<< Chapter < Page Chapter >> Page >

Log normal distribution.

The continuous random variable x has log normal distribution if y has a normal distribution and x = e y . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacqGH9aqpcaWGLbWaaWbaaSqabeaacaWG5baaaOGaaiOlaaaa@3ABD@ Thus, if y N ( μ , σ 2 ) , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhacqWI8iIocaWGobWaaeWaaeaacqaH8oqBcaGGSaGaeq4Wdm3aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaGaaiilaaaa@4038@ then the pdf of a log normal distribution is f ( x ) = { 1 x σ 2 π e ( ln ( x ) μ ) 2 2 σ 2 ,  for  x > 0 0     otherwise } . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6667@ The mean and variance of x are μ x = e μ + σ 2 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTnaaBaaaleaacaWG4baabeaakiabg2da9iaadwgadaahaaWcbeqaaiabeY7aTjabgUcaRmaaleaameaacqaHdpWCdaahaaqabeaacaaIYaaaaaqaaiaaikdaaaaaaaaa@4101@ and σ x 2 = ( e σ 2 1 ) e 2 μ + σ 2 . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaDaaaleaacaWG4baabaGaaGOmaaaakiabg2da9maabmaabaGaamyzamaaCaaaleqabaGaeq4Wdm3aaWbaaWqabeaacaaIYaaaaaaakiabgkHiTiaaigdaaiaawIcacaGLPaaacaWGLbWaaWbaaSqabeaacaaIYaGaeqiVd0Maey4kaSIaeq4Wdm3aaWbaaWqabeaacaaIYaaaaaaakiaac6caaaa@4975@ Because the distribution is skewed downward for variances over 1, the log normal distribution is sometimes used to describe income distributions (where there are relatively few very wealthy people and incomes generally are positive. Figure 4 shows the graphs of the pdf and cumulative functions for the log normal distributions for two values of σ .

The log-normal distribution.

Graph of the log-normal distribution for two values of σ.
The two panels illustrate the log-normal distribution for two values of σ ..

Gamma distribution.

A positive random variable x has a gamma distribution if its pdf is f ( x ) = 1 Γ ( α ) β α x α 1 e x β MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacqqHtoWrdaqadaqaaiabeg7aHbGaayjkaiaawMcaaiabek7aInaaCaaaleqabaGaeqySdegaaaaakiaadIhadaahaaWcbeqaaiabeg7aHjabgkHiTiaaigdaaaGccaWGLbWaaWbaaSqabeaacqGHsisldaWcbaadbaGaamiEaaqaaiabek7aIbaaaaaaaa@4C6C@ for x > 0 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacqGH+aGpcaaIWaaaaa@38A8@ and 0 elsewhere. Γ ( α ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfo5ahnaabmaabaGaeqySdegacaGLOaGaayzkaaaaaa@3A79@ is known as the gamma function and is defined to be Γ ( α ) = 0 y α 1 e y d y = ( α 1 ) ! . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfo5ahnaabmaabaGaeqySdegacaGLOaGaayzkaaGaeyypa0Zaa8qmaeaacaWG5bWaaWbaaSqabeaacqaHXoqycqGHsislcaaIXaaaaOGaamyzamaaCaaaleqabaGaeyOeI0IaamyEaaaakiaadsgacaWG5baaleaacaaIWaaabaGaeyOhIukaniabgUIiYdGccqGH9aqpdaqadaqaaiabeg7aHjabgkHiTiaaigdaaiaawIcacaGLPaaacaGGHaGaaiOlaaaa@5079@ The gamma function is often used to model waiting times like waiting for death. Its mean and variance are given by μ = α β MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTjabg2da9iabeg7aHjabek7aIbaa@3BE5@ and σ 2 = α β 2 . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaakiabg2da9iabeg7aHjabek7aInaaCaaaleqabaGaaGOmaaaakiaac6caaaa@3E8A@

Chi-square distribution.

