP 345. Modeling and Simulation. Spring Class 11
EE/PEP 345 Modeling and Simulation Class 11 11-1 Output Analysis for a Single Model Performance measures System being simulated Output Output analysis Stochastic character Types of simulations Output analysis
More informationarxiv: v1 [stat.me] 14 Jan 2019
arxiv:1901.04443v1 [stat.me] 14 Jan 2019 An Approach to Statistical Process Control that is New, Nonparametric, Simple, and Powerful W.J. Conover, Texas Tech University, Lubbock, Texas V. G. Tercero-Gómez,Tecnológico
More informationEC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix)
1 EC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix) Taisuke Otsu London School of Economics Summer 2018 A.1. Summation operator (Wooldridge, App. A.1) 2 3 Summation operator For
More information401 Review. 6. Power analysis for one/two-sample hypothesis tests and for correlation analysis.
401 Review Major topics of the course 1. Univariate analysis 2. Bivariate analysis 3. Simple linear regression 4. Linear algebra 5. Multiple regression analysis Major analysis methods 1. Graphical analysis
More informationReview of probability and statistics 1 / 31
Review of probability and statistics 1 / 31 2 / 31 Why? This chapter follows Stock and Watson (all graphs are from Stock and Watson). You may as well refer to the appendix in Wooldridge or any other introduction
More informationSimulation. Where real stuff starts
1 Simulation Where real stuff starts ToC 1. What is a simulation? 2. Accuracy of output 3. Random Number Generators 4. How to sample 5. Monte Carlo 6. Bootstrap 2 1. What is a simulation? 3 What is a simulation?
More informationSampling Distributions
In statistics, a random sample is a collection of independent and identically distributed (iid) random variables, and a sampling distribution is the distribution of a function of random sample. For example,
More informationVariance reduction techniques
Variance reduction techniques Lecturer: Dmitri A. Moltchanov E-mail: moltchan@cs.tut.fi http://www.cs.tut.fi/ moltchan/modsim/ http://www.cs.tut.fi/kurssit/tlt-2706/ OUTLINE: Simulation with a given confidence;
More informationStochastic Simulation
Stochastic Simulation Jan-Pieter Dorsman 1 & Michel Mandjes 1,2,3 1 Korteweg-de Vries Institute for Mathematics, University of Amsterdam 2 CWI, Amsterdam 3 Eurandom, Eindhoven University of Amsterdam,
More informationSimulation optimization via bootstrapped Kriging: Survey
Simulation optimization via bootstrapped Kriging: Survey Jack P.C. Kleijnen Department of Information Management / Center for Economic Research (CentER) Tilburg School of Economics & Management (TiSEM)
More informationEMPIRICAL EVALUATION OF DATA-BASED DENSITY ESTIMATION
Proceedings of the 2006 Winter Simulation Conference L. F. Perrone, F. P. Wieland, J. Liu, B. G. Lawson, D. M. Nicol, and R. M. Fujimoto, eds. EMPIRICAL EVALUATION OF DATA-BASED DENSITY ESTIMATION E. Jack
More informationChapter 11 Estimation of Absolute Performance
Chapter 11 Estimation of Absolute Performance Purpose Objective: Estimate system performance via simulation If q is the system performance, the precision of the estimator qˆ can be measured by: The standard
More informationSTAT 830 Non-parametric Inference Basics
STAT 830 Non-parametric Inference Basics Richard Lockhart Simon Fraser University STAT 801=830 Fall 2012 Richard Lockhart (Simon Fraser University)STAT 830 Non-parametric Inference Basics STAT 801=830
More informationSTAT 443 Final Exam Review. 1 Basic Definitions. 2 Statistical Tests. L A TEXer: W. Kong
STAT 443 Final Exam Review L A TEXer: W Kong 1 Basic Definitions Definition 11 The time series {X t } with E[X 2 t ] < is said to be weakly stationary if: 1 µ X (t) = E[X t ] is independent of t 2 γ X
More informationThis does not cover everything on the final. Look at the posted practice problems for other topics.
