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a)𝑋1,…,π‘‹π‘›βˆΌπ‘–.𝑖.𝑑.π–―π—ˆπ—‚π—Œπ—Œ(πœ†) for some unknown πœ†>0 ;

𝐻0:πœ†=πœ†0 v.s. 𝐻1:πœ†β‰ πœ†0where πœ†0>0.

(Type barX_n for π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘› , lambda_0 for πœ†0 . If applicable, type abs(x) for |π‘₯| , Phi(x) for Ξ¦(π‘₯)=𝐏(𝑍≀π‘₯) where π‘βˆΌN(0,1) , and q(alpha) for π‘žπ›Ό , the 1βˆ’π›Ό quantile of a standard normal variable, e.g. enter q(0.01) for π‘ž0.01 .)

Asymptotic 𝑝-value=?

b) 𝑋1,…,π‘‹π‘›βˆΌπ‘–.𝑖.𝑑.π–―π—ˆπ—‚π—Œπ—Œ(πœ†) for some unknown πœ†>0 ;

𝐻0:πœ†β‰₯πœ†0 v.s. 𝐻1:πœ†<πœ†0where πœ†0>0.

( Type barX_n for π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘› , lambda_0 for πœ†0. . If applicable, type abs(x) for |π‘₯| , Phi(x) for Ξ¦(π‘₯)=𝐏(𝑍≀π‘₯) where π‘βˆΌN(0,1) , and q(alpha) for π‘žπ›Ό , the 1βˆ’π›Ό quantile of a standard normal variable. )

Asymptotic 𝑝-value=?

c)𝑋1,…,π‘‹π‘›βˆΌπ‘–.𝑖.𝑑.𝖀𝗑𝗉(πœ†) for some unknown πœ†>0 ;

𝐻0:πœ†=πœ†0 v.s. 𝐻1:πœ†β‰ πœ†0where πœ†0>0.

(Type barX_n for π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘› , lambda_0 for πœ†0. If applicable, type abs(x) for |π‘₯| , Phi(x) for Ξ¦(π‘₯)=𝐏(𝑍≀π‘₯) where π‘βˆΌN(0,1) , and q(alpha) for π‘žπ›Ό , the 1βˆ’π›Ό quantile of a standard normal variable.)

Asymptotic 𝑝-value=?

a) 2*Phi((sqrt(n)*(barX_n-lambda_0))/sqrt(lambda_0))

b) 1-Phi((sqrt(n)*(lambda_0-barX_n))/sqrt(lambda_0))
c) 2*(1-Phi(sqrt(n)*(barX_n-(1/lambda_0))*lambda_0))

a) To calculate the asymptotic 𝑝-value for the test 𝐻0:πœ†=πœ†0 v.s. 𝐻1:πœ†β‰ πœ†0, where 𝑋1,…,𝑋𝑛 are 𝑖.𝑖.𝑑. π‘ƒπ‘œπ‘–π‘ π‘ (πœ†) random variables with πœ†>0, you can use the formula:

𝑝-value = 2 * (1 - Phi(sqrt(n) * abs(barX_n - lambda_0) / sqrt(lambda_0)))

where barX_n is the sample mean, lambda_0 is the assumed value of πœ†, abs(x) is the absolute value of x, Phi(x) is the cumulative distribution function of the standard normal distribution, and n is the sample size.

b) To calculate the asymptotic 𝑝-value for the test 𝐻0:πœ†β‰₯πœ†0 v.s. 𝐻1:πœ†<πœ†0, where 𝑋1,…,𝑋𝑛 are 𝑖.𝑖.𝑑. π‘ƒπ‘œπ‘–π‘ π‘ (πœ†) random variables with πœ†>0, you can use the formula:

𝑝-value = Phi(sqrt(n) * (barX_n - lambda_0) / sqrt(lambda_0))

where barX_n is the sample mean, lambda_0 is the assumed value of πœ†, abs(x) is the absolute value of x, Phi(x) is the cumulative distribution function of the standard normal distribution, and n is the sample size.

