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A class of tests based on U-statistic is proposed for two-sample scale problem. The U-statistic is function of extremes of subsamples taken from random samples of two absolutely continuous distributions. The asymptotic distribution, null distribution and efficacy of the class of tests are studied. A comparison of its performance with respect to other tests is studied in terms of Pitman ARE and small sample performance through its empirical power. Application of the class of tests is illustrated through an example.

Background: The mortality associated with the pandemic COVID-19 is a subject of intense scrutiny as COVID-19 can cause severe disease leading to hospitalization in ICU and potentially death, especially in the elderly with comorbidities. A statistical analysis is carried out to study the impact of age, gender and comorbidities on deaths among early one lakh infected population of Karnataka, a large state in south India.Methods: Daily case fatality rate and adjusted case fatality rate (CFR) (adjusted to median death time) are estimated. The impacts of age, gender and comorbidities on mortality outcomes of COVID patients are studied.Results: The daily CFR on 27th July for Karnataka is estimated from the dataset to be 1.93%. However, the adjusted CFR based on the median number of days from diagnosis to death was found to be 2.15% (95% confidence interval 2%-2.3%) on that day. The deaths among male patients outnumber those in females. As far as age of the patients is concerned, more than 50% of the deaths occurred in the age group 50-60 and 60-70 years. Majority of deaths reported in the state were associated with at least one of the comorbidity. Diabetes mellitus and hypertension were the most significant comorbidities.Conclusions: The daily adjusted CFR for the study region is found to be lower than the CFR of the whole nation. Also the age, gender and comorbidities were found to be associated with the deaths as opposed to the infection alone. It was also deduced that, patients with a history of diabetes or hypertension or ischemic heart disease or a combination of any of these were most likely to experience severe outcomes of the infection.

A class of distribution-free tests based on U-statistics with its kernel being function of subsample quantiles is proposed for a two-sample scale problem. The proposed class of tests is a general class of tests that includes numerous members which explore information from the tails of the distributions and tests resistant to outliers. This class of tests includes many existing classes of tests as its subclasses. The distribution of the proposed class of tests is derived and its relevance is discussed. One of its members, which is resistant to outliers in -sample is investigated in detail.

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