A chi-square distribution ( χ 2 ( k ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeE8aJnaaCaaaleqabaGaaGOmaaaakmaabmaabaGaam4AaaGaayjkaiaawMcaaiaac6caaaa@3BBE@ ) is the sum of k independent standard normal random variables and is a special case of the gamma distribution (with α = k 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHjabg2da9maalaaabaGaam4Aaaqaaiaaikdaaaaaaa@3A4A@ and β = 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIjabg2da9iaaikdaaaa@394C@ ). The pdf of a chi-square distribution with k degrees of freedom is f ( x ) = 1 2 k 2 Γ ( k 2 ) x k 2 1 e x 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaWaaWbaaSqabeaadaWcdaadbaGaam4AaaqaaiaaikdaaaaaaOGaeu4KdC0aaeWaaeaadaWcbaWcbaGaam4AaaqaaiaaikdaaaaakiaawIcacaGLPaaaaaGaamiEamaaCaaaleqabaWaaSqaaWqaaiaadUgaaeaacaaIYaaaaSGaeyOeI0IaaGymaaaakiaadwgadaahaaWcbeqaaiabgkHiTmaaleaameaacaWG4baabaGaaGOmaaaaaaaaaa@4B36@ where x >0. Its mean and variance are μ = k MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTjabg2da9iaadUgaaaa@3995@ and σ 2 = 2 k . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaakiabg2da9iaaikdacaWGRbGaaiOlaaaa@3C03@ If y = i = 1 k x i 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhacqGH9aqpdaaeWbqaaiaadIhadaqhaaWcbaGaamyAaaqaaiaaikdaaaaabaGaamyAaiabg2da9iaaigdaaeaacaWGRbaaniabggHiLdaaaa@4097@ where the x i 's are independently drawn from the standard normal distribution (N(1, 0)), then y i χ 2 ( k ) . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaGccqWI8iIocqaHhpWydaahaaWcbeqaaiaaikdaaaGcdaqadaqaaiaadUgaaiaawIcacaGLPaaacaGGUaaaaa@3F09@

Student's t-distribution.

Consider two random variables, x and v . Assume that x N ( 0 , 1 ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacqWI8iIocaWGobWaaeWaaeaacaaIWaGaaiilaiaaigdaaiaawIcacaGLPaaaaaa@3C90@ and v χ 2 ( r ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAhacqWI8iIocqaHhpWydaahaaWcbeqaaiaaikdaaaGcdaqadaqaaiaadkhaaiaawIcacaGLPaaaaaa@3D37@ and are stochastically independent. Then the random variable t = w v r MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshacqGH9aqpdaWcaaqaaiaadEhaaeaadaGcaaqaamaalaaabaGaamODaaqaaiaadkhaaaaaleqaaaaaaaa@3B11@ has the t-distribution with r degrees of freedom . The pdf and cumulative function of t are f ( t ) = Γ ( r + 1 2 ) r π Γ ( r 2 ) ( 1 + t 2 r ) ( r + 1 2 ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadshaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiabfo5ahnaabmaabaWaaSaaaeaacaWGYbGaey4kaSIaaGymaaqaaiaaikdaaaaacaGLOaGaayzkaaaabaWaaOaaaeaacaWGYbGaeqiWdahaleqaaOGaeu4KdC0aaeWaaeaadaWcaaqaaiaadkhaaeaacaaIYaaaaaGaayjkaiaawMcaaaaadaqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaadshadaahaaWcbeqaaiaaikdaaaaakeaacaWGYbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0YaaeWaaeaadaWcaaqaaiaadkhacqGHRaWkcaaIXaaabaGaaGOmaaaaaiaawIcacaGLPaaaaaaaaa@5466@ and F ( t ) = 1 2 + t Γ ( t 2 ) . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaqadaqaaiaadshaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiabgUcaRiaadshacqqHtoWrdaqadaqaamaalaaabaGaamiDaaqaaiaaikdaaaaacaGLOaGaayzkaaGaaiOlaaaa@4306@ The mean and variance of the distribution are 0 for r > 1 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkhacqGH+aGpcaaIXaaaaa@38A3@ and r r 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaamOCaaqaaiaadkhacqGHsislcaaIYaaaaaaa@3990@ for t > 2 , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshacqGH+aGpcaaIYaGaaiilaaaa@3956@ respectively. The mean of the t-distribution is undefined for t 1. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshacqGHKjYOcaaIXaGaaiOlaaaa@3A04@ The variance of the distribution is MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabg6HiLcaa@375A@ for 1 < r 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaigdacqGH8aapcaWGYbGaeyizImQaaGOmaaaa@3B10@ and undefined for r 1. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkhacqGHKjYOcaaIXaGaaiOlaaaa@3A02@ The t-distribution plays a prominent role in hypothesis testing that is well-known to all undergraduate economics majors.

F distribution.