c) To calculate the asymptotic 𝑝-value for the test 𝐻0:πœ†=πœ†0 v.s. 𝐻1:πœ†β‰ πœ†0, where 𝑋1,…,𝑋𝑛 are 𝑖.𝑖.𝑑. 𝐸π‘₯𝑝(πœ†) random variables with πœ†>0, you can use the formula:

𝑝-value = 2 * (1 - Phi(sqrt(n) * abs(barX_n - lambda_0) / sqrt(lambda_0)))

where barX_n is the sample mean, lambda_0 is the assumed value of πœ†, abs(x) is the absolute value of x, Phi(x) is the cumulative distribution function of the standard normal distribution, and n is the sample size.

a) To calculate the asymptotic p-value for testing 𝐻0:πœ†=πœ†0 versus 𝐻1:πœ†β‰ πœ†0, we can use the likelihood ratio test statistic.

The likelihood ratio test statistic is given by:

πœ†Μ‚ = -2 ln [(likelihood under 𝐻0)/(likelihood under 𝐻1)]

Under the null hypothesis 𝐻0, the likelihood function is 𝐿(πœ†0) = πœ†0^𝑛 * exp(-πœ†0 * π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘›), where π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘› is the sample mean.

Under the alternative hypothesis 𝐻1, the likelihood function is 𝐿(πœ†Μ‚) = πœ†Μ‚^𝑛 * exp(-πœ†Μ‚ * π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘›).

To calculate the p-value, we need to find the probability of observing a test statistic as extreme as πœ†Μ‚ under the null hypothesis.

The asymptotic p-value can be calculated using the chi-squared distribution with 1 degree of freedom. Since we have 1 parameter estimated under the null hypothesis, the test statistic follows a chi-squared distribution with 1 degree of freedom under the null hypothesis.

Therefore, the asymptotic p-value is given by:

𝑝-value = 𝐏(πœ†Μ‚ > πœ†* | 𝐻0) = 1 - Ξ¦(sqrt(πœ†Μ‚))

where Ξ¦(x) is the cumulative distribution function (CDF) of the standard normal distribution and πœ†* is the limit of the chi-squared distribution with 1 degree of freedom.

b) To calculate the asymptotic p-value for testing 𝐻0:πœ†β‰₯πœ†0 versus 𝐻1:πœ†<πœ†0, we can follow a similar approach as in part a).

The likelihood ratio test statistic is still given by:

πœ†Μ‚ = -2 ln [(likelihood under 𝐻0)/(likelihood under 𝐻1)]

Under the null hypothesis 𝐻0, the likelihood function is 𝐿(πœ†0) = πœ†0^𝑛 * exp(-πœ†0 * π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘›), where π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘› is the sample mean.

Under the alternative hypothesis 𝐻1, the likelihood function is 𝐿(πœ†Μ‚) = πœ†Μ‚^𝑛 * exp(-πœ†Μ‚ * π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘›).

To calculate the p-value, we need to find the probability of observing a test statistic as extreme as πœ†Μ‚ under the null hypothesis.

The asymptotic p-value can be calculated using the chi-squared distribution with 1 degree of freedom. Since we have 1 parameter estimated under the null hypothesis, the test statistic follows a chi-squared distribution with 1 degree of freedom under the null hypothesis.

Therefore, the asymptotic p-value is given by:

𝑝-value = 𝐏(πœ†Μ‚ > πœ†* | 𝐻0) = 𝐏(πœ†Μ‚ > πœ†* | 𝐻0:πœ†=πœ†0) = 1 - Ξ¦(sqrt(πœ†Μ‚))

where Ξ¦(x) is the cumulative distribution function (CDF) of the standard normal distribution and πœ†* is the limit of the chi-squared distribution with 1 degree of freedom.

c) To calculate the asymptotic p-value for testing 𝐻0:πœ†=πœ†0 versus 𝐻1:πœ†β‰ πœ†0 for exponential distribution, we can use a similar approach as in part a).

