a) Transformation method (for continuous distributions) U(0,1) : uniform distribution f(x) : arbitrary distribution f(x) dx = U(0,1)(u) du When inverse function of integral, F -1 (u), is known, then x = F -1 (u) distributed according to f(x) Example: Exponential distribution 4. MC Methods 4.2 Generators for arbitrary distributions K. Desch – Statistical methods of data analysis SS10

b) Transformation method (discrete distributions) 4. MC Methods 4.2 Generators for arbitrary distributions K. Desch – Statistical methods of data analysis SS10 c) Hit-or-miss method (brute force) Uniform distr. fr. 0 to c: u i Uniform distr. from x min to x max : x i when u i ≤ f(x i ) → accept x i, otherwise not - two random numbers per try - inefficient when f(x) « c - need to (conservatively) estimate c (maximum of f(x)) (can be done in “warm-up” run)

4. MC Methods 4.2 Generators for arbitrary distributions K. Desch – Statistical methods of data analysis SS10 Improvement: - search for analytical function s(x) close to f(x) - use c so that c s(x) >f(x) for all x 1.take u i in [0,1] and calculate x i = S -1 (u i ) 2.take u j in [0,c] 3.when u j s(x i ) ≤ f(x i ) accept x i, otherwise not

search for: 4. MC Methods 4.3 Monte Carlo Integration K. Desch – Statistical methods of data analysis SS10 Integration over one dimension: (E[g] = expectation value of g w.r.t. uniform distribution) Take x i uniformly distributed in [a,b] → Variance: (CLT)

4. MC Methods 4.3 Monte Carlo Integration K. Desch – Statistical methods of data analysis SS10 Alternative: hit-or-miss integration

- Variance of r(x): will be small when r is flat, so f ≈ g - The method takes care of (integrable) singularities (find f(x) with has the same singularity structure as g(x)) x i distributed as f(x) 4. MC Methods 4.3 Monte Carlo Integration K. Desch – Statistical methods of data analysis SS10 Variance-reduced methods a) importance sampling: If f(x) is a known p.d.f., which could be integrated and inverted, then: Expectation value of r(x) can be obtained with random numbers, which is distributed according to f(x):

4. MC Methods 4.3 Monte Carlo Integration K. Desch – Statistical methods of data analysis SS10 b) Control function (subtraction of an integrable analytical function) analytical MC c) Partitioning (split integration range into several more „flat“ regions)

let x be a random variable distributed according to f(x) n independent “measurements” of x, x = (x 1,…,x n ) is sample of a distribution f(x) of size n (outcome of an experiment) x = itself is a random variable with p.d.f. f sample (x) sample space: all possible values of x = (x 1,…,x n ) If all x i are independent f sample (x) = f(x 1 )f(x 2 ) … f(x n ) is the p.d.f. for x 5. Estimation 5.1 Sample space, Estimators K. Desch – Statistical methods of data analysis SS10

A central problem of (frequentist) statistics: Find the properties of f(x) when only a sample x = (x 1,…,x n ) has been measured Task: construct functions of x i to estimate the properties of f(x) (e.g. μ, σ 2, …) Often f depends on parameters θ j : f(x i ;θ j )  try to estimate the parameters θ j from measured sample x Functions of (x i ) are called a statistic. If a statistic is used to estimate parameters (μ, σ 2, θ, …), it called an estimator Notation: is an estimator for θ can be calculated; true value θ is unknown Estimation of p.d.f. parameters is also called a fit 5. Estimation 5.1 Sample Space, Estimators K. Desch – Statistical methods of data analysis SS10

in simple words: n→ ∞  θ → 2.Bias: itself is a random variable, distributed according to a p.d.f. This p.d.f. is called the sampling distribution Expectation value of the sampling distribution: (or “ “) 1 Consistency: an estimator is consistent if for each ε > 0 : 5. Estimation 5.2 Properties of Estimators K. Desch – Statistical methods of data analysis SS10 because

5. Estimation 5.2 Properties of estimators K. Desch – Statistical methods of data analysis SS10 The bias of an estimator is defined as An estimator is unbiased (or bias-free) if b=0 An estimator is asymptotically unbiased if Attentions Consistent: for large sample size Unbiased: for fixed sample size 3. Efficiency: One estimator is more efficient than another if its variance is smaller, or more precise if its mean squared error (MSE) is smaller and

5. Estimation 5.2 Properties of estimators K. Desch – Statistical methods of data analysis SS10 4. Robustness An estimator is robust if it does not strongly depend on single measurements (which might be systematically wrong) 5. Simplicity (subjective)

5. Estimation 5.3 Estimation of the mean K. Desch – Statistical methods of data analysis SS10 In principle one can construct an arbitrary number of different esitmators for the mean value of a pdf,  = E[x] Examples: mean of the sample mean of the first ten members of the sample median of the sample all have different (wanted and unwanted) properties

5. Estimation 5.3 Estimation of the mean K. Desch – Statistical methods of data analysis SS10 The mean of a sample provides an estimate of the true mean: a) is consistent: CLT: p.d.f. of approaches Gaussian with variance b) is unbiased c) Is efficient ?