Hypothesis TestingHypothesis Testing – Distribution
Hypothesis Testing – DistributionSTATISTICS – Lecture no. 12
Jirı Neubauer
Department of Econometrics FEM UO Brnooffice 69a, tel. 973 442029email:[email protected]
5. 1. 2010
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Hypothesis Testing
A statistical hypothesis is a claim (or statement) about
a population parameters (µ, σ2, π, λ, . . . ),
a distribution (normal, Poisson, . . . ).
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Hypothesis Testing
A statistical hypothesis is a claim (or statement) about
a population parameters (µ, σ2, π, λ, . . . ),
a distribution (normal, Poisson, . . . ).
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Hypothesis Testing
A null hypothesis H is a claim (or statement) abouta population parameter that is assumed to be true until it isdeclared false . . . for example
H : µ = µ0,
An alternative hypothesis A is a claim about a populationparameter that will be true if the null hypothesis is false
1. A : µ 6= µ0 → both-sided test,2. A : µ > µ0 → one-sided test,3. A : µ < µ0 → one-sided test.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Hypothesis Testing
A null hypothesis H is a claim (or statement) abouta population parameter that is assumed to be true until it isdeclared false . . . for example
H : µ = µ0,
An alternative hypothesis A is a claim about a populationparameter that will be true if the null hypothesis is false
1. A : µ 6= µ0 → both-sided test,
2. A : µ > µ0 → one-sided test,3. A : µ < µ0 → one-sided test.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Hypothesis Testing
A null hypothesis H is a claim (or statement) abouta population parameter that is assumed to be true until it isdeclared false . . . for example
H : µ = µ0,
An alternative hypothesis A is a claim about a populationparameter that will be true if the null hypothesis is false
1. A : µ 6= µ0 → both-sided test,2. A : µ > µ0 → one-sided test,
3. A : µ < µ0 → one-sided test.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Hypothesis Testing
A null hypothesis H is a claim (or statement) abouta population parameter that is assumed to be true until it isdeclared false . . . for example
H : µ = µ0,
An alternative hypothesis A is a claim about a populationparameter that will be true if the null hypothesis is false
1. A : µ 6= µ0 → both-sided test,2. A : µ > µ0 → one-sided test,3. A : µ < µ0 → one-sided test.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Hypothesis Testing
reality H is true H is false
decision about H prob. prob.
not reject correct decision 1− α type II error β
reject type I error α correct decision 1− β
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Hypothesis Testing
If we reject the null hypothesis which is true, we call this typeI error. The probability of this error is α ⇒ a significancelevel. A number 1− α is probability that we do not reject thetrue hypothesis H.
If we accept the null hypothesis although is false, we call thistype II error. The probability of this error is β. A number1− β ⇒ a power of the test is the probability that we rejectthe null hypothesis H, if it is false.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Hypothesis Testing
If we reject the null hypothesis which is true, we call this typeI error. The probability of this error is α ⇒ a significancelevel. A number 1− α is probability that we do not reject thetrue hypothesis H.
If we accept the null hypothesis although is false, we call thistype II error. The probability of this error is β. A number1− β ⇒ a power of the test is the probability that we rejectthe null hypothesis H, if it is false.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Hypothesis Testing
To test the null hypothesis we use a function of a random sampleT = T (x1, x2, . . . , xn), so called test statistic, which has under thenull hypothesis H known distribution (usually t, u, χ2,F ).We divide the all possible values of the test statistic into
W1−α - nonrejection region of H – the set of valuesconnected with the hypothesis H,
Wα - rejection region of H – the set of values connectedwith the hypothesis A.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Hypothesis Testing
To test the null hypothesis we use a function of a random sampleT = T (x1, x2, . . . , xn), so called test statistic, which has under thenull hypothesis H known distribution (usually t, u, χ2,F ).We divide the all possible values of the test statistic into
W1−α - nonrejection region of H – the set of valuesconnected with the hypothesis H,
Wα - rejection region of H – the set of values connectedwith the hypothesis A.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Hypothesis Testing
To test the null hypothesis we use a function of a random sampleT = T (x1, x2, . . . , xn), so called test statistic, which has under thenull hypothesis H known distribution (usually t, u, χ2,F ).We divide the all possible values of the test statistic into
W1−α - nonrejection region of H – the set of valuesconnected with the hypothesis H,
Wα - rejection region of H – the set of values connectedwith the hypothesis A.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Steps to Perform a Test of Hypothesis
1. State the null and alternative hypothesis H and A.
2. Select a significance level α (usually 0.05 a 0.01).
3. Choose the test statistic.
4. Determine the rejection region Wα.
5. Calculate the value of the test statistic.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Steps to Perform a Test of Hypothesis
1. State the null and alternative hypothesis H and A.
2. Select a significance level α (usually 0.05 a 0.01).
3. Choose the test statistic.
4. Determine the rejection region Wα.
5. Calculate the value of the test statistic.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Steps to Perform a Test of Hypothesis
1. State the null and alternative hypothesis H and A.
2. Select a significance level α (usually 0.05 a 0.01).
3. Choose the test statistic.
4. Determine the rejection region Wα.
5. Calculate the value of the test statistic.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Steps to Perform a Test of Hypothesis