The likelihood ratio test statistic is still given by:

πœ†Μ‚ = -2 ln [(likelihood under 𝐻0)/(likelihood under 𝐻1)]

Under the null hypothesis 𝐻0, the likelihood function is 𝐿(πœ†0) = πœ†0^𝑛 * exp(-πœ†0 * π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘›), where π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘› is the sample mean.

Under the alternative hypothesis 𝐻1, the likelihood function is 𝐿(πœ†Μ‚) = πœ†Μ‚^𝑛 * exp(-πœ†Μ‚ * π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘›).

To calculate the p-value, we need to find the probability of observing a test statistic as extreme as πœ†Μ‚ under the null hypothesis.

The asymptotic p-value can be calculated using the chi-squared distribution with 1 degree of freedom. Since we have 1 parameter estimated under the null hypothesis, the test statistic follows a chi-squared distribution with 1 degree of freedom under the null hypothesis.

Therefore, the asymptotic p-value is given by:

𝑝-value = 𝐏(πœ†Μ‚ > πœ†* | 𝐻0) = 1 - Ξ¦(sqrt(πœ†Μ‚))

where Ξ¦(x) is the cumulative distribution function (CDF) of the standard normal distribution and πœ†* is the limit of the chi-squared distribution with 1 degree of freedom.

To calculate the asymptotic p-value for these hypothesis tests, we can use the likelihood ratio test statistic.

a) For the hypothesis test 𝐻0:πœ†=πœ†0 v.s. 𝐻1:πœ†β‰ πœ†0, where πœ†0>0, we can calculate the test statistic as follows:

1. Compute the sample mean: barX_n = (𝑋1 + 𝑋2 + ... + 𝑋𝑛)/𝑛.
2. Calculate the likelihood ratio:
LR = 2 * (log(likelihood(πœ†0)) - log(likelihood(barX_n))).
The log-likelihood function for a Poisson distribution is given by log(likelihood(πœ†)) = 𝑛 * (πœ† * log(πœ†) - πœ† - log(𝑋1!) - log(𝑋2!) - ... - log(𝑋𝑛!)).
3. Calculate the p-value using the chi-square distribution:
p-value = 1 - Ξ¦(sqrt(LR)).

b) For the hypothesis test 𝐻0:πœ†β‰₯πœ†0 v.s. 𝐻1:πœ†<πœ†0, where πœ†0>0, we can calculate the test statistic as follows:

1. Compute the sample mean: barX_n = (𝑋1 + 𝑋2 + ... + 𝑋𝑛)/𝑛.
2. Calculate the likelihood ratio:
LR = 2 * (log(likelihood(πœ†_0)) - log(likelihood(barX_n))).
The log-likelihood function for a Poisson distribution is given by log(likelihood(πœ†)) = 𝑛 * (πœ† * log(πœ†) - πœ† - log(𝑋1!) - log(𝑋2!) - ... - log(𝑋𝑛!)).
3. Calculate the p-value using the chi-square distribution:
p-value = Ξ¦(sqrt(LR)).

c) For the hypothesis test 𝐻0:πœ†=πœ†0 v.s. 𝐻1:πœ†β‰ πœ†0, where πœ†0>0, we can calculate the test statistic as follows:

1. Compute the sample mean: barX_n = (𝑋1 + 𝑋2 + ... + 𝑋𝑛)/𝑛.
2. Calculate the likelihood ratio:
LR = 2 * (log(likelihood(πœ†0)) - log(likelihood(barX_n))).
The log-likelihood function for an exponential distribution is given by log(likelihood(πœ†)) = 𝑛 * (log(πœ†) - πœ† * 𝑋1 - πœ† * 𝑋2 - ... - πœ† * 𝑋𝑛).
3. Calculate the p-value using the chi-square distribution:
p-value = 1 - Ξ¦(sqrt(LR)).

Note: In all three cases, Ξ¦(x) represents the cumulative distribution function of the standard normal distribution, and q(alpha) represents the (1βˆ’alpha) quantile of the standard normal distribution.