Statistics, Data Analysis, and Simulation SS 2013

Transcription

1 Statistics, Data Analysis, and Simulation SS Statistik, Datenanalyse und Simulation Dr. Michael O. Distler Mainz, May 21, 2013

2 3. Parameter estimation 1 Consistency: lim n â = a 0. 2 Unbiasedness: E[â] = a 0. 3 Efficiency: Variance V[â] small. 4 Robustness: Insensitive to false data and false assumptions, e.g. of normality. 3.1 Robust estimate of the mean

3 3.2 The maximum likelihood method Our sample consists of n observations x i. The probability distribution function f (x a) is known and properly normalized f (x a) dx = 1. Likelihood function: L(a) = f (x 1 a) f (x 2 a)... f (x n a) = n f (x i a) The best estimation of â can be found by maximizing the likelihood function. dl(a) da or L(a k ) a k for all k

4 The maximum likelihood method For practical reasons one uses the logarithm of the likelihood function l(a) = ln L(a) or the negative logarithmus: F(a) = l(a) = ln f (x i a) of cause F(a) has to be minimized. negative log-likelihood-function

5 The maximum likelihood method 0 3m 4m 5m source detector (exp.1) detector (exp.2) 100 particles 5 particles 1 particle detector (exp.3) 0 particles Example: A particle source emits ν 0 = 100 particles. In a first experiment one detects 5 particles in 3 m distance from the source. In a second and third experiment one detects 1 (0) particles in 4m (5m) distance. Use the ML method to estimate the range parameter λ. Exponential decay: ν = ν 0 exp( λx) Poisson distribution: p(n ν) = νn n! exp( ν) p i p(n i ν i ) pi guess x = 3 m 4 m 5 m λ = 1.2 ν(λ, x) p(ν, n) λ = 1.1 ν(λ, x) p(ν, n) λ = ν(λ, x) p(ν, n)

6 The maximum likelihood method neg. log likelihood F(λ) F= ˆλ parameter λ Variance of the estimate: ( ) d 2 F 1 V[ˆλ] = dλ 2 = (66.59) 1 =

7 Combining measurements with maximum likelihood Consider an experiment in which one has n measured values of a random variable x distributed according to a Gaussian p.d.f. of unknown µ and varying, known σ 2 : Gaussian p.d.f.: f (x i, µ) = 1 ( ) e 1 xi µ 2 2 σ i 2πσi neg. log-likelihood function: ( xi µ ) 2 F(µ) = const df dµ = = x i µ σ 2 i x i σ 2 i µ = 0 1 σ 2 i σ i

8 Combining measurements with maximum likelihood Best estimate: ˆµ = xi σ 2 i 1 σ 2 i = wi x i wi (weighted mean) Standard deviation: ( d 2 F σ(ˆµ) = dµ 2 Equal variances: σ i = σ Weights: w i = 1 σ 2 i ˆµ) 1/2 = 1 1 / σi 2 ˆµ = 1 / σ 2 i x i 1 / σ 2 i 1 = 1 n xi = 1 wi σ(ˆµ) = σ n

9 Combining measurements: Bad example Radioactive decay: Poisson distribution Counting for 1h, results: 4 decays/h, 1/h, 2/h, 1/h Application of half knowledge: Counting experiment: error N for large N (min. N > 10) Mean number of decays per hour: 4 λ 1/h = = σ 1/h = 2 WRONG 11 λ = (1.45 ± 0.60) /h = 16 11

10 Mean of a Poisson distribution (ML estimator) Estimate parameter ˆµ (using n observations) of a Poisson distribution: f (x i µ) = µx i x i! e µ normalized! for all µ neg. log-likelihood function: F (µ) = (x i ln µ µ ln x i!) = ln µ df dµ = n 1 µ x i d 2 F dµ 2 = 1 µ 2 x i x i + nµ + const. Best estimate: ˆµ = 1 x i n ( V[ˆµ] = ˆµ n = d 2 1 F dµ 2 = µ=ˆµ) ( ) nˆµ 1 ˆµ 2 = ˆµ n

11 Combining measurements: correctly, this time Measurements: 4, 1, 2, 1 ˆµ = 1 ( ) = 2.0 (not 1.45) 4 V[ˆµ] = 2 4 = 1 2 σ(ˆµ) = 0.70 (not 0.60) neg. log likelihood F(µ) F= parameter µ