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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

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Berger describes sociologists as concerned with
Mueller Reply
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Source:  OpenStax, Econometrics for honors students. OpenStax CNX. Jul 20, 2010 Download for free at http://cnx.org/content/col11208/1.2
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