Question

Suppose \{X_1, X_2, ..., X_n\} is a random sample from N(\mu, \sigma^2). Let \overline{X} and S^2 be the sample mean and sample variance respectively. Show that (\overline{X}, S^2) is a minimal sufficient statistic if both \mu and \sigma^2 are not known.

Name: Suppose {X1, X2, ..., Xn} is a random sample from N(μ, σ^2). Let X and S^2 be the sample mean and sample variance respectively. Show that (X, S^2) is a minimal sufficient statistic if both μand σ^2 are not known.
Uploaded: 2023-06-13T15:49:17-08:00
Duration: 3 min 14 s
Channel: Sri K
Description: Suppose {X1, X2, ..., Xn} is a random sample from N(μ, σ^2). Let X and S^2 be the sample mean and sample variance respectively. Show that (X, S^2) is a minimal sufficient statistic if both μand σ^2 are not known.

          Suppose \{X_1, X_2, ..., X_n\} is a random sample from N(\mu, \sigma^2). Let \overline{X} and S^2 be the sample mean and sample variance respectively. Show that (\overline{X}, S^2) is a minimal sufficient statistic if both \mu and \sigma^2 are not known.

$Suppose {X1, X2, ..., Xn} is a random sample from N(μ, σ^2). Let X and S^2 be the sample mean and sample variance respectively. Show that (X, S^2) is a minimal sufficient statistic if both μand σ^2 are not known.$

Added by Tom-S W.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

06/13/2023

Step 1

.., x_n; \mu, \sigma^2) = \frac{1}{(2\pi\sigma^2)^{n/2}} exp\left[-\frac{1}{2\sigma^2}\sum_{i=1}^{n}(x_i - \mu)^2\right]$$ Show more…

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Suppose {X1, X2, ..., Xn} is a random sample from N(μ, σ²). Let X and S² be the sample mean and sample variance respectively. Show that (X, S²) is a minimal sufficient statistic if both μ and σ² are not known.