Let X (w) be a random function that maps from (2, F, P) to a measurable...

 

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Let X (w) be a random function that maps from (2, F, P) to a measurable space (X, Fx), for some X CR and an appropriate o-algebra Fx, where X has image measure Px < M (where M is either the Lebesgue or the counting measure). Let P= {PP (z)-p(z;), ES} be a set of density functions corresponding to some probability measure on (X. Fx), with respect to M, in the sense that P. (2) 20, for each E X, and 1 [p. (z) dM (a) — 1. Here SC R¹, for some q E N. We say that the density px (z) Pe (a) of Px is a g component mixture of densities from family P if (a) where g € N, ₂ > 0 for each [g], and . 1. We say that the measure Px is a g component mixture of measures with densities from P, or simply a g component finite mixture P. Here, 0 (₁ g. g) ET, and (b) >-{... T= 10 (7) - Σπερ (πψ.), ==1 (₁₂): ₂ > 0,= € [g], Show that if 1. Σ=₁ -1} x 8². Sº. 1 Prove that (i) p, (z) satisfies the usual definition of a density function (i.e., (1) holds for 0 instead of u), for any fixed ge I and class P. Further, prove that (ii) the set of mixtures of countable number of components of class P is convex: P = {1, (4) – Ĺw.p (=; 4.), w₂ > 0, V., € S., & € [g], Ĺæ, = 1,9 € N} =1 a=1 [5 Marks] Let n RoR be a convex function, and let a Ro be a fixed constant vector. 4 r (0) = n((a,0)), for 0(0,0) ETCR, then r (0) is majorized by m (0,7)=" 0₂T₂ (a. T) (1) 20.). T₂ where r = (T1, Tg) € T. [5 Marks] E.g. P(Zz(a)) = a where Z~ N(0, 1) (2) (c) (d) (e) Let Pro be defined as the family of Poisson density functions: exp(-) ProsPP(x) = * = {P=1 a! Use (2) to derive a majorizer for the loss function corresponding to the negative loga- rithm of the g component finite mixture of Pris, i.e, where 0 (₁) € T. T= {(10₁) Т lo (x) = -log po (x) = -log am1 0 1 2 Deaths per day Frequency 162 267 271 185 111 3 4 Wil (₁... wg): ₂ > 0. [g], =1 2=1 = R₂0}. z=1 ES-R₂ [5 Marks] We continue from Part (c). Given IID data (X.) where cach X, (i [n]) maps to (Z₂0, 2²20), and has identical image measure to X: Px « Mz, where Mz is the counting measure on Z. Further suppose that Px has a density function of the g component finite mixture of Proir form: P. (π) = ΣπP(π.), for some 8. (₁) ET. Construct an MM algorithm for com- puting a maximum likelihood estimator of 0., using the data (₁)ie[n] [5 Marks] -1}x₁ The following table describes the number of deaths of Women older than 80 years in London per day, between 1910 and 1912. 5 6 7 8 9 x R²0. Deaths per day Frequency 61 27 8 3 1 iming that the random sample (Xi)ie] (here n = 1096), corresponding to the data above, is IID, where each X, has the same density as X, as per Part (d), for g = 2 and generative parameter 8.. Write an R script using the algorithm developed in Part (d) to estimate .. Comment on the suitability of the assumption that has an image measure describable by a two component mixture of Poisson densities. [10 Marks] Let X (w) be a random function that maps from (2, F, P) to a measurable space (X, Fx), for some X CR and an appropriate o-algebra Fx, where X has image measure Px < M (where M is either the Lebesgue or the counting measure). Let P= {PP (z)-p(z;), ES} be a set of density functions corresponding to some probability measure on (X. Fx), with respect to M, in the sense that P. (2) 20, for each E X, and 1 [p. (z) dM (a) — 1. Here SC R¹, for some q E N. We say that the density px (z) Pe (a) of Px is a g component mixture of densities from family P if (a) where g € N, ₂ > 0 for each [g], and . 1. We say that the measure Px is a g component mixture of measures with densities from P, or simply a g component finite mixture P. Here, 0 (₁ g. g) ET, and (b) >-{... T= 10 (7) - Σπερ (πψ.), ==1 (₁₂): ₂ > 0,= € [g], Show that if 1. Σ=₁ -1} x 8². Sº. 1 Prove that (i) p, (z) satisfies the usual definition of a density function (i.e., (1) holds for 0 instead of u), for any fixed ge I and class P. Further, prove that (ii) the set of mixtures of countable number of components of class P is convex: P = {1, (4) – Ĺw.p (=; 4.), w₂ > 0, V., € S., & € [g], Ĺæ, = 1,9 € N} =1 a=1 [5 Marks] Let n RoR be a convex function, and let a Ro be a fixed constant vector. 4 r (0) = n((a,0)), for 0(0,0) ETCR, then r (0) is majorized by m (0,7)=" 0₂T₂ (a. T) (1) 20.). T₂ where r = (T1, Tg) € T. [5 Marks] E.g. P(Zz(a)) = a where Z~ N(0, 1) (2) (c) (d) (e) Let Pro be defined as the family of Poisson density functions: exp(-) ProsPP(x) = * = {P=1 a! Use (2) to derive a majorizer for the loss function corresponding to the negative loga- rithm of the g component finite mixture of Pris, i.e, where 0 (₁) € T. T= {(10₁) Т lo (x) = -log po (x) = -log am1 0 1 2 Deaths per day Frequency 162 267 271 185 111 3 4 Wil (₁... wg): ₂ > 0. [g], =1 2=1 = R₂0}. z=1 ES-R₂ [5 Marks] We continue from Part (c). Given IID data (X.) where cach X, (i [n]) maps to (Z₂0, 2²20), and has identical image measure to X: Px « Mz, where Mz is the counting measure on Z. Further suppose that Px has a density function of the g component finite mixture of Proir form: P. (π) = ΣπP(π.), for some 8. (₁) ET. Construct an MM algorithm for com- puting a maximum likelihood estimator of 0., using the data (₁)ie[n] [5 Marks] -1}x₁ The following table describes the number of deaths of Women older than 80 years in London per day, between 1910 and 1912. 5 6 7 8 9 x R²0. Deaths per day Frequency 61 27 8 3 1 iming that the random sample (Xi)ie] (here n = 1096), corresponding to the data above, is IID, where each X, has the same density as X, as per Part (d), for g = 2 and generative parameter 8.. Write an R script using the algorithm developed in Part (d) to estimate .. Comment on the suitability of the assumption that has an image measure describable by a two component mixture of Poisson densities. [10 Marks]