[5] Simultaneous Mean and Variance#

The functions in pysht.mean_variance test the mean and variance of one or two univariate normal populations in one decision. A significant result can reflect a mean departure, a variance departure, or both.

Question or calibration	Function
One population versus a specified mean and variance	`as_1samp`
Pearson–Neyman beta approximation for two populations	`pn_2samp`
Perng–Littell combination of pooled-t and F tests	`pl_2samp`
Muirhead second-order likelihood-ratio approximation	`muirhead_2samp`
Zhang–Xu–Chen exact likelihood-ratio calibration	`zxc_2samp`
Wilks asymptotic likelihood-ratio calibration	`lrt_2samp`

All methods require independent normal observations and positive within-sample variance. The scalar null variance for as_1samp is supplied with the keyword variance; for example:

from pysht.mean_variance import as_1samp

result = as_1samp(x, popmean=2.0, variance=1.5)
print(result)

The result prints in the same compact style as an R htest object: method, statistic, degrees of freedom where applicable, p-value, alternative, and calibration. Pearson–Neyman beta shapes, Perng–Littell component p-values, and Muirhead correction constants appear under immutable diagnostics; they are calibration metadata, not parameter estimates.

For small samples, prefer zxc_2samp when exact likelihood-ratio calibration is required. pn_2samp is a highly accurate moment approximation in the validated normal scenarios. The chi-square calibrations in as_1samp and lrt_2samp, and the finite Muirhead expansion, are asymptotic approximations. The validation ledger records the finite-sample regimes that passed the release-size gate and the smaller regimes that did not.

Functions#

pysht.mean_variance.as_1samp(x, *, popmean=0.0, variance=1.0)[source]#

Test one normal population mean and variance jointly.

This is the likelihood-ratio procedure discussed by Arnold and Shavelle (1998). It represents both mvar1.1998AS and mvar1.LRT from SHT, whose statistics are algebraically identical.

Parameters:

x: One-dimensional sample with positive sample variance.
popmean: Population mean under the joint null hypothesis.
variance: Positive population variance under the joint null hypothesis.

Parameters:

x (ArrayLike)
popmean (float)
variance (float)

Return type:

HypothesisTestResult

Notes

The chi-square calibration is asymptotic. Normality and independent observations are required.

pysht.mean_variance.pn_2samp(x, y)[source]#

Perform the Pearson–Neyman two-sample normal-population test.

The null distribution of the likelihood ratio is approximated by a beta distribution whose parameters match its first two exact null moments. Small likelihood ratios contradict equality, so the p-value is the lower beta tail. This corrects the reversed tail in SHT 0.1.9.

Parameters:

x (ArrayLike)
y (ArrayLike)

Return type:

HypothesisTestResult

pysht.mean_variance.pl_2samp(x, y)[source]#

Perform the Perng–Littell joint two-sample test.

Under independent normal samples and the joint null, the equal-variance pooled t statistic is independent of the sample-variance ratio. Fisher’s method therefore combines their two-sided p-values with a chi-square law having four degrees of freedom.

Parameters:

x (ArrayLike)
y (ArrayLike)

Return type:

HypothesisTestResult

pysht.mean_variance.muirhead_2samp(x, y)[source]#

Perform Muirhead’s corrected likelihood-ratio approximation.

The second-order null approximation is evaluated in the upper tail. The finite expansion can leave the probability interval in extreme tails, so the reported approximation is truncated to [0, 1].

Parameters:

x (ArrayLike)
y (ArrayLike)

Return type:

HypothesisTestResult

pysht.mean_variance.zxc_2samp(x, y)[source]#

Perform the exact Zhang–Xu–Chen two-sample normal test.

The likelihood ratio itself is reported. Its exact p-value is the lower tail because smaller ratios provide stronger evidence against equality. Computation stays in the log domain and uses the conditional-beta form of the null probability rather than exponentiating a two-dimensional quadrature integrand.

Parameters:

x (ArrayLike)
y (ArrayLike)

Return type:

HypothesisTestResult

pysht.mean_variance.lrt_2samp(x, y)[source]#

Perform the asymptotic likelihood-ratio equality test.

The procedure tests equality of both parameters of two independent normal populations. Its statistic is -2 log(Lambda) and its chi-square law is asymptotic.

Parameters:

x (ArrayLike)
y (ArrayLike)

Return type:

HypothesisTestResult