Class 7: Review Problems for Final Exam 8.5 Spring 7 This does not cover everything on the final. Look at the posted practice problems for other topics. To save time in class: set up, but do not carry
More informationTime series models in the Frequency domain. The power spectrum, Spectral analysis
ime series models in the Frequency domain he power spectrum, Spectral analysis Relationship between the periodogram and the autocorrelations = + = ( ) ( ˆ α ˆ ) β I Yt cos t + Yt sin t t= t= ( ( ) ) cosλ
More information7.1 Basic Properties of Confidence Intervals
7.1 Basic Properties of Confidence Intervals What s Missing in a Point Just a single estimate What we need: how reliable it is Estimate? No idea how reliable this estimate is some measure of the variability
More informationSummarizing Measured Data
Summarizing Measured Data 12-1 Overview Basic Probability and Statistics Concepts: CDF, PDF, PMF, Mean, Variance, CoV, Normal Distribution Summarizing Data by a Single Number: Mean, Median, and Mode, Arithmetic,
More informationSEQUENTIAL ESTIMATION OF THE STEADY-STATE VARIANCE IN DISCRETE EVENT SIMULATION
SEQUENTIAL ESTIMATION OF THE STEADY-STATE VARIANCE IN DISCRETE EVENT SIMULATION Adriaan Schmidt Institute for Theoretical Information Technology RWTH Aachen University D-5056 Aachen, Germany Email: Adriaan.Schmidt@rwth-aachen.de
More informationPoint and Interval Estimation II Bios 662
Point and Interval Estimation II Bios 662 Michael G. Hudgens, Ph.D. mhudgens@bios.unc.edu http://www.bios.unc.edu/ mhudgens 2006-09-13 17:17 BIOS 662 1 Point and Interval Estimation II Nonparametric CI
More informationChapter 10 Verification and Validation of Simulation Models. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation
Chapter 10 Verification and Validation of Simulation Models Banks, Carson, Nelson & Nicol Discrete-Event System Simulation Purpose & Overview The goal of the validation process is: To produce a model that
More informationStatistics II Lesson 1. Inference on one population. Year 2009/10
Statistics II Lesson 1. Inference on one population Year 2009/10 Lesson 1. Inference on one population Contents Introduction to inference Point estimators The estimation of the mean and variance Estimating
More informationInference on distributions and quantiles using a finite-sample Dirichlet process
Dirichlet IDEAL Theory/methods Simulations Inference on distributions and quantiles using a finite-sample Dirichlet process David M. Kaplan University of Missouri Matt Goldman UC San Diego Midwest Econometrics
More informationThe bootstrap. Patrick Breheny. December 6. The empirical distribution function The bootstrap
Patrick Breheny December 6 Patrick Breheny BST 764: Applied Statistical Modeling 1/21 The empirical distribution function Suppose X F, where F (x) = Pr(X x) is a distribution function, and we wish to estimate
More informationResampling and the Bootstrap
Resampling and the Bootstrap Axel Benner Biostatistics, German Cancer Research Center INF 280, D-69120 Heidelberg benner@dkfz.de Resampling and the Bootstrap 2 Topics Estimation and Statistical Testing
More informationDiagnostics and Remedial Measures
Diagnostics and Remedial Measures Yang Feng http://www.stat.columbia.edu/~yangfeng Yang Feng (Columbia University) Diagnostics and Remedial Measures 1 / 72 Remedial Measures How do we know that the regression
More informationStatistical Inference
Statistical Inference Bernhard Klingenberg Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Outline Estimation: Review of concepts
More informationProbability Theory and Statistics. Peter Jochumzen
Probability Theory and Statistics Peter Jochumzen April 18, 2016 Contents 1 Probability Theory And Statistics 3 1.1 Experiment, Outcome and Event................................ 3 1.2 Probability............................................
More informationOptimal Linear Combinations of Overlapping Variance Estimators for Steady-State Simulation
Optimal Linear Combinations of Overlapping Variance Estimators for Steady-State Simulation Tûba Aktaran-Kalaycı, Christos Alexopoulos, David Goldsman, and James R. Wilson Abstract To estimate the variance
More informationSampling Distributions
Sampling Distributions In statistics, a random sample is a collection of independent and identically distributed (iid) random variables, and a sampling distribution is the distribution of a function of
More informationSTAT100 Elementary Statistics and Probability
STAT100 Elementary Statistics and Probability Exam, Sample Test, Summer 014 Solution Show all work clearly and in order, and circle your final answers. Justify your answers algebraically whenever possible.