Consider two stochastically independent chi-square random variable such that u χ 2 ( r 1 ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacqWI8iIocaqGhpWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaWGYbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaaaaa@3DBD@ and v χ 2 ( r 2 ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAhacqWI8iIocaqGhpWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaWGYbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaaaa@3DBF@ and u , v > 0. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacaGGSaGaamODaiabg6da+iaaicdacaGGUaaaaa@3B02@ The new random variable f = u r 1 v r 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacqGH9aqpdaWcaaqaamaaliaabaGaamyDaaqaaiaadkhadaWgaaWcbaGaaGymaaqabaaaaaGcbaWaaSGaaeaacaWG2baabaGaamOCamaaBaaaleaacaaIYaaabeaaaaaaaaaa@3DCA@ has a F-distribution with r 1 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkhadaWgaaWcbaGaaGymaaqabaaaaa@37C7@ and r 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkhadaWgaaWcbaGaaGOmaaqabaaaaa@37C8@ degrees of freedom. The pdf for the F-distribution is g ( f ) = Γ ( r 1 + r 2 2 ) ( r 1 r 2 ) Γ ( r 1 2 ) Γ ( r 2 2 ) f r 1 2 1 ( 1 + r 1 f r 2 ) r 1 + r 2 2 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6771@ The F-distribution is used in testing if population variances are equal and in performing likelihood ratio tests.

Multinomial distribution.

Consider the n random variables x 1 , x 2 , , x n MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaacYcacqWIVlctcaGGSaGaamiEamaaBaaaleaacaWGUbaabeaaaaa@3FE0@ where each variable has a normal distribution—that is, x i N ( μ i , σ i 2 ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccqWI8iIocaWGobWaaeWaaeaacqaH8oqBdaWgaaWcbaGaamyAaaqabaGccaGGSaGaeq4Wdm3aa0baaSqaaiaadMgaaeaacaaIYaaaaaGccaGLOaGaayzkaaaaaa@42BD@ and the covariance between of the variables is σ i j = E [ ( x i μ i ) ( x j μ j . ) ] MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWGfbWaamWaaeaadaqadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccqGHsislcqaH8oqBdaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaadaqadaqaaiaadIhadaWgaaWcbaGaamOAaaqabaGccqGHsislcqaH8oqBdaWgaaWcbaGaamOAaaqabaaakiaawIcacaGLPaaaaiaawUfacaGLDbaaaaa@4C65@ We can arrange the variances and covariances into a n -by- n matrix where Σ = [ σ 1 2 σ 12 σ 1 n σ 21 σ 2 2 σ 2 n σ n 1 σ n 2 σ n 2 ] MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6816@ that is known as the variance-covariance matrix. Define the vector ( x μ ) = ( x 1 μ 1 x n μ n ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaCiEaiabgkHiTiaahY7aaiaawIcacaGLPaaacqGH9aqpdaqadaqaauaabeqadeaaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaeqiVd02aaSbaaSqaaiaaigdaaeqaaaGcbaGaeSO7I0eabaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkHiTiabeY7aTnaaBaaaleaacaWGUbaabeaaaaaakiaawIcacaGLPaaaaaa@4AA8@ and ( x μ ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaCiEaiabgkHiTiaahY7aaiaawIcacaGLPaaadaahaaWcbeqaaOGamai4gkdiIcaaaaa@3DC7@ as its transpose. Then, ( x μ ) Σ ( x μ ) = i = 1 n j = 1 n ( x i μ i ) ( x j μ j ) σ i j , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@63E3@ where σ i i = σ i 2 . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamyAaaqabaGccqGH9aqpcqaHdpWCdaqhaaWcbaGaamyAaaqaaiaaikdaaaaaaa@3E5E@ If | Σ | MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaaC4OdaGaay5bSlaawIa7aaaa@3A3A@ is the determinant of the variance-covariance matrix, then the pdf for the joint distribution of these random variables is f ( x 1 , x 2 , , x n ) = 1 ( 2 π ) n / 2 | Σ | 1 2 e 1 2 ( x μ ) Σ ( x μ ) . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@61FC@ If the random variables are stochastically independent the covariances are equal to 0 and the pdf becomes f ( x 1 , x 2 , , x n ) = 1 ( 2 π ) n / 2 ( i = 1 n σ 1 2 ) 1 2 e 1 2 i = 1 n ( x i μ i ) 2 σ i 2 . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6CB6@ If the n random variables are all drawn from the same normal distribution with a mean of μ and a variance of σ 2 , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaakiaacYcaaaa@394F@ then the pdf simplifies to f ( x 1 , x 2 , , x n ) = 1 ( 2 π σ 2 ) n / 2 e 1 2 σ 2 i = 1 n ( x i μ ) 2 . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@60C6@

Characteristics of an estimator of a population parameter θ

Finite estimators

Bias.

The bias of an estimator is defined to be B ( θ ^ ) = E ( θ ^ ) θ . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkeadaqadaqaaiqbeI7aXzaajaaacaGLOaGaayzkaaGaeyypa0JaamyramaabmaabaGafqiUdeNbaKaaaiaawIcacaGLPaaacqGHsislcqaH4oqCcaGGUaaaaa@4273@ An estimator is unbiased if and only if B ( θ ^ ) = 0. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkeadaqadaqaaiqbeI7aXzaajaaacaGLOaGaayzkaaGaeyypa0JaaGimaiaac6caaaa@3C71@

Mean square error.

