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Likelihood

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The use of the probability density instead of a probability in specifying the likelihood function above is justified as follows. The likelihood that an observation x{\displaystyle x} lies in the interval [xj,xj+h]{\displaystyle [x_{j},x_{j}+h]}, where xj{\displaystyle x_{j}} is a specific observed value and h>0{\displaystyle h>0} a constant, is given by L(θ|x∈[xj,xj+h]){\displaystyle {\mathcal {L}}(\theta |x\in [x_{j},x_{j}+h])}. Observe that arg⁡maxθL(θ|x∈[xj,xj+h])=arg⁡maxθ1hL(θ|x∈[xj,xj+h]){\displaystyle \arg \max _{\theta }{\mathcal {L}}(\theta |x\in [x_{j},x_{j}+h])=\arg \max _{\theta }{\frac {1}{h}}{\mathcal {L}}(\theta |x\in [x_{j},x_{j}+h])}, since h{\displaystyle h} is positive and constant. Because arg⁡maxθ1hL(θ|x∈[xj,xj+h])=arg⁡maxθ1hPr(xj≤x≤xj+h|θ)=arg⁡maxθ1h∫xjxj+hf(x|θ)dx{\displaystyle \arg \max _{\theta }{\frac {1}{h}}{\mathcal {L}}(\theta |x\in [x_{j},x_{j}+h])=\arg \max _{\theta }{\frac {1}{h}}\mathrm {Pr} (x_{j}\leq x\leq x_{j}+h|\theta )=\arg \max _{\theta }{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x|\theta )\,dx}, where f(x|θ){\displaystyle f(x|\theta )} is the probability density function of the variable x{\displaystyle x}, it follows that arg⁡maxθL(θ|x∈[xj,xj+h])=arg⁡maxθ1h∫xjxj+hf(x|θ)dx{\displaystyle \arg \max _{\theta }{\mathcal {L}}(\theta |x\in [x_{j},x_{j}+h])=\arg \max _{\theta }{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x|\theta )\,dx}. The first fundamental theorem of calculus and the l'Hôpital's rule together provide that limh→0+1h∫xjxj+hf(x|θ)dx=limh→0+ddh∫xjxj+hf(x|θ)dxdhdh=limh→0+f(xj+h|θ)1=f(xj|θ){\displaystyle \lim _{h\to 0^{+}}{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x|\theta )\,dx=\lim _{h\to 0^{+}}{\frac {{\frac {d}{dh}}\int _{x_{j}}^{x_{j}+h}f(x|\theta )\,dx}{\frac {dh}{dh}}}=\lim _{h\to 0^{+}}{\frac {f(x_{j}+h|\theta )}{1}}=f(x_{j}|\theta )}. Then, arg⁡maxθL(θ|xj)=arg⁡maxθ[limh→0+L(θ|x∈[xj,xj+h])]=arg⁡maxθ[limh→0+1h∫xjxj+hf(x|θ)dx]=arg⁡maxθf(xj|θ){\displaystyle \arg \max _{\theta }{\mathcal {L}}(\theta |x_{j})=\arg \max _{\theta }\left[\lim _{h\to 0^{+}}{\mathcal {L}}(\theta |x\in [x_{j},x_{j}+h])\right]=\arg \max _{\theta }\left[\lim _{h\to 0^{+}}{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x|\theta )\,dx\right]=\arg \max _{\theta }f(x_{j}|\theta )}. Therefore,

late 14c., "resemblance, similarity," from likely + -hood . Meaning "probability" is from mid-15c.What made you want to look up likelihood ? Please tell us where you read or heard it (including the quote, if possible).


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