More informationSummary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1)
Summary of Chapter 7 (Sections 7.2-7.5) and Chapter 8 (Section 8.1) Chapter 7. Tests of Statistical Hypotheses 7.2. Tests about One Mean (1) Test about One Mean Case 1: σ is known. Assume that X N(µ, σ
More informationGauge Plots. Gauge Plots JAPANESE BEETLE DATA MAXIMUM LIKELIHOOD FOR SPATIALLY CORRELATED DISCRETE DATA JAPANESE BEETLE DATA
JAPANESE BEETLE DATA 6 MAXIMUM LIKELIHOOD FOR SPATIALLY CORRELATED DISCRETE DATA Gauge Plots TuscaroraLisa Central Madsen Fairways, 996 January 9, 7 Grubs Adult Activity Grub Counts 6 8 Organic Matter
More informationSimulation. Where real stuff starts
Simulation Where real stuff starts March 2019 1 ToC 1. What is a simulation? 2. Accuracy of output 3. Random Number Generators 4. How to sample 5. Monte Carlo 6. Bootstrap 2 1. What is a simulation? 3
More informationLinear models and their mathematical foundations: Simple linear regression
Linear models and their mathematical foundations: Simple linear regression Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/21 Introduction
More informationRedacted for Privacy
AN ABSTRACT OF THE THESIS OF Lori K. Baxter for the degree of Master of Science in Industrial and Manufacturing Engineering presented on June 4, 1990. Title: Truncation Rules in Simulation Analysis: Effect
More informationWISE International Masters
WISE International Masters ECONOMETRICS Instructor: Brett Graham INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This examination paper contains 32 questions. You are
More informationModel-free prediction intervals for regression and autoregression. Dimitris N. Politis University of California, San Diego
Model-free prediction intervals for regression and autoregression Dimitris N. Politis University of California, San Diego To explain or to predict? Models are indispensable for exploring/utilizing relationships
More informationComparing two independent samples
In many applications it is necessary to compare two competing methods (for example, to compare treatment effects of a standard drug and an experimental drug). To compare two methods from statistical point
More informationGlossary. The ISI glossary of statistical terms provides definitions in a number of different languages:
Glossary The ISI glossary of statistical terms provides definitions in a number of different languages: http://isi.cbs.nl/glossary/index.htm Adjusted r 2 Adjusted R squared measures the proportion of the
More informationCH.8 Statistical Intervals for a Single Sample
CH.8 Statistical Intervals for a Single Sample Introduction Confidence interval on the mean of a normal distribution, variance known Confidence interval on the mean of a normal distribution, variance unknown
More informationK. Model Diagnostics. residuals ˆɛ ij = Y ij ˆµ i N = Y ij Ȳ i semi-studentized residuals ω ij = ˆɛ ij. studentized deleted residuals ɛ ij =
K. Model Diagnostics We ve already seen how to check model assumptions prior to fitting a one-way ANOVA. Diagnostics carried out after model fitting by using residuals are more informative for assessing
More informationOutput Analysis for a Single Model
Output Analysis for a Single Model Output Analysis for a Single Model Output analysis is the examination of data generated by a simulation. Its purpose is to predict the performance of a system or to compare
More informationSGN Advanced Signal Processing: Lecture 8 Parameter estimation for AR and MA models. Model order selection
SG 21006 Advanced Signal Processing: Lecture 8 Parameter estimation for AR and MA models. Model order selection Ioan Tabus Department of Signal Processing Tampere University of Technology Finland 1 / 28
More informationLectures on Simple Linear Regression Stat 431, Summer 2012
Lectures on Simple Linear Regression Stat 43, Summer 0 Hyunseung Kang July 6-8, 0 Last Updated: July 8, 0 :59PM Introduction Previously, we have been investigating various properties of the population
More informationRegression Estimation Least Squares and Maximum Likelihood
Regression Estimation Least Squares and Maximum Likelihood Dr. Frank Wood Frank Wood, fwood@stat.columbia.edu Linear Regression Models Lecture 3, Slide 1 Least Squares Max(min)imization Function to minimize
More informationStatistics - Lecture One. Outline. Charlotte Wickham 1. Basic ideas about estimation
Statistics - Lecture One Charlotte Wickham wickham@stat.berkeley.edu http://www.stat.berkeley.edu/~wickham/ Outline 1. Basic ideas about estimation 2. Method of Moments 3. Maximum Likelihood 4. Confidence
More informationEstimation theory. Parametric estimation. Properties of estimators. Minimum variance estimator. Cramer-Rao bound. Maximum likelihood estimators
Estimation theory Parametric estimation Properties of estimators Minimum variance estimator Cramer-Rao bound Maximum likelihood estimators Confidence intervals Bayesian estimation 1 Random Variables Let
More informationAPPM/MATH 4/5520 Solutions to Exam I Review Problems. f X 1,X 2. 2e x 1 x 2. = x 2
APPM/MATH 4/5520 Solutions to Exam I Review Problems. (a) f X (x ) f X,X 2 (x,x 2 )dx 2 x 2e x x 2 dx 2 2e 2x x was below x 2, but when marginalizing out x 2, we ran it over all values from 0 to and so
More informationSTAT 520: Forecasting and Time Series. David B. Hitchcock University of South Carolina Department of Statistics
David B. University of South Carolina Department of Statistics What are Time Series Data? Time series data are collected sequentially over time. Some common examples include: 1. Meteorological data (temperatures,
More informationz and t tests for the mean of a normal distribution Confidence intervals for the mean Binomial tests
z and t tests for the mean of a normal distribution Confidence intervals for the mean Binomial tests Chapters 3.5.1 3.5.2, 3.3.2 Prof. Tesler Math 283 Fall 2018 Prof. Tesler z and t tests for mean Math
More informationMore on Input Distributions
More on Input Distributions Importance of Using the Correct Distribution Replacing a distribution with its mean Arrivals Waiting line Processing order System Service mean interarrival time = 1 minute mean
More informationChapter 4 - Fundamentals of spatial processes Lecture notes
TK4150 - Intro 1 Chapter 4 - Fundamentals of spatial processes Lecture notes Odd Kolbjørnsen and Geir Storvik January 30, 2017 STK4150 - Intro 2 Spatial processes Typically correlation between nearby sites
More informationSimple linear regression
Simple linear regression Biometry 755 Spring 2008 Simple linear regression p. 1/40 Overview of regression analysis Evaluate relationship between one or more independent variables (X 1,...,X k ) and a single
More informationMathematical statistics
October 18 th, 2018 Lecture 16: Midterm review Countdown to mid-term exam: 7 days Week 1 Chapter 1: Probability review Week 2 Week 4 Week 7 Chapter 6: Statistics Chapter 7: Point Estimation Chapter 8:
More informationStatistical inference
Statistical inference Contents 1. Main definitions 2. Estimation 3. Testing L. Trapani MSc Induction - Statistical inference 1 1 Introduction: definition and preliminary theory In this chapter, we shall
More informationComputer Science, Informatik 4 Communication and Distributed Systems. Simulation. Discrete-Event System Simulation. Dr.