The mean square error (MSE) of an estimator is defined to be M S E ( θ ^ ) = E [ ( θ ^ θ ) 2 ] . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eacaWGtbGaamyramaabmaabaGafqiUdeNbaKaaaiaawIcacaGLPaaacqGH9aqpcaWGfbWaamWaaeaadaqadaqaaiqbeI7aXzaajaGaeyOeI0IaeqiUdehacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaGccaGLBbGaayzxaaGaaiOlaaaa@4705@ It is relatively easy to show that M S E ( θ ^ ) = V ( θ ^ ) + ( B ( θ ^ ) ) 2 . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eacaWGtbGaamyramaabmaabaGafqiUdeNbaKaaaiaawIcacaGLPaaacqGH9aqpcaWGwbWaaeWaaeaacuaH4oqCgaqcaaGaayjkaiaawMcaaiabgUcaRmaabmaabaGaamOqamaabmaabaGafqiUdeNbaKaaaiaawIcacaGLPaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccaGGUaaaaa@4902@ Often a biased estimator with a smaller MSE may be preferred to an unbiased estimator with a relatively larger MSE.

Efficiency.

An estimator θ ^ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbeI7aXzaajaaaaa@37AF@ is relatively more efficient than θ ˜ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbeI7aXzaaiaaaaa@37AE@ if and only if V ( θ ^ ) < V ( θ ˜ ) . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAfadaqadaqaaiqbeI7aXzaajaaacaGLOaGaayzkaaGaeyipaWJaamOvamaabmaabaGafqiUdeNbaGaaaiaawIcacaGLPaaacaGGUaaaaa@3FF2@ Generally, we would prefer to use the most efficient estimator available (if it is unbiased).

Asymtoptic estimators

Plim.

x n MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaamOBaaqabaaaaa@3805@ converges to a constant, c , if lim n Pr ( | x n c | > ε ) = 0 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacYgacaGGPbGaaiyBamaaBaaaleaacaWGUbGaeyOKH4QaeyOhIukabeaakiGaccfacaGGYbWaaeWaaeaadaabdaqaaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGJbaacaGLhWUaayjcSdGaeyOpa4JaeqyTdugacaGLOaGaayzkaaGaeyypa0JaaGimaaaa@4C21@ for any positive ε . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew7aLjaac6caaaa@3842@ We can write this relationship as p lim x n = c . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchaciGGSbGaaiyAaiaac2gacaWG4bWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0Jaam4yaiaac6caaaa@3E74@

Greene Greene, William H. (1990). Econometric Analysis (New York: Macmillan Publishing Company): 103. offers this example of plim: Suppose x n MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaamOBaaqabaaaaa@3805@ equals 0 with probability 1 ( 1 n ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaigdacqGHsisldaqadaqaamaalaaabaGaaGymaaqaaiaad6gaaaaacaGLOaGaayzkaaaaaa@3AD8@ and n with probability ( 1 n ) . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaWaaSaaaeaacaaIXaaabaGaamOBaaaaaiaawIcacaGLPaaacaGGUaaaaa@39E2@ As n increases, the second point becomes more remote from the first point. However, at the same time the probability of observing the second point becomes more and more unlikely. This effect is shown in Figure 5 where as n increases the probability distribution concentrates more and more on 1.

Example of plim.

This is an illustration of the concept of plim.
The probability x = 1 is the area of the gray box centered on 1 for n = 5; the gray area plus the blue area for n = 10; and the sum of the gray, blue, and red areas for n = 20; the probability x = n is the area of the box centered on n.

Consistency.

The estimator θ ^ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbeI7aXzaajaaaaa@37AF@ is a consistent estimator of θ if and only if p lim θ ^ = θ . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchaciGGSbGaaiyAaiaac2gacuaH4oqCgaqcaiabg2da9iabeI7aXjaac6caaaa@3EE2@

Asymmtotically unbiased.

An estimator θ ^ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbeI7aXzaajaaaaa@37AF@ is an asymtotically unbiased estimator of θ if lim n E [ θ ^ ] = θ . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacYgacaGGPbGaaiyBamaaBaaaleaacaWGUbGaeyOKH4QaeyOhIukabeaakiaadweadaWadaqaaiqbeI7aXzaajaaacaGLBbGaayzxaaGaeyypa0JaeqiUdeNaaiOlaaaa@4530@

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Econometrics for honors students. OpenStax CNX. Jul 20, 2010 Download for free at http://cnx.org/content/col11208/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Econometrics for honors students' conversation and receive update notifications?

Ask