1. State the null and alternative hypothesis H and A.
2. Select a significance level α (usually 0.05 a 0.01).
3. Choose the test statistic.
4. Determine the rejection region Wα.
5. Calculate the value of the test statistic.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Steps to Perform a Test of Hypothesis
1. State the null and alternative hypothesis H and A.
2. Select a significance level α (usually 0.05 a 0.01).
3. Choose the test statistic.
4. Determine the rejection region Wα.
5. Calculate the value of the test statistic.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Steps to Perform a Test of Hypothesis
6. Make a decision:
If the value of the test statistic falls in the rejection region, wereject the null hypothesis H and say that we accept thealternative hypothesis A with the probability 1− α.If the value of the test statistic falls in the nonrejection region,we do not reject the null hypothesis H.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Steps to Perform a Test of Hypothesis
6. Make a decision:
If the value of the test statistic falls in the rejection region, wereject the null hypothesis H and say that we accept thealternative hypothesis A with the probability 1− α.
If the value of the test statistic falls in the nonrejection region,we do not reject the null hypothesis H.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Steps to Perform a Test of Hypothesis
6. Make a decision:
If the value of the test statistic falls in the rejection region, wereject the null hypothesis H and say that we accept thealternative hypothesis A with the probability 1− α.If the value of the test statistic falls in the nonrejection region,we do not reject the null hypothesis H.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Chi-Square Goodness of Fit TestTests of Skewness and KurtosisCompound Tests of Skewness and Kurtosis
Chi-Square Goodness of Fit Test
We divide values of a random sample x1, x2, . . . , xn into k disjunctclasses, where nj , j = 1, 2, . . . , k, is frequency of the class j and πj
is a probability that the random variable X has value from theclass j , calculated on condition that X has an assumed distribution.
The main idea of the test is to compare relative frequencies nj/nwith theoretical probabilities πj .
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Chi-Square Goodness of Fit TestTests of Skewness and KurtosisCompound Tests of Skewness and Kurtosis
Chi-Square Goodness of Fit Test
We state the null and alternative hypothesis:H : the random X has an assumed distribution → A : the randomX has not an assumed distribution.The test statistic is
χ2 =k∑
j=1
(nj − nπj)2
nπj,
which has under the null hypothesis H for large n (asymptotically)a Pearson χ2-distribution with ν = k − c − 1 degrees of freedom,where c is a number of estimated parameters of the assumeddistribution. A rejection region is
Wα ={χ2, χ2 ≥ χ2
1−α(ν)}
,
where χ21−α(ν) is a quantile of the Pearson χ2-distribution.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Chi-Square Goodness of Fit TestTests of Skewness and KurtosisCompound Tests of Skewness and Kurtosis
Chi-Square Goodness of Fit Test
Recommendation:
nπj > 5, j = 1, 2, . . . , k.
If this condition is not satisfied, it is necessary to join the classes.
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Chi-Square Goodness of Fit TestTests of Skewness and KurtosisCompound Tests of Skewness and Kurtosis
Tests of Skewness and Kurtosis
The normal distribution has α3 = 0 a α4 = 0. We can use theseproperties to test normality. We calculate a sample skewness andkurtosis (they are estimates of α3 and α4)
α3 = a3 =1
ns3n
n∑i=1
(xi − x)3, α3 = a4 =1
ns4n
n∑i=1
(xi − x)4 − 3.
We state hypothesis:H1 : α3 = 0 → A1 : α3 6= 0H2 : α4 = 0 → A2 : α4 6= 0
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Chi-Square Goodness of Fit TestTests of Skewness and KurtosisCompound Tests of Skewness and Kurtosis
Tests of Skewness and Kurtosis
1. H1 : α3 = 0 → A1 : α3 6= 0Test statistic is
u3 =a3√D(a3)
, where D(a3) =6(n − 2)
(n + 1)(n + 3),
which has under the null hypothesis H1 asymptotically normaldistribution N(0, 1). A rejection region is
Wα ={
u3, |u3| ≥ u1−α2
},
where u1−α2
is a quantile of N(0, 1).
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Chi-Square Goodness of Fit TestTests of Skewness and KurtosisCompound Tests of Skewness and Kurtosis
Tests of Skewness and Kurtosis
2. H2 : α4 = 0 → A2 : α4 6= 0Test statistic is
u4 =a4 + 6
n+1√D(a4)
, where D(a4) =24n(n − 2)(n − 3)
(n + 1)2(n + 3)(n + 5),
which has under the null hypothesis H2 asymptotically normaldistribution N(0, 1). A rejection region is
Wα ={
u4, |u4| ≥ u1−α2
},
where u1−α2
is a quantile of N(0, 1).
Jirı Neubauer Hypothesis Testing – Distribution
Hypothesis TestingHypothesis Testing – Distribution
Chi-Square Goodness of Fit TestTests of Skewness and KurtosisCompound Tests of Skewness and Kurtosis
Compound Tests of Skewness and Kurtosis
We state hypothesis:H : a random variable X has a normal distribution → A : a randomvariable X has not a normal distribution.Test statistic is
C = u23 + u2
4 ,
which has under the null hypothesis H approximately χ2
distribution with two degrees of freedom.u3 and u4 are test statistics defined above. A rejection region is
Wα ={C ,C ≥ χ2
1−α(2)}
,
where χ21−α(2) is a quantile of the Pearson χ2-distribution.
Jirı Neubauer Hypothesis Testing – Distribution
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