Simulation Discrete-Event System Simulation Chapter Comparison and Evaluation of Alternative System Designs Purpose Purpose: comparison of alternative system designs. Approach: discuss a few of many statistical
More informationOutline. Simulation of a Single-Server Queueing System. EEC 686/785 Modeling & Performance Evaluation of Computer Systems.
EEC 686/785 Modeling & Performance Evaluation of Computer Systems Lecture 19 Outline Simulation of a Single-Server Queueing System Review of midterm # Department of Electrical and Computer Engineering
More informationSTAT 4385 Topic 01: Introduction & Review
STAT 4385 Topic 01: Introduction & Review Xiaogang Su, Ph.D. Department of Mathematical Science University of Texas at El Paso xsu@utep.edu Spring, 2016 Outline Welcome What is Regression Analysis? Basics
More informationVariance reduction techniques
Variance reduction techniques Lecturer: Dmitri A. Moltchanov E-mail: moltchan@cs.tut.fi http://www.cs.tut.fi/kurssit/elt-53606/ OUTLINE: Simulation with a given accuracy; Variance reduction techniques;
More informationSimple Linear Regression
Simple Linear Regression In simple linear regression we are concerned about the relationship between two variables, X and Y. There are two components to such a relationship. 1. The strength of the relationship.
More informationFinal Examination Statistics 200C. T. Ferguson June 11, 2009
Final Examination Statistics 00C T. Ferguson June, 009. (a) Define: X n converges in probability to X. (b) Define: X m converges in quadratic mean to X. (c) Show that if X n converges in quadratic mean
More informationLecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2
Lecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2 Fall, 2013 Page 1 Random Variable and Probability Distribution Discrete random variable Y : Finite possible values {y
More informationCh 8. MODEL DIAGNOSTICS. Time Series Analysis
Model diagnostics is concerned with testing the goodness of fit of a model and, if the fit is poor, suggesting appropriate modifications. We shall present two complementary approaches: analysis of residuals
More informationReview of Statistics
Review of Statistics Topics Descriptive Statistics Mean, Variance Probability Union event, joint event Random Variables Discrete and Continuous Distributions, Moments Two Random Variables Covariance and
More informationAsymptotic Statistics-VI. Changliang Zou
Asymptotic Statistics-VI Changliang Zou Kolmogorov-Smirnov distance Example (Kolmogorov-Smirnov confidence intervals) We know given α (0, 1), there is a well-defined d = d α,n such that, for any continuous
More informationBIOS 6649: Handout Exercise Solution
BIOS 6649: Handout Exercise Solution NOTE: I encourage you to work together, but the work you submit must be your own. Any plagiarism will result in loss of all marks. This assignment is based on weight-loss
More informationEconometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018
Econometrics I KS Module 2: Multivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: April 16, 2018 Alexander Ahammer (JKU) Module 2: Multivariate
More informationIntroduction to statistics
Introduction to statistics Literature Raj Jain: The Art of Computer Systems Performance Analysis, John Wiley Schickinger, Steger: Diskrete Strukturen Band 2, Springer David Lilja: Measuring Computer Performance:
More information[Chapter 6. Functions of Random Variables]
[Chapter 6. Functions of Random Variables] 6.1 Introduction 6.2 Finding the probability distribution of a function of random variables 6.3 The method of distribution functions 6.5 The method of Moment-generating
More information