Prof. Dr. Christine Müller Dipl.-Math. Christoph Kustosz Eercises to Statistics of Material Fatigue No. 5 E. 9 (5 a Show, that a Fisher information matri for a two dimensional parameter θ (θ,θ 2 R 2, can be calculated by (( [ ] θ ln h(x,θ E ln h(x,θ, ln h(x,θ, θ ln h(x,θ θ 2 θ 2 where h(,θ f (,θ F(,θ is the hazard function under appropriate conditions. b Derive the information matri for X W (α,, i.e. X having a Weibull distribution with unknown parameters α and. Solution: We consider variables with density function θ ( 2 θ2 ( ep( θ θ θ θ2, R +. Further we have F θ (. First we remind some properties of the Gamma function, which will be helpful in b. We have Γ( t e t dt, Γ( : Γ( t e t log(tdt, Γ( (n : n n Γ( t e t log(t n dt. a To prove a we show, that [ ( ( ] I X (θ E θ log f θ (X θ log f θ (X [ ( ( ]! E θ θ (X θ θ (X ( ] 2 E[ θ θ (X ( ] E[( θ 2 θ (X θ θ (X You do not need to simplify the Gamma function! ( E[( θ θ (X E[ ( θ 2 θ (X ] θ 2 θ (X ] 2. Deadline: Tuesday.2.5; : Uhr
( ] Hence we calculate E[( θ i θ (X θ j θ (X for i, j {,2} and show [( ( ] [( ( ] E θ (X θ (X E log f θ (X log f θ (X. θ i θ j θ i θ j We have [( E ( log deriv. ( ] θ (X θ (X θ i θ j ( log d θ j θ i θ j d ( θ i log ( a θ i ( b θ i c d a θ j θ j θ j θ j θ j θ j θ j ( F θ ( ( F θ ( ( θ i t θ j d dt ( F θ ( d θ i d dt ( θ i t d dt θ i ( θ i ( θ i θ i ( Fθ( ( F θ ( dt In the steps we use the following additional calculations. In a we use: ( fθ ( θ i F θ ( θ i log θ i For b we use: θ i θ i ( F θ ( ( F θ ( F θ ( F θ ( dt dt, dt θ i F θ ( θ i ( F θ ( F θ ( Deadline: Tuesday.2.5; : Uhr
and hence ( F θ ( θ i θ i dt reg dt θ i θ i dt. For c we use F θ ( and Fubini. Further we rename t to and vice versa. For d we use t d θ i θ i t d t θ i log( F θ (d θ i ( F θ (t F θ (t b To calculate the Fisher Matri we remind, that for a Weibull distribution with Parameters α and the Hazard function is given by (, >. α α Hence the derivatives are α log (log( α α + ( log( α (log( log(α + ( log( ( log(α α α ( α α, log + log( log(α. Now we can calculate the elements of the Fisher Matri by application of a. [ ( ] [ 2 ( IX α (θ E α θ (X E ] 2 ( 2, α α I α, X [( ( ] (θ I,α X (θ E α θ (X θ (X [( E ( ] + log(x log(α α α + log(α α α E[log(X], [ ( ] I 2 X (θ E + log(x log(α [ (( E log(α [ ( E log(α ( log(α ] 2 + log(x 2 ( ] + 2 log(x log(α + log(x 2 log(α E[log(X] + E [ log(x 2]. 2 ( + 2. Deadline: Tuesday.2.5; : Uhr
We need to calculate the epectations. E[log(X] e log( α log(α log(tlog(t ( α ep ( ( α d log(αtt ep(t dt (log(α + log(t (t ep(t dt t ep(t dt + log(tt ep(t dt log(α t ep(t dt + log(t t ep(t dt f log(α ep(udu + log(uep(udu g log(α + Γ (. In e we substitute t α dt. Hence d α and therefore d αdt. This eliminates α. Further tα is used in the logarithm. In f we substitute u t. Hence du dt t and dt du. t In g we integrate the first term as usual and notice, that the second integral is the derivative Deadline: Tuesday.2.5; : Uhr
of the Gamma function, evaluated in. This completes the first calculation. E [ log(x 2] log( 2 ( α α e (log(t + log(α 2 t ep(t dt log(α 2 t ep(t dt + 2log(α + seeabove log(tt ep(t dt log(t 2 t ep(t dt log(α 2 t ep(t dt + 2log(α Γ ( + log(t 2 t ep(t dt h log(α 2 + 2log(α Γ ( + st log(α 2 + 2log(α Γ ( + 2 log(t 2 t ep(t dt log(α 2 + 2log(α Γ ( + Γ ( 2. log(s 2 ep(sds In h we substitute u t for the first integral and etend by 2 in the last integral. 2 Now we can plug the epecations into the calculations for the Fisher Matri and get I α, X (θ α + log(α ( log(α + α α Γ ( α α Γ ( ( + Γ ( α and ( I 2 ( ( X (θ log(α + 2 log(α log(α + Γ ( The Fisher Matri hence is + log(α 2 + 2log(α Γ ( + Γ ( 2 2 + 2 2 Γ ( + 2 Γ ( 2 ( + 2Γ ( + Γ (. ( α 2 α ( + Γ ( I X (θ. α ( + Γ ( ( + 2Γ ( + Γ ( 2 Deadline: Tuesday.2.5; : Uhr
E. (5 a Consider the problem of Eercise 6 on Sheet 3, part (i and calculate prediction intervals based on Theorems 2.7. and 2.7.3 assuming eponentially distributed random variables. b Compare the asymptotic prediction intervals with eact prediction intervals as calculated in Eercise 6. How do the interval scores change? Deadline: Tuesday.2.5; : Uhr